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BIT - SERIAL NEURAL NETWORKS Alan F. Murray, Anthony V. W. Smith and Zoe F. Butler. Department of Electrical Engineering, University of Edinburgh, The King's Buildings, Mayfield Road, Edinburgh, Scotland, EH93JL. ABSTRACT 573 A bit - serial VLSI neural network is described from an initial architecture for a synapse array through to silicon layout and board design. The issues surrounding bit - serial computation, and analog/digital arithmetic are discussed and the parallel development of a hybrid analog/digital neural network is outlined. Learning and recall capabilities are reported for the bit - serial network along with a projected specification for a 64 - neuron, bit - serial board operating at 20 MHz. This technique is extended to a 256 (2562 synapses) network with an update time of 3ms, using a "paging" technique to time - multiplex calculations through the synapse array. 1. INTRODUCTION The functions a synthetic neural network may aspire to mimic are the ability to consider many solutions simultaneously, an ability to work with corrupted data and a natural fault tolerance. This arises from the parallelism and distributed knowledge representation which gives rise to gentle degradation as faults appear. These functions are attractive to implementation in VLSI and WSI. For example, the natural fault - tolerance could be useful in silicon wafers with imperfect yield, where the network degradation is approximately proportional to the non-functioning silicon area. To cast neural networks in engineering language, a neuron is a state machine that is either "on" or "off', which in general assumes intermediate states as it switches smoothly between these extrema. The synapses weighting the signals from a transmitting neuron such that it is more or less excitatory or inhibitory to the receiving neuron. The set of synaptic weights determines the stable states and represents the learned information in a system. The neural state, VI' is related to the total neural activity stimulated by inputs to the neuron through an activation junction, F. Neural activity is the level of excitation of the neuron and the activation is the way it reacts in a response to a change in activation. The neural output state at time t, V[, is related to x[ by V[ = F (xf) (1) The activation function is a "squashing" function ensuring that (say) Vi is 1 when Xi is large and -1 when Xi is small. The neural update function is therefore straightforward: . i-n-l ,+1 -, + ~ ~ T V' XI XI • •••• 0 ~ ii J J-O where 8 represents the rate of change of neural activity, Tij and n is the number of terms giving an n - neuron array [1]. (2) is the synaptic weight Although the neural function is simple enough, in a totally interconnected n - neuron network there are n 2 synapses requiring n 2 multiplications and summations and © American Institute of Physics 1988 574 a large number of interconnects. The challenge in VLSI is therefore to design a simple, compact synapse that can be repeated to build a VLSI neural network with manageable interconnect. In a network with fixed functionality, this is relatively straightforward. H the network is to be able to learn, however, the synaptic weights must be programmable, and therefore more complicated. 2. DESIGNING A NEURAL NETWORK IN VLSI There are fundamentally two approaches to implementing any function in silicon digital and analog. Each technique has its advantages and disadvantages, and these are listed below, along with the merits and demerits of bit - serial architectures in digital (synchronous) systems. Digital vs. analog: The primary advantage of digital design for a synapse array is that digital memory is well understood, and can be incorporated easily. Learning networks are therefore possible without recourse to unusual techniques or technologies. Other strengths of a digital approach are that design techniques are advanced, automated and well understood and noise immunity and computational speed can be high. Unattractive features are that digital circuits of this complexity need to be synchronous and all states and activities are quantised, while real neural networks are asynchronous and unquantised. Furthermore, digital multipliers occupy a large silicon area, giving a low synapse count on a single chip. The advantages of analog circuitry are that asynchronous behaviour and smooth neural activation are automatic. Circuit elements can be small, but noise immunity is relatively low and arbitrarily high precision is not possible. Most importantly, no reliable analog, non - volatile memory technology is as yet readily available. For this reason, learning networks lend themselves more naturally to digital design and implementation. Several groups are developing neural chips and boards, and the following listing does not pretend to be exhaustive. It is included, rather, to indicate the spread of activity in this field. Analog techniques have been used to build resistor I operational amplifier networks [2,3] similar to those proposed by Hopfield and Tank [4]. A large group at Caltech is developing networks implementing early vision and auditory processing functions using the intrinsic nonlinearities of MaS transistors in the subthreshold regime [5,6]. The problem of implementing analog networks with electrically programmable synapses has been addressed using CCDIMNOS technology [7]. Finally, Garth [8] is developing a digital neural accelerator board ("Netsim") that is effectively a fast SIMD processor with supporting memory and communications chips. Bit - serial vs. bit - parallel: Bit - serial arithmetic and communication is efficient for computational processes, allowing good communication within and between VLSI chips and tightly pipelined arithmetic structures. It is ideal for neural networks as it minimises the interconnect requirement by eliminating multi - wire busses. Although a bit - parallel design would be free from computational latency (delay between input and output), pipelining makes optimal use of the high bit rates possible in serial systems, and makes for efficient circuit usage. 2.1 An asynchronous pulse stream VLSI neural network: In addition to the digital system that forms the substance of this paper, we are developing a hybrid analOg/digital network family. This work is outlined here, and has been reported in greater detail elsewhere [9, 10, 11]. The generic (logical and layout) architecture of a single network of n totally interconnected neurons is shown 575 schematically in figure 1. Neurons are represented by circles, which signal their states, Vi upward into a matrix of synaptic operators. The state signals are connected to a n - bit horizontal bus running through the synaptic array, with a connection to each synaptic operator in every column. All columns have n operators (denoted by squares) and each operator adds its synaptic contribution, Tij V j , to the running total of activity for the neuron i at the foot of the column. The synaptic function is therefore to multiply the signalling neuron state, Vj , by the synaptic weight, Tij , and to add this product to the running total. This architecture is common to both the bit - serial and pulse - stream networks. Synapse States { Vj } Neurons Figure 1. Generic architecture for a network of n totally interconnected neurons. This type of architecture has many attractions for implementation in 2 - dimensional j=II -1 silicon as the summation 2 Tij Vj is distributed in space. The interconnect j=O requirement (n inputs to each neuron) is therefore distributed through a column, reducing the need for long - range wiring. The architecture is modular, regular and can be easily expanded. In the hybrid analog/digital system, the circuitry uses a "pulse stream" signalling method similar to that in a natural neural system. Neurons indicate their state by the presence or absence of pulses on their outputs, and synaptic weighting is achieved by time - chopping the presynaptic pulse stream prior to adding it to the postsynaptic activity summation. It is therefore asynchronous and imposes no fundamental limitations on the activation or neural state. Figure 2 shows the pulse stream mechanism in more detail. The synaptic weight is stored in digital memory local to the operator. Each synaptic operator has an excitatory and inhibitory pulse stream input and output. The resultant product of a synaptic operation, Tij Vj , is added to the running total propagating down either the excitatory or inhibitory channel. One binary bit (the MSBit) of the stored Tij determines whether the contribution is excitatory or inhibitory. The incoming excitatory and inhibitory pulse stream inputs to a neuron are integrated to give a neural activation potential that varies smoothly from 0 to 5 V. This potential controls a feedback loop with an odd number of logic inversions and 576 . • • V , .u.u, • XT •• Figure 2. Pulse stream arithmetic. Neurons are denoted by 0 and synaptic operators by D. thus forms a switched "ring - oscillator". H the inhibitory input dominates, the feedback loop is broken. H excitatory spikes subsequently dominate at the input, the neural activity rises to 5V and the feedback loop oscillates with a period determined by a delay around the loop. The resultant periodic waveform is then converted to a series of voltage spikes, whose pulse rate represents the neural state, Vi' Interestingly, a not dissimilar technique is reported elsewhere in this volume, although the synapse function is executed differently [12]. 3. A 5 - STATE BIT - SERIAL NEURAL NETWORK The overall architecture of the 5 - state bit - serial neural network is identical to that of the pulse stream network. It is an array of n2 interconnected synchronous synaptic operators, and whereas the pulse stream method allowed Vj to assume all values between "off' and "on", the 5 - state network VJ is constrained to 0, ±0.5 Qr ± 1. The resultant activation function is shown in Figure 3. Full digital multiplication is costly in silicon area, but multiplication of Tij by Vj = 0.5 merely requires the synaptic weight to be right - shifted by 1 bit. Similarly, multiplication by 0.25 involves a further right - shift of Til' and multiplication by 0.0 is trivially easy. VJ < 0 is not problematic, as a switchable adder/subtractor is not much more complex than an adder. Five neural states are therefore feasible with circuitry that is only slightly more complex than a simple serial adder. The neural state expands from a 1 bit to a 3 bit (5 - state) representation, where the bits represent "add/subtract?", "shift?" and "multiply by O?". Figure 4 shows part of the synaptic array. Each synaptic operator includes an 8 bit shift register memory block holding the synaptic weight, Til' A 3 bit bus for the 5 neural states runs horizontally above each synaptic row. Single phase dynamic CMOS has been used with a clock frequency in excess of 20 MHz [13). Details of a synaptic operator are shown in figure 5. The synaptic weight Til cycles around the shift register and the neural state Vj is present on the state bus. During the first clock CYCle, the synaptic weight is multiplied by the neural state and during the second, the most significant bit (MSBit) of the resultant Tij Vj is sign - extended for lHRESHOLD State VJ ..... -------=-------.. Activity sJ s· "5 STATE" "Sharper" "Smoother" ~.....::~-"'--x.&..t------ Activity "J Figure 3. "Hard - threshold", 5 - state and sigmoid activation functions. J-a-1T v ~ .. J J-li v, v, Figure 4. Section of the synaptic array of the 5 - state activation function neural network. 8 bits to allow for word growth in the running summation. A least significant bit (LSBit) signal running down the synaptic columns indicates the arrival of the LSBit of the Xj running total. If the neural state is ±O.5 the synaptic weight is right shifted by 1 bit and then added to or subtracted from the running total. A multiplication of ± 1 adds or subtracts the weight from the total and multiplication by 0 577 578 .0.5 .0.0 Add! Subtract Add/Subtract Carry Figure S. The synaptic operator with a 5 - state activation function. does not alter the running summation. The final summation at the foot of the column is thresholded externally according to the 5 - state activation function in figure 3. As the neuron activity Xj' increases through a threshold value x" ideal sigmoidal activation represents a smooth switch of neural state from -1 to 1. The 5 - state "staircase" function gives a superficially much better approximation to the sigmoid form than a (much simpler to implement) threshold function. The sharpness of the transition can be controlled to "tune" the neural dynamics for learning and computation. The control parameter is referred to as temperature by analogy with statistical functions with this sigmoidal form. High "temperature" gives a smoother staircase and sigmoid, while a temperature of 0 reduces both to the ''Hopfield'' - like threshold function. The effects of temperature on both learning and recall for the threshold and 5 - state activation options are discussed in section 4. 4. LEARNING AND RECALL WITH VLSI CONSTRAINTS Before implementing the reduced - arithmetic network in VLSI, simulation experiments were conducted to verify that the 5 - state model represented a worthwhile enhancement over simple threshold activation. The "benchmark" problem was chosen for its ubiquitousness, rather than for its intrinsic value. The implications for learning and recall of the 5 - state model, the threshold (2 - state) model and smooth sigmoidal activation ( 00 - state) were compared at varying temperatures with a restricted dynamic range for the weights Tij • In each simulation a totally interconnected 64 node network attempted to learn 32 random patterns using the delta rule learning algorithm (see for example [14]). Each pattern was then corrupted with 25% noise and recall attempted to probe the content addressable memory properties under the three different activation options. During learning, individual weights can become large (positive or negative). When weights are "driven" beyond the maximum value in a hardware implementation, 579 which is determined by the size of the synaptic weight blocks, some limiting mechanism must be introduced. For example, with eight bit weight registers, the limitation is -128 S Tij S 127. With integer weights, this can be seen to be a problem of dynamic range, where it is the relationship between the smallest possible weight (± 1) and the largest (+ 127/-128) that is the issue. Results: Fig. 6 shows examples of the results obtained, studying learning using 5 state activation at different temperatures, and recall using both 5 - state and threshold activation. At temperature T=O, the 5 - state and threshold models are degenerate, and the results identical. Increasing smoothness of activation (temperature) during learning improves the quality of learning regardless of the activation function used in recall, as more patterns are recognised successfully. Using 5 - state activation in recall is more effective than simple threshold activation. The effect of dynamic range restrictions can be assessed from the horizontal axis, where T/j:6. is shown. The results from these and many other experiments may be summarised as follows:5 - State activation vs. threshold: 1) Learning with 5 - state activation was protracted over the threshold activation, as binary patterns were being learnt, and the inclusion of intermediate values added extra degrees of freedom. 2) Weight sets learnt using the 5 - state activation function were "better" than those learnt via threshold activation, as the recall properties of both 5 - state and threshold networks using such a weight set were more robust against noise. 3) Full sigmoidal activation was better than 5 - state, but the enhancement was less significant than that incurred by moving from threshold - 5 - state. This suggests that the law of diminishing returns applies to addition of levels to the neural state Vi' This issue has been studied mathematically [15], with results that agree qualitatively with ours. Weight Saturation: Three methods were tried to deal with weight saturation. Firstly, inclusion of a decay, or "forgetting" term was included in the learning cycle [1]. It is our view that this technique can produce the desired weight limiting property, but in the time available for experiments, we were unable to "tune" the rate of decay sufficiently well to confirm it. Renormalisation of the weights (division to bring large weights back into the dynamic range) was very unsuccessful, suggesting that information distributed throughout the numerically small weights was being destroyed. Finally, the weights were allowed to "clip" (ie any weight outside the dynamic range was set to the maximum allowed value). This method proved very successful, as the learning algorithm adjusted the weights over which it still had control to compensate for the saturation effect. It is interesting to note that other experiments have indicated that Hopfield nets can "forget" in a different way, under different learning control, giving preference to recently acquired memories [16]. The results from the saturation experiments were:1) For the 32 pattemJ64 node problem, integer weights with a dynamic range greater than ±30 were necessary to give enough storage capability. 2) For weights with maximum values TiJ = 50-70, "clipping" occurs, but network performance is not seriously degraded over that with an unrestricted weight set. 580 15 "0 10 c = .2 en e u 5 -~ 0 0 ,- .... ---------., ... e ~ ;A ....... ;.. f:'-:' :::::7.:::.::-:::-: f'-. I ".' , ,. i ! ! , i I I , 20 30 40 50 60 70 Limit 5 . state activation function recal1 15 T=30 _._.-.T=20 T=10 T=O ,.. .•. -..... -.•. _ .•. .. -.-._.-.. , i j''''-,,'i ~------------- . . .,. '" j ••••••• •••••••••••••••• •••••• j I O~~~~--~~ __ ~~ __ o 20 30 40 50 60 70 Limit tlHopficld" activation function recall Figure 6. Recall of patterns learned with the 5 . state activation function and subsequently restored using the 5-state and the hard - threshold activation functions. T is the "temperature", or smoothness of the activation function, and "limit" the value ofTI;· These results showed that the 5 - state model was worthy of implementation as a VLSI neural board, and suggested that 8 - bit weights were sufficient. S. PROJECTED SPECIFICATION OF A HARDWARE NEURAL BOARD The specification of a 64 neuron board is given here, using a 5 - state bit - serial 64 x 64 synapse array with a derated clock speed of 20 MHz. The synaptic weights are 8 bit words and the word length of the running summation XI is 16 bits to allow for growth. A 64 synapse column has a computational latency of 80 clock cycles or bits, giving an update time of 4 .... s for the network. The time to load the weights into the array is limited to 6O .... s by the supporting RAM, with an access time of 12Ons. These load and update times mean that the network is executing 1 x 10' operations/second, where one operation is ± Tlj Vj • This is much faster than a natural neural network, and much faster than is necessary in a hardware accelerator. We have therefore developed a "paging" architecture, that effectively "trades off" some of this excessive speed against increased network size. A "moving - patch" neural board: An array of the 5 - state synapses is currently being fabricated as a VLSI integrated circuit. The shift registers and the adderlsubtractor for each synapse occupy a disappointingly large silicon area, allowing only a 3 x 9 synaptic array. To achieve a suitable size neural network from this array, several chips need to be included on a board with memory and control circuitry. The "moving patch" concept is shown in figure 7, where a small array of synapses is passed over a much larger n x n synaptic array. Each time the array is "moved" to represent another set of synapses, new weights must be loaded into it. For example, the first set of weights will be T 11 •. , T;J ... T 21 ... T 2j to Tjj , the second set Tj + 1,l to T u etc.. The final weight to be loaded will be Smaller "Patch" n neurons .. om synaptic array moves over array rr~ _____ ) __ -.. ~'> Figure 7. The "moving patch" concept, passing a small synaptic "patch" over a larger run synapse array. TNt· Static, off - the - shelf RAM is used to store the weights and the whole operation is pipelined for maximum efficiency. Figure 8 shows the board level design for the network. Control Synaptic Accelerator Chips HOST Figure 8. A "moving patch" neural network board. The small "patch" that moves around the array to give n neurons comprises 4 VLSI synaptic accelerator chips to give a 6 x 18 synaptic array. The number of neurons to be simulated is 256 and the weights for these are stored in 0.5 Mb of RAM with a load time of 8ms. For each "patch" movement, the partial runnin~ summatinn ;. 581 582 calculated for each column, is stored in a separate RAM until it is required to be added into the next appropriate summation. The update time for the board is 3ms giving 2 x 107 operations/second. This is slower than the 64 neuron specification, but the network is 16 times larger, as the arithmetic elements are being used more efficiently. To achieve a network of greater than 256 neurons, more RAM is required to store the weights. The network is then slower unless a larger number of accelerator chips is used to give a larger moving "patch". 6. CONCLUSIONS A strategy and design method has been given for the construction of bit - serial VLSI neural network chips and circuit boards. Bit - serial arithmetic, coupled to a reduced arithmetic style, enhances the level of integration possible beyond more conventional digital, bit - parallel schemes. The restrictions imposed on both synaptic weight size and arithmetic precision by VLSI constraints have been examined and shown to be tolerable, using the associative memory problem as a test. While we believe our digital approach to represent a good compromise between arithmetic accuracy and circuit complexity, we acknowledge that the level of integration is disappointingly low. It is our belief that, while digital approaches may be interesting and useful in the medium term, essentially as hardware accelerators for neural simulations, analog techniques represent the best ultimate option in 2 - dimensional silicon. To this end, we are currently pursuing techniques for analog pseudo - static memory, using standard CMOS technology. In any event, the full development of a nonvolatile analog memory technology, such as the MNOS technique [7], is key to the long - term future of VLSI neural nets that can learn. 7. ACKNOWLEDGEMENTS The authors acknowledge the support of the Science and Engineering Research Council (UK) in the execution of this work. References 1. S. Grossberg, "Some Physiological and Biochemical Consequences of Psychological Postulates," Proc. Natl. Acad. Sci. USA, vol. 60, pp. 758 - 765, 1968. 2. H. P. Graf, L. D. Jackel, R. E. Howard, B. Straughn, J. S. Denker, W. Hubbard, D. M. Tennant, and D. Schwartz, "VLSI Implementation of a Neural Network Memory with Several Hundreds of Neurons," Proc. AlP Conference on Neural Networks for Computing. Snowbird, pp. 182 - 187, 1986. 3. W. S. Mackie, H. P. Graf, and J. S. Denker, "Microelectronic Implementation of Connectionist Neural Network Models," IEEE Conference on Neural Information Processing Systems. Denver, 1987. 4. J. J. Hopfield and D. W. Tank, "Neural" Computation of Decisions in Optimisation Problems," BioI. Cybern., vol. 52, pp. 141 - 152, 1985. 5. M. A. Sivilotti, M. A. Mahowald, and C. A. Mead, Real - Time Visual Computations Using Analog CMOS Processing Arrays, 1987. To be published 6. C. A. Mead, "Networks for Real - Time Sensory Processing," IEEE Conference on Neural Information Processing Systems, Denver, 1987. 583 7. J. P. Sage, K. Thompson. and R. S. Withers, "An Artificial Neural Network Integrated Circuit Based on MNOSlCCD Principles," Proc. AlP Conference on Neural Networlcs for Computing, Snowbird, pp. 381 - 385, 1986. 8. S. C. J. Garth, "A Chipset for High Speed Simulation of Neural Network Systems," IEEE Conference on Neural Networlc.s, San Diego, 1987. 9. A. F. Murray and A. V. W. Smith, "A Novel Computational and Signalling Method for VLSI Neural Networks," European Solid State Circuits Conference , 1987. 10. A. F. Murray and A. J. W. Smith, "Asynchronous Arithmetic for VLSI Neural Systems," Electronics Letters, vol. 23, no. 12, p. 642, June, 1987. 11. A. F. Murray and A. V. W. Smith, "Asynchronous VLSI Neural Networks using Pulse Stream Arithmetic," IEEE Journal of Solid-State Circuits and Systems, 1988. To be published 12. M. E. Gaspar, "Pulsed Neural Networks: Hardware, Software and the Hopfield AID Converter Example," IEEE Conference on Neural Information Processing Systems. Denver, 1987. 13. M. S. McGregor, P. B. Denyer, and A. F. Murray, "A Single - Phase Clocking Scheme for CMOS VLSI," Advanced Research in VLSI " Proceedings of the 1987 Stanford Conference, 1987. 14. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, "Learning Internal Representations by Error Propagation," Parallel Distributed Processing " Explorations in the Microstructure of Cognition, vol. 1, pp. 318 - 362, 1986. 15. M. Fleisher and E. Levin, "The Hopfiled Model with Multilevel Neurons Models," IEEE Conference on Neural Information Processing Systems. Denver, 1987. 16. G. Parisi, "A Memory that Forgets," J. Phys. A .' Math. Gen., vol. 19, pp. L617 - L620, 1986.
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474 OPTIMIZA nON WITH ARTIFICIAL NEURAL NETWORK SYSTEMS: A MAPPING PRINCIPLE AND A COMPARISON TO GRADIENT BASED METHODS t Harrison MonFook Leong Research Institute for Advanced Computer Science NASA Ames Research Center 230-5 Moffett Field, CA, 94035 ABSTRACT General formulae for mapping optimization problems into systems of ordinary differential equations associated with artificial neural networks are presented. A comparison is made to optimization using gradient-search methods. The perfonnance measure is the settling time from an initial state to a target state. A simple analytical example illustrates a situation where dynamical systems representing artificial neural network methods would settle faster than those representing gradientsearch. Settling time was investigated for a more complicated optimization problem using computer simulations. The problem was a simplified version of a problem in medical imaging: determining loci of cerebral activity from electromagnetic measurements at the scalp. The simulations showed that gradient based systems typically settled 50 to 100 times faster than systems based on current neural network optimization methods. INTRODUCTION Solving optimization problems with systems of equations based on neurobiological principles has recently received a great deal of attention. Much of this interest began when an artificial neural network was devised to find near-optimal solutions to an np-complete problem 13. Since then, a number of problems have been mapped into the same artificial neural network and variations of it 10.13,14,17.18,19.21,23.24. In this paper, a unifying principle underlying these mappings is derived for systems of first to nth -order ordinary differential equations. This mapping principle bears similarity to the mathematical tools used to generate optimization methods based on the gradient. In view of this, it seemed important to compare the optimization efficiency of dynamical systems constructed by the neural network mapping principle with dynamical systems constructed from the gradient. . THE PRINCIPLE This paper concerns itself with networks of computational units having a state variable V, a function! that describes how a unit is driven by inputs, a linear ordinary differential operator with constant coefficients D (v) that describes the dynamical response of each unit, and a function g that describes how the output of a computational unit is detennined from its state v. In particular, the paper explores how outputs of the computational units evolve with time in tenns of a scalar function E, a single state variable for the whole network. Fig. I summarizes the relationships between variables, functions, and operators associated with each computational unit. Eq. (1) summarizes the equations of motion for a network composed of such units: "-+(M) 1 D (v) = (g 1 (v I)' ...• gN (VN ) ) (I) where the i th element of jJ(M) is D(M)(Vj), superscript (M) denotes that operator D is Mth order, the i th element of 1 is !i(gl(VI) • ...• gN(VN», and the network is comprised of N computational units. The network of Hopfield 12 has M=I, functions 1 are weighted linear sums, and functions 1 (where the ith element of 1 is gj(Vj) ) are all the same sigmoid function. We will examine two ways of defining functions 1 given a function F. Along with these definitions will be t Work supported by NASA Cooperative Agreement No. NCC 2-408 © American Institute of Physics 1988 475 defined corresponding functions E that will be used to describe the dynamics of Eq. (1). The first method corresponds to optimization methods introduced by artificial neural network research. It will be referred to as method V y ("dell gil): ! == VyF (2a) with associated E function tN[ dv'(S)jdg .(S) E"j = F("g)-JL D(M)(v·(S»- -' ' ds. i ' dt dt (2b) Here, V xR denotes the gradient of H, where partials are taken with respect to variables of X, and E7 denotes the E function associated with gradient operator V 7' With appropriate operator D and functions 1 and g, Er is simply the "energy function" of Hopfield 12. Note that Eq. (2a) makes explicit that we will only be concerned with 1 that can be derived from scalar potential functions. For example, this restriction excludes artificial neural networks that have connections between excitatory and inhibitory units such as that of Freeman 8. The second method corresponds to optimization methods based on the gradient. It will be referred to as method V if ("dell v"): 1 == VyoF (3a) with associated E function t N [ dv· (s) 1 dv · (s ) Ev> = FCg) -JL D(M)(v.(s»--' ' ds i I dt dt (3b) where notation is analogous to that for Eqs. (2). The critical result ~_ •• that allows us to map \\ optimization problems into networks described by Eq. (1) is that conditions on the constituents of the equation can be chosen so that along any solution trajectory, the E function corresponding to the system will be a monotonic function of time. For method V"j' here are the conditions: all func/ tions g are 1) differentiable /gl(V 1) 'Tg2(v:z) computational unit i : transform that detennines unit i's output from state variable Vi differential operator specifying the dynamical characteristics of unit i function governing how inputs to unit i are combined to drive it and 2) monotonic in the I' same sense. Only the first Figure 1: Schematic of a computational unit i from which netcondition is needed to works considered in this paper are constructed. Triangles suggest make a similar assertion for connections between computational units. method V v- When these conditions are met and when solutions of Eq. (1) exist, the dynamical systems can be used for optimization. The appendix contains proofs for the monotonicity of function E along solution trajectories and references necessary existence theorems. In conclusion, mapping optimization problems onto dynamical systems summarized by Eq. (l) can be reduced to a matter of differentiation if a scalar function representation of the problem can be found and the integrals of Eqs. (2b) and (3b) are ignorable. This last assumption is certainly upheld for the case where operator D has no derivatives less than M'h order. In simulations below, it will be observed to hold for the case M =1 with a nonzero O'h order derivative in D . (Also see Lapedes and Falber 19.) PERSPECTIVES OF RECENT WORK 476 The fonnulations above can be used to classify the neural network optimization techniques used in several recent studies. In these studies, the functions 1 were all identical. For the most part, following Hopfield's fonnulation, researchers 10.13.14.17.23.24 have used method Vy to derive fonns of Eq. (1) that exhibit the ability to find extrema of E-t with Ey quadratic in functions 1 and all functions 1 describable by sigmoid functions such as tanh (x ). However, several researchers have written about artificial neural networks associated with non-quadratic E functions. Method Vy has been used to derive systems capable of finding extrema of non-quadrntic Ey 19. Method V v has been used to derive systems capable of optimizing Ev where Ev were not necessarily quadratic in variables V 21. A sort of hybrid of the two methods was used by Jeffery and Rosner 18 to find extrema of functions that were not quadratic. The important distinction is that their functions j were derived from a given function Fusing Eq. (3a) where, in addition, a sign definite diagonal matrix was introduced; the left side of Eq. (3a) was left multiplied by this matrix. A perspective on the relationship between all three methods to construct dynamical systems for optimization is summarized by Eq. (4) which describes the relationship between methods Vyand Vyo: V? = <liag [a~~;ll-l V,J' (4) where diag [ Xi] is a diagonal matrix with Xi as the diagonal element of row i. (A similar equation has been derived for quadratic F s.) The relationship between the method of Jeffery and Rosner and Vv is simply Eq. (4) with the time dependent diagonal matrix replaced by a constant diagonal matrix of free parameters. It is noted that Jeffery and Rosner presented timing results that compared simulated annealing. conjugate-gradient, and artificial neural network methods for optimization. Their results are not comparable to the results reported below since they used computation time as a perfonnance measure, not settling times of analog systems. The perspective provided by Eq. (4) will be useful for anticipating the relative performance of methods V ~ and V v in the analytical example below and will aid in understanding the results of computer simulations. COMPARISON OF METHODS Vt AND Vv When M =1 and operator D has no Ofh order derivatives, method V v is the basis of gradientsearch methods of optimization. Given the long history of of such methods. it is important to know what possible benefits could be achieved by the relatively ne,w optimization scheme. method Vy. In the following. the optimization efficiency of methods V t and V v is compared by comparing settling times. the time required for dynamical systems described by Eq. (1) to traverse a continuous path to local optima. To qualify this perfonnance measure. this study anticipates application to the creation of analog devices that would instantiate Eq. (1); hence, we are not interested in estimating the number of discrete steps that would be required to find local optima, an appropriate performance measure if the point was to develop new numerical methods. An analytical example will serve to illustrate the possibility of improvements in settling time by using method V t instead of method V V' Computer simulations will be reported for more complicated problems following this example. For the analytical example, we will examine the case where all functions 1 are identical and g(v) = tanhG(v -Th) (5) where G > 0 is the gain and Th is the threshold. Transforms similar to this are widely used in artificial neural network research. Suppose we wish to use such computational units to search a multi-dimensional binary solution space. We note that !li.. = G sech 2G(v -Th) (6) dv is near 0 at valid solution states (comers of a hypercube for the case of binary solution spaces). We see from Eq. (4) that near a valid solution state. a network based on method Vy will allow computational units to recede from incorrect states and approach correct states comparatively faster. Does 477 this imply faster settling time for method V"t? To obtain an analytical comparison of settling times, consider the case where M =1 and operator D has no Om order derivatives and 1 F = ~('.·(tanhGv·)(tanhGv · ) 2~'J • J 'oJ where matrix S is symmetric. Method V y gives network equations dV =StanhGv dt and method V v gives network equations ~ = diag [G sech 2Gvj 1 S tanhGV (7) (8) (9) where tanhGY denotes a vector with i'" component tanhGv;. For method V r there is one stable point, i.e. where ':: = 0, at V = O . For method V v the stable points are V = 0 and V € V where V is the set of vectors with component values that are either +- or -. Further trivialization allows for comparing estimates of settling times: Suppose S is diagonal. For this case, if Vj = 0 is on the trajectory of any computational unit i for one method, Vj = 0 is on the trajectory of that unit for the other method; hence, a comparison of settling times can be obtained by comparing time estimates for a computational unit to evolve from near 0 to near an extremum or, equivalently, the converse. Specifically, let the interval be [Bo, I-a] where 0< Bo<l-a and o<a<1. For method V.., integrating velocity over time gives the estimate 1 [1 [1 1 1 [1-a ~ lJ T Vi = G '2 5(2-5) - l-aJ + In "5(2-a) 00 (10) and for method V y the estimate is T,,;= ~ln [~~~) ~l (11) From these estimates, method V v will always take longer to satisfy the criterion for convergence: Note that only with the largest value for Bo, Bo = 1-5, is the first term of Eq. (10) zero; for any smaller Bo, this term is positive. Unfortunately, this simple analysis cannot be generalized to nondiagonal S. With diagonal S, all computational units operate independently. Hence, the derivation of ':: is irrelevant with respect to convergence rates; convergence rate depends only on the diagonal element of S having the smallest magnitude. In this sense, the problem is one dimensional. But for non-diagonal S, the problem would be, in general, multi-dimensional and, hence, the direction of ':: becomes relevant To compare settling times for non-diagonal S, computer simulations were done. 'These are described below. COMPUTER SIMULA nONS Methods The problem chosen for study was a much simplified version of a problem in medical imaging: Given electromagnetic field measurements taken from the human scalp, identify the location and magnitude of cerebral activity giving rise to the fields. This problem has received much attention in the last 20 years 3,6.7. The problem, sufficient for our purposes here, was reduced to the following problem: given a few samples of the electric potential field at the surface of a spherical conductor within which reside several static electric dipoles, identify the dipole locations and moments. For this situation, there is a closed form solution for electric potential fields at the 478 spherical surface: (12) where ~ is the electric potential at the spherical conductor surface, 'Xsamp/~ is the location of the sample point ( x denotes a vector, i the corresponding unit vector, and x the corresponding vector magnitude), j1; is the dipole moment of dipole i, and d; is the vector from dipole i to X:ampl~ (This equation can be derived from one derived by Brody, Terry, and Ideker 4 ). Fig. 2 facilitates picturing these relationships. Figure 2: Vectors of Eq. (12). With this analytical solution, the problem was formulated as a least squares minimization problem where the variables were dipole moments. In short, the following process was used: A dipole model was chosen. This model was used with Eq. (12) to calculate potentials at points on a sphere which covered about 60% of the surface. A cluster of internal locations that encompassed the locations of the model was specified. The two optimization techniques were then required to determine dipole moment values at cluster locations such that the collection of dipoles at cluster locations accurately reflected the dipole distribution specified by the model. This was to be done given only the potential values at the sample points and an initial guess of dipole moments at cluster locations. The optimization systems were to accomplish the task by minimizing the sum of squared differences between potentials calculated using the dipole model and potentials calculated using a guess of dipole moments at cluster locations where the sum is taken over all sample points. Further simplifications of the problem included 1) choosing the dipole model locations to correspond exactly to various locations of the cluster, 2) requiring dipole model moments to.be I, 0, or -I, and 3) representing dipole moments at cluster locations with two bit binary numbers. To describe the dynamical systems used, it suffices to specify operator D and functions '( of Eq. (1) and function F used in Eqs. (2a) and (3a). Operator D was d D = dt + 1. (13) Eq. (5) with a multiplicative factor of 112 was used for all functions '(. Hence, regarding simplification 3) above, each cluster location was associated with two computational units. Considering simplification 2) above, dipole moment magnitude 1 would be represented by both computational units being in the high state, for -I, both in the low state, and for 0, one in the high state and one in the low state. Regarding function F , F = ~ [~lMaSlll'~d(X:) - <Ilcillomr ('Xs) r -c ~ g (v)2 all samp/~ all compu,ariOflal (14) poims s u"irs j where ~_as""~d is calculated from the dipole model and Eq. (12) (The subscript measured is used because the role of the dipole model is to simulate electric potentials that would be measured in a real world situation. In real world situations, we do not know the source distribution underlying ~_asar~d .), C is an experimentally detennined constant (.002 was used), and ~clJIS'~r is Eq. (12) where the sum of Eq. (12) is taken over all cluster locations and the k,h coordinate of the i,h cluster location dipole moment is • Pi#: = ~ g (Vil:b)' (15) all bits b 479 Index j of Eq. (14) corresponds to one combination of indices ikb. Sample points, 100 of them, were scattered semi-uniformly over the spherical surface emphasized by horizontal shading in Fig. 3. Ouster locations, 11, and model dipoles, 5, were scattered within the subset of the sphere emphasized by vertical shading. For the dipole model used, 10 dipole moment components were non-zero; hence, optimization techniques needed to hold 56 dipole moment components at zero and set 10 components to correct non-zero values in order to correctly identify the dipole model underlying ~_Qs"'~d' I I , ' I I I , I I 0.8 I , I I I , I I I relative radii The dynamical systems corresponding to methods V,. and Vv' were integrated using the forward Euler method (e.g. Press, Flannery, Teukolsky, and Vetterling 22). Numerical methods were observed to be convergent experimentally: settling time and path length were observed to asymtotically approach stable values as step size of the numerical integrator was decreased over two orders of magnitude. Figure 3: illustration of the distribution of sample points on the surface of the sphericll conductor (horizontal shading) and the distribution of model dipole locations and cluster locations within the conductor (verticll shading). Settling times, path lengths, and relative directions of travel were calculated for the two optimization methods using several different initial bit patterns at the cluster locations. In other words. the search was started at different corners of the hypercube comprising the space of acceptable solutions. One corner of the hypercube was chosen to be the target solution. (Note that a zero dipole moment has a degenerate two bit representation in the dynamical systems explored; the target corner was arbitrarily chosen to be one of the degenerate solutions.) Note from Eq. (5) that for the network to reach a hypercube corner, all elements of v would have to be singular. For this reason, settling time and other measures were studied as a function of the proximity of the computational units to their extremum states. Computations were done on a Sequent Balance. Results Graph 1 shows results for exploring settling time as a function of extremum depth, the minimum of the deviations of variables v from the threshold of functions g. Extremum depth is reported in multiples of the width of functions g. The term transition, used in the caption of Graph 1 and below, refers to the movement of a computational unit from one extremum state to the other. The calculations were done for two initial states, one where the output of 1 computational unit was set to zero and one where outputs of 13 computational units were set to zero; bence, 1 and 13, respectively, half transitions were required to reach the target hypercube comer. It can be observed that settling time increases faster for method V v' than that for method V y just as we would expect from considering Eqs. (4) and (5). However, it can be observed that method V v is still an order of magnitude faster even wben extremum depth is 3 widths of functions g. For the purpose of unambiguously identifying what hypercube corner the dynamical system settles 5 +,1 I - ~ I ~ I " ~ 4 3 # '" ... t---. =- o o 1 2 3 4 extremum depth Graph 1: settling time as a function of extremum depth. #: method V r- 1 half transition required. .: method V r 13 half transitions required. +: method V.... 1 half transition required. -: V .... 13 half transitions required. 480 to, this extremum depth is more than adequate. Table 1 displays results for various initial conditions. Angles are reported in degrees. These measures refer to the angle between directions of travel in v-space as specified by the two optimization methods. The average angle reported is taken over all trajectory points visited by the numerical integrator. Initial angle is the angle at the beginning of the path. Parasite cost percentage is a measure that compares parasite cost, the integral in Eqs. (2b) and (3b), to the range of function F over the path: . parasite cost parasite cost % = 100x IFF I fi",," ;,udal transitions time relative path initial Mean angle extremum parasite reauired time len2th anlZle (std dev) deoth cost % 1 0.16 100 6.1 68 76 (3.8) 2.3 0.22 0.0016 1.9 76 (3.5) 2.3 0.039 2 0.14 78 4.7 75 72 (4.3) 2.5 0.055 0.0018 1.9 73 (4.1) 2.5 0.016 3 0.15 71 4.7 74 71 (3.7) 2.3 0.051 0.0021 2.1 72 (3.0) 2.5 0.0093 7 0.19 59 4.6 63 69 (4.1) 2.4 0.058 0.0032 2.4 71 (7.0) 2.7 0.0033 10 0.17 49 3.8 60 63 (2.8) 2.5 0.030 0.0035 2.5 64 (4.7) 2.8 O.OOO6{) 13 0.80 110 9.2 39 77 (11) 2.3 0.076 0.0074 3.2 71 (8.9) 2.7 0.0028 Table 1: Settling time and other measurements for various required transitions. For each transition case, the upper row is for V y and the lower row is for V vStd deY denotes standard deviation. See text for definition of measurement terms and units. (16) Noting the differences in path length and angles reported, it is clear that the path taken to the target hypercube comer was quite different for the two methods. Method V v settles from 1 to 2 orders of magnitude faster than method V -r and usually takes a path less than half as long. These relationships did not change significantly for different values for c of Eq. (14) and coefficients of Eq. (13) (both unity in Eq. (13». Values used favored method V r Parasite cost is consistently less significant for method V v and is quite small for both methods. To further compare the ability of the optimization methods to solve the brain imaging problem, a large variety of initial hypercube comers were tested. Table 2 displays results that suggest the ability of each method to locate the target comer or to converge to a solution that was consistent with the dipole model. Initial comers were chosen by randomly selecting a number of computational units and setting them to eXtI"emwn states opposite to that required by the target solution. Five cases were run for each case of required transitions. It can be observed that the system based on method Vv is better at finding the target comer and is much better at finding a solution that is consistent with the dipole model. DISCUSSION The simulation results seem to contradict settling time predictions of the second analytical example. It is intuitively clear that there is no contradiction when considering the analytical example as a one dimensional search and the simulations as multi-dimensional searches. Consider Fig. 4 which illustrates one dimensional search starting at point I. Since both optimization methods must decrease function E monotonically, both must head along the same path to the minimum point A. Now consider Fig. 5 which illustrates a two dimensional search starting at point I: Here, the two methods needn't follow the same paths. The two dashed paths suggest that method V." can still be transitions I V .. Vv required ~erent dipole different target different dipole different target solution comer comer, solution comer comer 3 1 0 4 0 0 5 4 1 1 3 0 1 4 5 I 0 1 4 0 1 I 4 6 2 1 2 0 1 4 7 4 0 1 0 I 1 I 4 13 5 0 0 1 3 1 20 5 0 0 0 5 I 0 26 5 0 0 2 3 0 33 5 0 0 3 2 0 40 5 0 0 3 I 2 I 0 46 I 5 0 0 2 3 0 53 I 5 0 0 4 1 ! 0 Table 2: Solutions found starting from various initial conditions, five cases for each transition case. Different dipole solution indicates that the system assigned non-zero dipole moments at cluster locations that did not correspond to locations of the dipole model sources. Different corner indicates the solution was consistent with the dipole model but was not the target hypercube comer. Target corner indicates that the solution was the target solution. v 481 monotonically decreasing E while traversing a more circuitous route to minimum B or traversing a path to minimum A. The longer path lengths reported in Table 1 for method V ~ suggest the occurrence of the fonner. The data of Table 2 verifies the occurrence of the latter: Note that for many cases where the system based on method V v settled to the . Figure 4: One dimensional search target comer, the system based on method V ~ settled to some other minimum. for minima. I E Would we observe similar differences in optimization efficiency for other optimization problems that also have binary solution spaces? A view that supports the plausibility of the affirmative is the following: Consider Eq. (4) and Eq. (5). We have already made the observation that method V v would slow convergence into extrema of functions g. We have observed this experimentally via Graph 1. These observations suggest that computational units of V v systems tend to stay closer to the transition regions of functions g compared to computational units of V'I systems. It seems plausible that this property may allow V v systems to avoid advancing too deeply toward ineffective solutions and, hence, allow the systems to approach effective solutions more Figure 5: Two dimensional search efficiently. 1bis behavior might also be the explanation for for minima. the comparative success of method V v revealed in Table 2. Regarding the construction of electronic circuitry to instantiate Eq. (l), systems based on method V v would require the introduction of a component implementing multiplication by the derivative of functions g. This additional complexity may binder the use of method V v for the 482 (a) Output (b) Input construction of analog circuits for optimization. To illustrate the extent of this additional complexity, Fig. 6a shows a schematized circuit for a computational unit of method V -r and Fig. 6b shows a schematized circuit for a computational unit of method V T The simulations reported above suggest that there may be problems for which improvements in settling time may offset complications that might come with added circuit complexity. On the problem of imaging cerebral activity, the results above suggest the possibility of constructing analog devices to do the job. Consider the problem of analyzing electric potentials from the scalp of one perOutput son: It is noted that the measured electric potentials, Figure 6: Schematized circuits for a com~_as"rcd' appear as linear coefficients in F of Eq. (14); hence, they would appear as constant terms in 1 putational unit Notation is consistem with Horowitz and Hill IS. Shading of of Eq. (1). Thus. cf)_asllrcd would be implemented as amplifiers is to e3IIllark components amplifier biases in the circuits of Figs. 6. This is a referred to in the text. a) Computational significant benefit. To understand this. note that funcunit for method V r b) Computational tion Ij of Fig. 1 corresponding to the optimization of . ti thod V function F of Eq. (14) would involve a weighted umt or me ... linear sum of inputs g 1 (v 1 ), ••• , gN (VN). The weights would be the nonlinear coefficients of Eq. (14) and correspond to the strengths of the connections shown in Fig. 1. These connection strengths need only be calculated once for the person ar!d Car! then be set in hardware using, for example, a resistor network. Electric potential measurements could then be ar!alyzed by simply using the measurements to bias the input to shaded amplifiers of Figs. 6. For initialization, the system can be initialized with all dipole moments at zero (the 10 transition case in Table 1). This is a reasonable first guess if it is assumed that cluster locations are far denser than the loci of cerebral activity to be observed. For subsequent measurements, the solution for immediately preceding measurements would be a reasonable initial state if it is assumed that cerebral activity of interest waxes and wanes continuously. Might non-invasive real time imaging of cerebral activity be possible using such optimization devices? Results of this study are far from adequate for answering this question. Many complexities that have been avoided may nUllify the practicality of the idea. Among these problems are: 1) The experiment avoided the possibility of dipole sources actually occurring at locations other than cluster locations. The minimization of function F of Eq. (14) may circumvent this problem by employing the superposition of dipole moments at neighboring cluster locations to give a sufficient model in the mear!. 2) The experiment asswned a very restricted range of dipole strengths. This might be dealt with by increasing the number of bits used to represent dipole moments. 3) The conductor model, a homogeneously conducting sphere, may not be sufficient to model the hwnan head 16. Non-sphericity ar!d major inhomogeneities in conductivity Car! be dealt with, to a certain extent, by replacing Eq. (12) with a generalized equation based on a numerical approximation of a boundary integral equation 20 4) The cerebral activity of interest may not be observable at the scalp. 5) Not all forms of cerebral activity give rise to dipolar sources. (For example, this is well known in olfactory cortex 8.) 6) Activity of interest may be overwhelmed by irrelevant activity. Many methods have been devised to contend with this problem (For example, Gevins and Morgan 9.) Clearly, much theoretical work is left to be done. CONCLUDING REMARKS 483 In this study. the mapping principle underlying the application of artificial neural networks to the optimization of multi-dimensional scalar functions has been stated explicitly. Hopfield 12 has shown that for some scalar functions. i.e. functions F quadratic in functions 1. this mapping can lead to dynamical systems that can be easily implemented in hardware. notably. hardware that requires electronic components common to semiconductor technology. Here. mapping principles that have been known for a considerably longer period of time. those underlying gradient based optimization, have been shown capable of leading to dynamical systems that can also be implemented using semiconductor hardware. A problem in medical imaging which requires the search of a multi-dimensional surface full of local extrema has suggested the superiority of the latter mapping principle with respect to settling time of the corresponding dynamical system. 1bis advantage may be quite significant when searching for global extrema using techniques such as iterated descent 2 or iterated genetic hill climbing 1 where many searches for local extrema are required. This advantage is further emphasized by the brain imaging problem: volumes of measurements can be analyzed without reconfiguring the interconnections between computational units; hence, the cost of developing problem specific hardware for finding local extrema may be justifiable. Finally. simulations have contributed plausibility to a possible scheme for non-invasively imaging cerebral activity. APPENDIX To show that for a dynamical system based on method V r E,. is a monotonic function of time given that all functions g are differentiable and monotonic in the same sense, we need to show that the derivative of ET with respect to time is semi-definite: dET N dF T dgj N [M dVj ] dg, = L-- - L D( )(Vj)-- -. dt j dgj dt i dt dt (Ala) Substituting Eq. (2a), dET N [ dV'] dg· == I, f· -D(M)(v·)+-' -'. dt j' 'dt dt (Alb) Using Eq. (1), d~ = N [dVi ]2 dgi ~O dt ~ dt av· s , , (Alc) as needed. The appropriate inequality depends on the sense in which functions 1 are monotonic. In a similar manner, the result can be obtained for method V v>- With the condition that functions 1 are differentiable, we can show that the derivative of 4 is semi-definite: dE.". N dFv dv· N [ dV'] dv· _v = I,--' - I, D(M)(Vj)_-' -'. dt j dVj dt j dt dt Using Eqs. (3a) and (1), dEv N [dVj ]2~ --~0 dt - ~ dt S , as needed. (A2a) (A2b) In order to use the results derived above to conclude that Eq. (1) can be used for optimization of functions 4 and Et in the vicinity of some point vo. we need to show that there exists a neighborhood of Vo in which there exist solution trajectories to Eq. (1). The necessary existence theorems and transformations of Eq. (1) needed in order to apply the theorems can be found in many texts on ordinary differential equations; e.g. Guckenheimer and Holmes 11. Here, it is mainly important to state that the theorems require that functions ,£c(1), functions g are differentiable, and initial conditions are specified for all derivatives of lower order than M. 484 ACKNOWLEDGEMENTS I would like to thank Dr. Michael Raugh and Dr. Pentti Kanerva for constructive criticism and support. I would like to thank Bill Baird and Dr. James Keeler for reviewing this work. I would like to thank Dr. Derek Fender, Dr. John Hopfield, and Dr. Stanley Klein for giving me opportunities that fostered this conglomeration of ideas. REFERENCES [1] Ackley D.H., "Stochastic iterated genetic bill climbing", PhD. dissertation, Carnegie Mellon U.,1987. [2] Bawn E., Neural Networks for Computing, ed. Denker 1.S. (AlP Confrnc. Proc. 151, ed. Lerner R.G.), p53-58, 1986. [3] Brody D.A., IEEE Trans. vBME-32, n2, pl06-110, 1968. [4] Brody D.A., Terry F.H., !deker RE., IEEE Trans. vBME-20, p141-143, 1973. [5] Cohen M.A., Grossberg S., IEEE Trans. vSMC-13, p815-826, 1983. [6] Cuffin B.N., IEEE Trans. vBME-33, n9, p854-861. 1986. [7] Darcey T.M., AIr J.P., Fender D.H., Prog. Brain Res., v54, pI28-134, 1980. [8] Freeman W J., "Mass Action in the Nervous System", Academic Press, Inc., 1975. [9] Gevins A.S., Morgan N.H., IEEE Trans., vBME-33, n12, pl054-1068, 1986. [10] Goles E., Vichniac G.Y., Neural Networks for Computing, ed. Denker J.S. (AlP Confrnc. Proc. 151, ed. Lerner R.G.), p165-181, 1986. [11] Guckenheimer J., Holmes P., "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields", Springer Verlag, 1983. [12] Hopfield J.I., Proc. Nat!. Acad. Sci., v81, p3088-3092, 1984. [13] Hopfield 1.1., Tank D.W., Bio. Cybrn., v52, p141-152, 1985. [14] Hopfield 1.J., Tank D.W., Science, v233, n4764, p625-633, 1986. [15] Horowitz P., Hill W., "The art of electronics", Cambridge U. Press, 1983. [16] Hosek RS., Sances A., Jodat RW., Larson S.I., IEEE Trans., vBME-25, nS, p405-413, 1978. [171 Hutchinson J.M., Koch C., Neural Networks for Computing, ed. Denker J.S. (AlP Confrnc. Proc. 151, ed. Lerner RG.), p235-240, 1986. [18] Jeffery W., Rosner R, Astrophys. I., v310, p473-481, 1986. [19] Lapedes A., Farber R., Neural Networks for Computing, ed. Denker 1.S. (AlP Confrnc. Proc. lSI, ed. Lerner RG.), p283-298, 1986. [20] Leong H.M.F., ''Frequency dependence of electromagnetic fields: models appropriate for the brain", PhD. dissertation, California Institute of Technology, 1986. [21] Platt I.C., Hopfield J.J., Neural Networks for Computing, ed. Denker I.S. (AlP Confrnc. Proc. 151, ed. Lerner RG.), p364-369, 1986. [22] Press W.H., Flannery B.P., Teukolsky S.A., Vetterling W.T., "Numerical Recipes", Cambridge U. Press, 1986. [23] Takeda M., Goodman J.W., Applied Optics, v25. n18, p3033-3046, 1986. [24] Tank D.W., Hopfield I.J., "Neural computation by concentrating infornation in time", preprint, 1987.
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OPTIMAL NEURAL SPIKE CLASSIFICATION Abstract Amir F. Atiya(*) and James M. Bower(**) (*) Dept. of Electrical Engineering (**) Division of Biology California Institute of Technology Ca 91125 Being able to record the electrical activities of a number of neurons simultaneously is likely to be important in the study of the functional organization of networks of real neurons. Using one extracellular microelectrode to record from several neurons is one approach to studying the response properties of sets of adjacent and therefore likely related neurons. However, to do this, it is necessary to correctly classify the signals generated by these different neurons. This paper considers this problem of classifying the signals in such an extracellular recording, based upon their shapes, and specifically considers the classification of signals in the case when spikes overlap temporally. Introduction How single neurons in a network of neurons interact when processing information is likely to be a fundamental question central to understanding how real neural networks compute. In the mammalian nervous system we know that spatially adjacent neurons are, in general, more likely to interact, as well as receive common inputs. Thus neurobiologists are interested in devising techniques that allow adjacent groups of neurons to be sampled simultaneously. Unfortunately, the small scale of real neural networks makes inserting one recording electrode per cell impractical. Therefore, one is forced to use single electrodes designed to sample neural signals evoked by several cells at once. While this approach provides the multi-neuron recordings being sought, it also presents a rather serious waveform classification problem because the actual temporal sequence of action potentials in each individual neuron must be deciphered. This paper describes a method for classifying the activities of several individual neurons recorded simultaneously using a single electrode. Description of the Problem 95 Over the last two decades considerable attention 1-8 has been devoted to the problem of classification of action potentials in multi-neuron recordings. These action potentials (also referred to as "spikes") are the extracellularly recorded signal produced by a single neuron when it is passing information to other neurons (Fig. 1). Fortunately, spikes recorded from the same cell are more or less similar in shape, while spikes coming from different neurons usually have somewhat different shapes, depending on the neuron type, electrode characteristics, the distance between the electrode and the neuron, and the intervening medium. Fig. 1 illustrates some representative variations in spike shapes. It is our objective to detect and classify different spikes based on their shapes. However, relying entirely on the shape of the spikes presents difficulties. For example spikes from different neurons can overlap temporally producing novel waveforms (see Fig. 2 for an example of an overlap). To deal with these overlaps, one has first to detect the occurrence of an overlap, and then estimate the constituent spikes. Unfortunately, only a few of the available spike separation algorithms consider these events, even though they are potentially very important in understanding neural networks. Those few tend to rely © American Institute of Physics 1988 96 on heuristic rules and subtractive methods to resolve overlap cases. No currently published method we are aware of attempts to use knowledge of the likelihood of overlap events for detecting them, which is at the basis of the method we will describe. Fig. 1 An example of a multi-neuron recording overlapping spikes Fig. 2 An example of a temporal overlap of action potentials General Approach The first step in classifying neural waveforms is obviously to identify the typical spike shapes occurring in a particular recording. To do this we have applied a learning algorithm on the beginning portion of the recording, which in an unsupervised fashion (i.e. without the intervention of a human operator) estimates the shapes. After the learning stage we have the classification stage, which is applied on the remaining portion of the recording. A new classification method is proposed, which gives minimum probability of error, even in case of the occurrence of overlapping spikes. Both the learning and the classification algorithms require a preprocessing step to detect the position of the spike candidate in the data record. Detection: For the first task of detection most researchers use a simple level detecting algorithm, that signals a spike when recorded voltage levels cross a certain voltage threshold. However, variations in recording position due to natural brain movements during recording (e.g. respiration) can cause changes in relative height of the positive to the negative peak. Thus, a level detector (using either a positive or a negative threshold) can miss some spikes. Alternatively, we have chosen to detect an event by sliding a window of fixed length until a time when the peak to peak value within the window exceeds a certain threshold. Learning: Learning is performed on the beginning portion of the sampled data using the Isodata clustering algorithm 9. The task is to estimate the number of neurons n whose spikes are represented in the waveform and learn the different shapes of the spikes of the various neurons. For that purpose we apply the clustering algorithm choosing only one feature 97 from the spike, the peak to peak value which we have found to be quite an effective feature. Note that using the peak to peak value in the learning stage does not necessitate using it for classification (one might need additional or different features, especially for tackling the case of spike overlap) . The Optimal Olassification Rule: Once we have identified the number of different events present, the classification stage is concerned with estimating the identities of the spikes in the recording, based on the typical spike shapes obtained in the learning stage. In our classification scheme we make use of the information given by the shape of the detected spike as well as the firing rates of the different neurons. Although the shape plays in general the most important role in the classification, the rates become a more significant factor when dealing with overlapping events. This is because in general overlap is considerably less frequent than single spikes. The shape information is given by a set of features extracted from the waveform. Let x be the feature vector of the detected spike (e.g. the samples of the spike waveform). Let N I , ... , Nn represent the different neurons. The detection algorithm tells us only that at least one spike occurred in the narrow interval (t - TI,t + T2) (= say 1) where t is the instant of the peak of the detected spike, TI and T2 are constants chosen subjectively according to the smallest possible time separation between two consecutive spikes, identifiable as two separate (nonoverlapping) spikes. By definition, if more than one spike occurs in the interval I, then we have an overlap. As a matter of convention, the instant of the occurrence of a spike i. .. taken to be that of the spike peak. For simplicity, we will consider the case of two possibly overlapping spikes, though the method can be extended easily to more. The classification rule which results in minimum probability of error is the one which chooses the neuron (or pair of neurons in case of overlap) which has the maximum likelihood. We have therefore to compare the Pi'S and the P,/s, defined as ~ = P(Ni fired in Ilx, A), i = 1, ... ,n P,j = P(N, and Nj fired in Ilx, A), l,j=I, ... ,n, j<l where A represents the event that one or two spikes occurred in the interval I. In other words Pi the probability that what has been detected is a single spike from neuron i, whereas P,j is the probability that we have two overlapping spikes from neurons land j (note that spikes from the same neuron never overlap). Henceforth we will use I to denote probability density. For the purpose of abbreviation let Bi(t) mean "neuron Ni fired at t". The classification problem can be reduced to comparing the following likelihood functions: i = 1, ... ,n (la) " j = 1, ... , n, j < I (lb) (for a derivation refer to Appendix). Let Ii be the density of the inter-spike interval and Ti be the most recent firing instant of neuron Ni . IT we are given the fact that neuron Ni has been idle for at least a period of duration t Ti, we get A disadvantage of using (2) is that the available /i's and T&,S are only estimates, which depend on the previous classification results, Further, for reliable estimation of the densities Ii, one needs a large number of spikes and therefore a long learning period since we are estimating a 98 whole function. Therefore, we have not used this form, but instead have used the following two schemes. In the first one, we ignore the knowledge about the previous firing pattern except for the estimated firing rates >'1, ... , >'n of the different neurons Nl, ... , Nn respectively. Then the probability of a spike coming from neuron Ni in an interval of duration dt is simply >'idt. Hence In the second scheme we do not use any previous knowledge except for the total firing rate (of all neurons), say a. Then Although the second scheme does not use as much of the information about the firing pattern as the first scheme does, it has the advantage of obtaining and using a more reliable estimate of the firing rate, because in general the overall firing rate changes less with time than the individual rates and because the estimate of a does not depend on previous classification results. However, it is useful mostly when the firing rates of the different neurons do not vary much, otherwise the firt scheme is preferred. In real recording situations, sometimes one encounters voltage signals which are much different than any of the previously learned typical spike shapes or their pairwise overlaps. This can happen for example due to a falsely detected noise event, a spike from a class not encountered in the learning stage, or to the overlap of three or more spikes. To cope with these cases we use the reject option. This means that we refuse to classify the detected spike because of the unlikeliness of the assumed event A. The reject option is therefore employed whenever P(Alx) is smaller than a certain threshold. We know that P(Alx) = J(A,x)/[J(A,x) + J(AC ,x)] where AC is the complement of the event A. The density f(AC,x) can be approximated as uniform (over the possible values of x) because a large variety of cases are covered by the event AC. It follows that one can just compare J(A,x) to a threshold. Hence the decision strategy becomes finally: Reject if the sum of the likelihood functions is less than a threshold. Otherwise choose the neuron (or pair of neurons) corresponding to the largest likelihood functions. Note that the sum of the likelihood functions equals J(A,x) (refer to Appendix). Now, let us evaluate the integrals in (1). Overlapping spikes are assumed to add linearly. Since we intend to handle the overlap case, we have to use a set of features Xm which obeys the following. Given the features of two of the waveforms, then one can compute those of their overlap. A good such candidate is the set of the samples of the spike (or possibly also just part of the samples). The added noise, partly thermal noise from the electrode and partly due to firings from distant neurons, can usually be approximated as white Gaussian. Let the variance be a 2 • The integrals in the likelihood functions can be approximated as summations (note in fact that we have samples available, not a continuous waveform). Let yi represent the typical feature vector (template) associated with neuron Ni , with the mth component being y;". Then M J(xIB/(kI), Bj(kd) = (21r)~/2aM exp[ - 2~2 '~l (x m - y!n-k 1 y~_k2)2] 99 where Xm is the mth component of x, and M is the dimension of x. This leads to the following likelihood functions M~ M L~ = f(Bd k)) L exp[- 2~2 :L (xm - Y:n_kJ2] kl=-M 1 m=l M) Ml M LL· = f(B,(k))f(Bj(k)) L L exp[- 2~2 L (xm - y!n-k 1 y~_kl)2] kl=-Mlkl=-Ml m=l where k is the spike instant, and the interval from -Ml to M2 corresponds to the interval I defined at the beginning of the Section. Implementation The techniques we have just described were tested in the following way. For the first experiment we identified two spike classes in a recording from the rat cerebellum. A signal is created, composed of a number of spikes from the two classes at random instants, plus noise. To make the situation as realistic as possible, the added noise is taken from idle periods (i.e. non-spiking) of a real recording. The reason for using such an artificially generated signal is to be able to know the class identities of the spikes, in order to test our approach quantitatively. We implement the detection and classification techniques on the obtained signal, with various values of noise amplitude. In our case the ratio of the peak to peak values of the templates turns out to be 1.375. Also, the spike rate of one of the clases is twice that of the other class. Fig.3a shows the results with applying the first scheme (i.e. using Eq. 3). The overall percentage correct classification for all spikes (solid curve) and the percentage correct classification for overlapping spikes (dashed curve) are plotted versus the standard deviation of the noise (]" normalized with respect to the peak h of the large template. Notice that the overall classification accuracy is near 100% for (]" I h less than 0.15, which is actually the range of noise amplitudes we mostly encountered in our work with real recordings. Observe also the good results for classifying overlapping events. We have applied also the second scheme (i.e. using Eq. 4) and obtained similar results. We wish to mention that the thresholds for detection and for the reject option are set up so as to obtain no more than 3% falsely detected spikes. A similar experiment is performed with three waveforms (three classes), where two of the waveforms are the same as those used in the first experiment . The third is the average of the first two. All the three neurons have the same spike rate (i.e. ..\1 = ..\2 = ..\3)' Hence both classification schemes are equivalent in this case. Fig. 3b shows the overall as well as the sub-category of overlap classification results. One observes that the results are worse than those for the two-class case. This is because the spacings between the templates are in general smaller. Notice also that the accuracy in resolving overlapping events is now tangibly less than the overall accuracy. However, one can say that the results are acceptable in the range of (]" Ih less than 0.1. The following experiment is also performed using the same data. We would like to investigate the importance of the information given by the (overall) firing rate on the problem of classifying overlapping events. In our method the summation in the likelihood functions for single spikes is multiplied by Otln, while that for overlapping spikes is multiplied by (Otln)2 . Usually Otln is considerably less than one. Hence we have a factor which gives less weight for overlapping events. Now, consider the case of ignoring completely the information given by the firing rate and relying solely on sha.pe information. We assume that overlapping spikes from any two given classes represent "new" class of waveforms and that each of these overlap classes has the same rate as that of a single-spike cla.ss. In that case we can obtain expressions for the likelihood functions as consisting just the summations, i.e. free of the rate 100 1 •• -.. -; • -; 51.• C • d._ e ; ... It._ I . I. ....S 1 •• .. 51." C . ... . ; ... It._ ... " 1.111 I.ISZ '.l,' I. I. ... .. 1.1" ••••• /t. • I ••• ,1. a 1 • • .,-; ~ • 51.· c C ... • • ; ... It._ I. I. ..... 1.1" 1.11t 1.I!Il ••••• /t. c Fig. 3 a) Overall (solid curve) and overlap (dashed curve) classification accuracy for a two class case b b) Overall (solid curve) and overlap (dashed curve) classification accuracy for a three class case c)Percent of incorrect classification of single spikes as overlap solid curve: scheme utilzing the spike rate dashed curve: scheme not utilising the spike rate I .ISZ l .ltI factor Olin (refer to Appendix). An experiment is performed using that scheme (on the same three class data). One observes that the method classifies a number of single spikes wrongly as overlaps, much more than our original scheme does (see Fig. 3c), especially for the large noise case. On the other hand, the number of overlaps which are classified wrongly as single spikes is near zero for both schemes. Finally, in the last experiment the techniques are implemented on real recordings from the rat cerebellum. The recorded signal is band-pass-filtered in the frequency range 300 Hz - 10 KHz, then sampled with a rate of 20KHz. For classification, we take 20 samples per spike as features. Fig. 4 shows the results ofthe proposed method, using the first scheme (Eq. 3). The number of neurons whose spikes are represented in the waveform is estimated to be four. The 101 detection threshold is set up so that spikes which are too small are disregarded, because they come from several neurons far away from the electrode and are hard to distinguish. Notice the overlap of classes 1 and 2, which was detected. We used the second scheme also on the same portion and it gave similar results as those of the first scheme (only one of the spikes is classified differently). Overall, the discrepancies between classifications done by the proposed method and an experienced human observer were found to be small. 3 2 3 3 3 2 1 4 1 1 3 1,2 3 3 2 2 3 4 Fig. 4 Classification results for a recording from the rat cerebellum Conclusion Many researchers have considered the problem of spike classification in multi-neuron recordings, but only few have tackled the case of spike overlap, which could occur frequently, particularly if the group of neurons under study is stimulated. In this work we propose a method for spike classification, which can also aid in detecting and classifying overlapping spikes. By taking into account the statistical properties of the discharges of the neurons sampled, this method minimizes the probability of classification error. The application of the method to artificial as well as real recordings confirm its effectiveness. Appendix Consider first P'i' We can write 102 We can also obtain R. = It+T2[t+T2 f(x,AIBI(td,Bj (t2))f(B (t ) B ·(t ))dt dt IJ f( A) I 1, J 2 1 2 · t-T1 t-Tl x, Now, consider the two events B1(td and Bj (t2 ). In the absense of any information about their dependence, we assume that they are independent. We get Within the interval I, both f(B/(tt)) and f(Bj (t2)) hardly vary because the duration of I is very small compared to a typical inter-spike interval. Therefore we get the following approximation: f(B/(td) ~ f(B,(t)) f(B j (t2)) ~ f(Bj(t)). The expression for P"j becomes f(B,(t))f(B ·(t)) [t+T2 [t- T2 P"j ~ () J f(xIB/(td, B j (t2))dt 1dt2 • f X , A t-Tl t-Tl Notice that the term A was omitted from the argument of the density inside the integral, because the occurrence of two spikes at tl and t2El implies the occurrence of A. A similar derivation for ~ results in The term f(x, A) is common to all the Pils and the Pi's. Hence one can simply compare the following likelihood functions: Aeknow ledgement Our thanks to Dr. Yaser Abu-Mostafa for his assistance with this work. This project was supported by the Caltech Program of Advanced Technology (sponsored by Aerojet,GM,GTE, and TRW), and the Joseph Drown Foundation. References II] M. Abeles and M. Goldstein, Proc. IEEE, 65, pp.762-773, 1977. 12] G. Dinning and A. Sanderson, IEEE Trans. Bio - M ed. Eng., BME-28, pp. 804-812, 1981. 13] E. D'Hollander and G. Orban, IEEE Trans . Bio-Med. Eng., BME-26, pp. 279-284, 1979. 14] D. Mishelevich, IEEE Trans. Bio-Med. Eng., BMFr17, pp. 147-150, 1970. Is] V. Prochazka and H. Kornhuber, Electroenceph. din. Neurophysiol., 32, pp. 91-93, 1973. 16] W. Roberts, Bioi. Gybernet., 35, pp. 73-80, 1979. 17] W. Roberts and D. Hartline, Brain Res., 94, pp. 141-149, 1975. 18] E. Schmidt, J. Neurosci. Methods, 12, pp. 95-111, 1984. 19] R. Duda and P. Hart, Pattern Classification and Scene Analysis, John Wiley, 1973.
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495 REFLEXIVE ASSOCIATIVE MEMORIES Hendrlcus G. Loos Laguna Research Laboratory, Fallbrook, CA 92028-9765 ABSTRACT In the synchronous discrete model, the average memory capacity of bidirectional associative memories (BAMs) is compared with that of Hopfield memories, by means of a calculat10n of the percentage of good recall for 100 random BAMs of dimension 64x64, for different numbers of stored vectors. The memory capac1ty Is found to be much smal1er than the Kosko upper bound, which Is the lesser of the two dimensions of the BAM. On the average, a 64x64 BAM has about 68 % of the capacity of the corresponding Hopfield memory with the same number of neurons. Orthonormal coding of the BAM Increases the effective storage capaCity by only 25 %. The memory capacity limitations are due to spurious stable states, which arise In BAMs In much the same way as in Hopfleld memories. Occurrence of spurious stable states can be avoided by replacing the thresholding in the backlayer of the BAM by another nonl1near process, here called "Dominant Label Selection" (DLS). The simplest DLS is the wlnner-take-all net, which gives a fault-sensitive memory. Fault tolerance can be improved by the use of an orthogonal or unitary transformation. An optical application of the latter is a Fourier transform, which is implemented simply by a lens. I NTRODUCT ION A reflexive associative memory, also called bidirectional associative memory, is a two-layer neural net with bidirectional connections between the layers. This architecture is implied by Dana Anderson's optical resonator 1, and by similar configurations2,3. Bart KoSk04 coined the name "Bidirectional Associative Memory" (BAM), and Investigated several basic propertles4- 6. We are here concerned with the memory capac1ty of the BAM, with the relation between BAMs and Hopfleld memories7, and with certain variations on the BAM. © American Institute of Physics 1988 496 BAM STRUCTURE We will use the discrete model In which the state of a layer of neurons Is described by a bipolar vector. The Dirac notationS will be used, In which I> and <I denote respectively column and row vectors. <al and la> are each other transposes, <alb> Is a scalar product, and la><bl is an outer product. As depicted in Fig. 1, the BAM has two layers of neurons, a front layer of N neurons w tth state vector If>, and a back layer back layer. P neurons back of P neurons with state vector state vector b stroke Ib>. The bidirectional connecsignal flow In two directions. 1 1 tlons between the layers allow frOnt1ay~r. 'N ~eurons forward The front stroke gives Ib>= state vector f stroke s(Blf», where B 15 the connecFig. 1. BAM structure tlon matrix, and s( ) Is a threshold function, operating at zero. The back stroke results 1n an u~graded front state <f'I=s( <biB), whIch also may be wr1tten as !r'>=s(B Ib> >. where the superscr1pt T denotes transpos1t10n. We consider the synchronous model. where all neurons of a layer are updated s1multaneously. but the front and back layers are UPdated at d1fferent t1mes. The BAM act10n 1s shown 1n F1g. 2. The forward stroke entalls takIng scalar products between a front state vector If> and the rows or B, and enter1ng the thresholded results as elements of the back state vector Ib>. In the back stroke we take threshold ing f & reflection lID NxP FIg. 2. BAM act 10n threshold ing & reflection b v ~ ~hreShOlding 4J feedback & NxN V Ftg. 3. Autoassoc1at1ve memory act10n scalar products of Ib> w1th column vectors of B, and enter the thresholded results as elements of an upgraded state vector 1('>. In contrast, the act10n of an autoassoc1at1ve memory 1s shown 1n F1gure 3. The BAM may also be described as an autoassoc1at1ve memory5 by 497 concatenating the front and back vectors tnto a s1ngle state vector Iv>=lf,b>,and by taking the (N+P)x(N+P) connection matrtx as shown in F1g. 4. This autoassoclat1ve memory has the same number of neurons as our f . b'----"" BAM, viz. N+P. The BAM operat1on where ----!' initially only the front state 1s specizero [IDT lID zero f thresholding & feedback f1ed may be obtained with the corresb ponding autoassoc1ative memory by initially spectfying Ib> as zero, and by Fig. 4. BAM as autoassoarranging the threshold1ng operat1on ctative memory such that s(O) does not alter the state vector component. For a Hopfteld memory 7 the connection matrix 1s M H=( I 1m> <mD -MI , m=l (1) where 1m>, m= 1 to M, are stored vectors, and I is the tdentity matr1x. Writing the N+P d1mens1onal vectors 1m> as concatenations Idm,cm>, (1) takes the form M H-( I (ldm><dml+lcm><cml+ldm><cml+lcm><dmD)-MI , (2) m=l w1th proper block plactng of submatr1ces understood. Writing M K= Llcm><dml , (3) M m=l M Hd=(Lldm><dmD-MI, Hc=( L'lcm><cml>-MI, (4) m=l m=l where the I are identities in appropriate subspaces, the Hopfield matrix H may be partitioned as shown in Fig. 5. K is just the BAM matrix given by Kosko5, and previously used by Kohonen9 for linear heteroassoclatjve memories. Comparison of Figs. 4 and 5 shows that in the synchronous discrete model the BAM with connection matrix (3) is equivalent to a Hopfield memory in which the diagonal blocks Hd and Hc have been 498 deleted. Since the Hopfleld memory is robust~ this "prun1ng" may not affect much the associative recall of stored vectors~ if M is small; however~ on the average~ pruning will not improve the memory capaclty. It follows that, on the average~ a discrete synchronous BAM with matrix (3) can at best have the capacity of a Hopfleld memory with the same number of neurons. We have performed computations of the average memory capacity for 64x64 BAMs and for corresponding 128x 128 Hopfleld memories. Monte Carlo calculations were done for 100 memories) each of which stores M random bipolar vectors. The straight recall of all these vectors was checked) al10wtng for 24 Iterations. For the BAMs) the iterations were started with a forward stroke in which one of the stored vectors Idm> was used as input. The percentage of good recall and its standard deviation were calculated. The results plotted in Fig. 6 show that the square BAM has about 68~ of the capacity of the corresponding Hopfleld memory. Although the total number of neurons is the same) the BAM only needs 1/4 of the number of connections of the Hopfield memory. The storage capacity found Is much smaller than the Kosko 6 upper bound) which Is min (N)P). JR[= Fig. 5. Partitioned Hopfield matrix 10 20 30 40 50 60 M. number of stored vectors Fig. 6. ~ of good recall versus M CODED BAM So far) we have considered both front and back states to be used for data. There is another use of the BAM in which only front states are used as data) and the back states are seen as providing a code) label, or pOinter for the front state. Such use was antiCipated in our expression (3) for the BAM matrix which stores data vectors Idm> and their labels or codes lem>. For a square BAM. such an arrangement cuts the Information contained in a single stored data vector jn half. However, the freedom of 499 choosing the labels fCm> may perhaps be put to good use. Part of the problem of spurious stable statesl which plagues BAMs as well as Hopf1eld memories as they are loaded up, is due to the lack of orthogonality of the stored vectors. In the coded BAM we have the opportunity to remove part of this problem by choosing the labels as orthonorma1. Such labels have been used previously by Kohonen9 1n linear heteroassociative memories. The question whether memory capacity can be Improved In this manner was explored by taking 64x64 BAt1s In which the labels are chosen as Hadamard vectors. The latter are bipolar vectors with Euclidean norm ,.fp, which form an orthonormal set. These vectors are rows of a PxP Hadamard matrix; for a discussion see Harwtt and Sloane 1 0. The storage capacity of such Hadamard-coded BAMs was calculated as function of the number M of stored vectors for 100 cases for each value of M, in the manner discussed before. The percentage of good recall and its standard deviation are shown 1n Fig. 6. It Is seen that the Hadamard coding gives about a factor 2.5 in M, compared to the ordinary 64x64 BAM. However, the coded BAM has only half the stored data vector dimension. Accounting for this factor 2 reduction of data vector dimension, the effective storage capacity advantage obtained by Hadamard coding comes to only 25 ~. HALF BAt1 WITH HADAMARD CODING For the coded BAM there is the option of deleting the threshold operation In the front layer. The resulting architecture may be called "half BAt1". In the half BAM, thresholding Is only done on the labels, and consequently, the data may be taken as analog vectors. Although such an arrangement diminishes the robustness of the memory somewhat, there are applications of interest. We have calculated the percentage of good recall for 1 00 cases, and found that giving up the data thresholding cuts the storage capacity of the Hadamard-coded BAt1 by about 60 %. SELECTIVE REFLEXIVE MEMORY The memory capacity limitations shown in Fig. 6 are due to the occurence of spurious states when the memories are loaded up. Consider a discrete BAM with stored data vectors 1m>, m= 1 to M, orthonormal labels Icm>, and the connection matrix 500 (5) For an input data vector Iv> which is closest to the stored data vector 11 >, one has 1n the forward stroke M Ib>=s(clc 1 >+ L amlcm» , (6) m=2 where c=< llv> • and am=<mlv> (7) M Although for m# 1 am<c, for some vector component the sum L amlcm> m=2 may accumulate to such a large value as to affect the thresholded result Ib>. The problem would be avoided jf the thresholding operation s( ) in the back layer of the BAM were to be replaced by another nonl1near operation which selects, from the I inear combination M clc 1 >+ L amlcm> m=2 (8) the dominant label Ic 1 >. The hypothetical device which performs this operation is here called the "Dominant Label Selector" (DLS) 11, and we call the resulting memory architecture "Selective Reflexive Memory" (SRM). With the back state selected as the dominant label Ic 1 >, the back stroke gives <f'I=s( <c ,IK)=s(P< 1 D=< 11, by the orthogonal ity of the labels Icm>. It follows 11 that the SRM g1ves perfect assoc1attve recall of the nearest stored data vector, for any number of vectors stored. Of course, the llnear independence of the P-dimensionallabel vectors Icm>, m= 1 to M, requires P>=M. The DLS must select, from a linear combination of orthonormal labels, the dominant label. A trivial case is obtained by choosing the 501 labels Icm> as basis vectors Ium>, which have all components zero except for the mth component, which 1s unity. With this choice of labels, the f DLS may be taken as a winner~ winner b take-all net Flg.7. Simplest reflexive memory with DLS take-all net W, as shown in Fig. 7. This case appears to be Included in Adapt Ive Resonance Theory (ART) 12 as a special sjmpllf1ed case. A relationship between the ordinary BAM and ART was pOinted out by KoskoS. As in ART, there Is cons1derable fault sensitivity tn this memory, because the stored data vectors appear in the connectton matrix as rows. A memory with better fault tolerance may be obtained by using orthogonal labels other than basis vectors. The DLS can then be taken as an orthogonal transformation 6 followed by a winner-take-an net, as shown 1n Fig. 8. 6 is to be chosen such that 1t transforms the labels Icm> f I 1 1[ (G i u l tnto vectors proportional to the rthogonal 1 transforbasts vectors um>. This can always ,.0 mation winner/' take-all net be done by tak1ng p (9) F1g. 8. Select1ve reflex1ve memory G= [Iup> <cpl , p=l where the Icp>, p= 1 to P, form a complete orthonormal set which contains the labels Icm>, m=l to M. The neurons in the DLS serve as grandmother cells. Once a single winning cell has been activated, I.e., the state of the layer Is a single basis vector, say lu I ) J this vector must be passed back, after appllcation of the transformation G- 1, such as to produce the label IC1> at the back of the BAM. Since G 1s orthogonal. we have 6- 1 =6 T, so that the reQu1red 1nverse transformation may be accompl1shed sfmply by sending the bas1s vector back through the transformer; this gives P <u 116=[ <u 1 IUp><cpl=<c 11 p=l (10) 502 as required. HAlF SRM The SRM may be modified by deleting the thresholding operation in the front layer. The front neurons then have a I inear output, which is reflected back through the SRM, as shown in Fig. 9. In this case, the f I i near neurons / orthogonal 1 .1 ~ U (G T I ,- transformation winner'/' take-all net Fig. 9. Half SRM with l1near neurons in front layer stored data vectors and the input data vectors may be taken as analog vectors, but we reQu1re all the stored vectors to have the same norm. The act i on of the SRM proceeds in the same way as described above, except that we now require the orthonormal labels to have unit norm. It follows that, just l1ke the full SRM, the half SRM gives perfect associative recall to the nearest stored vector, for any number of stored vectors up to the dimension P of the labels. The latter condition 1s due to the fact that a P-dimensional vector space can at most conta1n P orthonormal vectors. In the SRM the output transform Gis 1ntroduced in order to improve the fauJt tolerance of the connection matrix K. This is accomplished at the cost of some fault sensitivity of G, the extent of which needs to be investigated. In this regard 1t is noted that in certatn optical implementat ions of reflexive memories, such as Dana Anderson's resonator I and Similar conflgurations2,3, the transformation G is a Fourier transform, which is implemented simply as a lens. Such an implementation ts quite insentive to the common semiconductor damage mechanisms. EQUIVALENT AUTOASSOCIATIVE MEMORIES Concatenation of the front and back state vectors allows description of the SRMs tn terms of autoassociative memories. For the SRM which uses basis vectors as labels the corresponding autoassociative memory js shown tn Fjg. 10. This connect jon matrtx structure was also proposed by Guest et. a1. 13. The wtnner-take-all net W needs to be /' /' f b zero I[T !r WI " [~ " ~ slow thres holding & feedback "I' f blJ h ast thres olding& feedback Fig. 10. Equivalent autoassociat lve memory 503 given t1me to settle on a basis vector state before the state Ib> can influence the front state If>. This may perhaps be achieved by arranging the W network to have a thresholding and feedback which are fast compared with that of the K network. An alternate method may be to equip the W network w1th an output gate which is opened only after the W net has sett led. These arrangements present a compUcatlon and cause a delay, which in some appllcations may be 1nappropriate, and In others may be acceptable in a trade between speed and memory density. For the SRM wtth output transformer and orthonormal1abels other fb, w ~eedback (OJ [T I[ (OJ (Q) (G (OJ (GT WI f thresholded b linear W thresholded + output gate Fig. 11. Autoassoc1at1ve memory equivalent to SRM with transform output gate wr ~ winner-take-all .......... Woutput :t@ b back layer, linear '--___ -' f front layer II = BAM connections @ =orthogonal transformat i on W! ~ winner-take-all net Fig. 12. Structure of SRM than basis vectors, a corresponding autoassoclat1ve memory may be composed as shown In Fig.l1. An output gate in the w layer is chosen as the device which prevents the backstroke through the BAM to take place before the w1nner-take-al net has settled. The same effect may perhaps be achieved by choosing different response times for the neuron layers f and w. These matters require investigation. Unless the output transform G 1s already required for other reasons, as in some optical resonators, the DLS with output transform is clumsy. I t would far better to combine the transformer G and the net W into a single network. To find such a DLS should be considered a cha 11 enge. 504 The wort< was partly supported by the Defense Advanced Research projects Agency, ARPA order -5916, through Contract DAAHOI-86-C -0968 with the U.S. Army Missile Command. REFERENCES 1. D. Z. Anderson, "Coherent optical eigenstate memory", Opt. Lett. 11, 56 (1986). 2. B. H. Soffer, G. J. Dunning, Y. Owechko, and E. Marom, "Associative holographic memory with feedback using phase-conjugate mirrors", Opt. Lett. II, 1 18 ( 1986). 3. A. Yarrtv and S. K. Wong, "Assoctat ive memories based on messagebearing optical modes In phase-conjugate resonators", Opt. Lett. 11, 186 (1986). 4. B. Kosko, "Adaptive Cognitive ProceSSing", NSF Workshop for Neural Networks and Neuromorphlc Systems, Boston, Mass., Oct. &-8, 1986. 5. B. KOSKO, "Bidirectional Associative Memories", IEEE Trans. SMC, In press, 1987. 6. B. KOSKO, "Adaptive Bidirectional Associative Memories", Appl. Opt., 1n press, 1987. 7. J. J. Hopfleld, "Neural networks and physical systems with emergent collective computational ablJ1tles", Proc. NatJ. Acad. Sct. USA 79, 2554 ( 1982). 8. P. A. M. Dirac, THE PRINCI PLES OF QUANTLt1 MECHANICS, Oxford, 1958. 9. T. Kohonen, "Correlation Matrix Memories", HelsinsKi University of Technology Report TKK-F-A 130, 1970. 10. M. Harwit and N. J. A Sloane, HADAMARD TRANSFORM OPTICS, Academic Press, New York, 1979. 11. H. G. Loos, It Adaptive Stochastic Content-Addressable Memory", Final Report, ARPA Order 5916, Contract DAAHO 1-86-C-0968, March 1987. 12. G. A. Carpenter and S. Grossberg, "A Massively Parallel Architecture for a Self-Organizing Neural Pattern Recognition Machine", Computer Vision, Graphics, and Image processing, 37, 54 (1987). 13. R. D. TeKolste and C. C. Guest, "Optical Cohen-Grossberg System with Ali-Optical FeedbaCK", IEEE First Annual International Conference on Neural Networks, San Diego, June 21-24, 1987.
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534 The Performance of Convex Set projection Based Neural Networks Robert J. Marks II, Les E. Atlas, Seho Oh and James A. Ritcey Interactive Systems Design Lab, FT-IO University of Washington, Seattle, Wa 98195. ABSTRACT We donsider a class of neural networks whose performance can be analyzed and geometrically visualized in a signal space environment. Alternating projection neural networks (APNN' s) perform by alternately projecting between two or more constraint sets. Criteria for desired and unique convergence are easily established. The network can be configured in either a homogeneous or layered form. The number of patterns that can be stored in the network is on the order of the number of input and hidden neurons. If the output neurons can take on only one of two states, then the trained layered APNN can be easily configured to converge in one iteration. More generally, convergence is at an exponential rate. Convergence can be improved by the use of sigmoid type nonlinearities, network relaxation and/or increasing the number of neurons in the hidden layer. The manner in which the network responds to data for which it was not specifically trained (i.e. how it generalizes) can be directly evaluated analytically. 1. INTRODUCTION In this paper, we depart from the performance analysis techniques normally applied to neural networks. Instead, a signal space approach is used to gain new insights via ease of analysis and geometrical interpretation. Building on a foundation laid elsewherel - 3 , we demonstrate that alternating projecting neural network's (APNN's) formulated from such a viewpoint can be configured in layered form or homogeneously. Significiantly, APNN's have advantages over other neural network architectures . For example, (a) APNN's perform by alternatingly projecting between two or more constraint sets. Criteria can be established for proper iterative convergence for both synchronous and asynchronous operation. This is in contrast to the more conventional technique of formulation of an energy metric for the neural networks, establishing a lower energy bound and showing that the energy reduces each iteration4- 7 • Such procedures generally do not address the accuracy of the final solution. In order to assure that such networks arrive at the desired globally minimum energy, computationaly lengthly procedures such as simulated annealing are usedB - 10 • For synchronous networks, steady state oscillation can occur between two states of the same energyll (b) Homogeneous neural networks such as Hopfield's content addressable memory4,12-14 do not scale well, i.e. the capacity © American Institute of Physics 1988 535 of Hopfield's neural networks less than doubles when the number of neurons is doubled 15-16. Also, the capacity of previously proposed layered neural networks17 ,18 is not well understood. The capacity of the layered APNN'S, on the other hand, is roughly equal to the number of input and hidden neurons19 • (c) The speed of backward error propagation learning 17-18 can be painfully slow. Layered APNN's, on the other hand, can be trained on only one pass through the training data 2 • If the network memory does not saturate, new data can easily be learned without repeating previous data. Neither is the effectiveness of recall of previous data diminished. Unlike layered back propagation neural networks, the APNN recalls by iteration. Under certain important applications, however, the APNN will recall in one iteration. (d) The manner in which layered APNN's generalizes to data for which it was not trained can be analyzed straightforwardly. The outline of this paper is as follows. After establishing the dynamics of the APNN in the next section, sufficient criteria for proper convergence are given. The convergence dynamics of the APNN are explored. Wise use of nonlinearities, e.g. the sigmoidal type nonlinearities 2 , improve the network's performance. Establishing a hidden layer of neurons whose states are a nonlinear function of the input neurons' states is shown to increase the network's capacity and the network's convergence rate as well. The manner in which the networks respond to data outside of the training set is also addressed. 2. THE ALTERNATING PROJECTION NEURAL NETWORK In this section, we Nonlinear modificiations established the to the network performance attributes are considered later. notation for the APNN. made to impose certain Consider a set of N continuous level linearly independent library vectors (or patterns) of length L> N: {£n I OSnSN}. We form the library matrix !:. = [£1 1£2 I ... I£N ] and the neural network interconnect matrixa T = F (!:.T !:. )-1 FT where the superscript T denotes transposition. We divide the L neurons into two sets: one in which the states are known and the remainder in which the states are unknown. This partition may change from application to application. Let Sk (M) be the state of the kth node at time M. If the kth node falls into the known catego~, its state is clamped to the known value (i.e. Sk (M) = Ik where I is some library vector). The states of the remaining floating neurons are equal to the sum of the inputs into the node. That is, Sk (M) = i k , where L i k = r tp k sp (1) p = 1 a The interconnect matrix is better trained iteratively2. To include a new library vector £, the interconnects are updated as ~T ~T~ ~ ~ ! + (EE ) / (E E) where E = (.!. - !) f. 536 If all neurons change state simultaneously (i.e. sp = sp (M-l) ), then the net is said to operate synchronously. If only one neuron changes state at a time, the network is operating asynchronously. Let P be the number of clamped neurons. We have provenl that the neural states converge strongly to the extrapolated library vector if the first P rows of ! (denoted KP) form a matrix of full column rank. That is, no column of ~ can be expressed as a linear combination of those remainin.,v. 2 By strong convergenceb , we mean lim II 1 (M) - t II == 0 where II x II == iTi. M~OO Lastly, note that subsumed in the criterion that ~ be full rank is the condition that the number of library vectors not exceed the number of known neural states (P ~ N). Techniques to bypass this restriction by using hidden neurons are discussed in section 5. Partition Notation: that neurons 1 through floating. We adopt the Without loss of generality, we will assume P are clamped and the remaining neurons are vectOr partitioning notation 7 IIp] 1 = ~ io where Ip is the P-tuple of the first P elements of 1. and 10 is a vector of the remaining Q = L-P. We can thus write, for example, ~ [ f~ If~ I ... If: ]. Using this partition notation, we can define the neural clamping operator by: 7 _ IL] !l ~ 7 10 Thus, the first P elements of I are clamped to l P • The remaining Q nodes "float". Partition notation useful. Define for the interconnect matrix will also prove T r!2 I !lJ L~ where ~2 is a P by P and !4 a Q by Q matrix. 3. STEADY STATE CONVERGENCE PROOFS For purposes of later reference, we address convergence of the network for synchronous operation. Asynchronous operation is addressed in reference 2. For proper convergence, both cases require that ~ be full rank. For synchronous operation, the network iteration in (1) followed by clamping can be written as: ~ ~ s(M+l) =!l ~ sCM) (2) As is illustrated in l - 3, this operation can easily be visualized in an L dimensional signal space. b The referenced convergence proofs prove strong convergence in an infinite dimensional Hilbert space. In a discrete finite dimensional space, both strong and weak convergence imply uniform convergencel9 • 2D , i.e. 1(M)~t as M~oo. 537 For a given partition with P clamped neurons, (2) can be written in partitioned form as [ ;'(M+J l*J[ I' J !l (3) !3!4 ~o (M) The states of the P clamped neurons are not affected by their input sum. Thus, there is no contribution to the iteration by ~1 and ~2. We can equivalently write (3) as -+0 -;tp-+o s (M+ 1) = !3 f +!4 s (M) (4 ) We show in that if fp is full rank, then the spectral radius (magnitude of the maximum eigenvalue) of ~4 is strictly less than one19 • It follows that the steady state solution of (4) is: (5 ) where, since fp is full rank, we have made use of our claim that -+0 -;to S (00) = f (6) 4. CONVERGENCE DYNAMICS In this section, we explore different convergence dynamics of the APNN when fp is full column rank. If the library matrix displays certain orthogonality characteristics, or if there is a single output (floating) neuron, convergence can be achieved in a single iteration. More generally, convergence is at an exponential rate. Two techniques are presented to improve convergence. The first is standard relaxation. Use of nonlinear convex constraint at each neuron is discussed elsewhere2 ,19. One Step Convergence: There are at least two important cases where the APNN converges other than uniformly in one iteration. Both require that the output be bipolar (±1). Convergence is in one step in the sense that -;to • -+0 f = Slgn s (1) (7) where the vector operation sign takes the sign of each element of the vector on which it operates. CASE 1: If there is a single output neuron, then, from (4), (5) and (6), sO (1) (1 t LL ) ,0 . Since the eigenvalue of the (scalar) matrix, !4 = tL L lies between zero and one 1 9, we conclude that 1t LL > O. Thus, if ,0 is restricted to ±1, (7) follows immediately. A technique to extend this result to an arbitrary number of output neurons in a layered network is discussed in section 7. CASE 2: For certain library matrices, the APNN can also display one step convergence. We showed that if the columns of K are orthogonal and the columns of fp are also orthogonal, then one synchronous iteration results in floating states proportional to the steady 538 state values19 • Specifically, for the floating neurons, tP 2 ~o (1) II II 10 111112 (8) An important special case of (8) is when the elements of Fare all ±1 and orthogonal. If each element were chosen by a 50-50 coin flip, for example, we would expect (in the statistical sense) that this would be the case. Exponential Convergence: More generally, the convergence rate of the APNN is exponential and is a function of the eigenstructure of .!4. Let {~r I 1 ~ r ~ Q } denote the eigenvectors of .!4 and {Ar } the corresponding eigenvalues. Define ~ = [ ~l 1~2 I ... I~o] and the diagonal matrix A4 such that diag ~ = [AI A2 ... Ao] T • Then we can . A T • -+ T-+ -. T • 1 f Wrl.te :!.4.=~ _4 ~. Defl.ne x (M) =~ s (M). S.;nce ~ ~ = I, \t...,. fol ows T ro~ the--+differe-ace equatJ-on i~ ('Up that x(M+l)=~:!.4 ~ ~ sCM) + ~ .!31 =~4 x (M) + g where g = ~.!3 t. The solution to this difference equation is M 't' "r [ 1 _ "kM + 1 ] ,,- 1 1J /\ok g k = /\0 ( 1 /\ok) g k (9) r = 0 Since the spectral radius of !4 is less than one19 , ~: ~ 0 as M ~ ~. Our steady state result is thus xk (~) = (1 Ak ) gk. Equation . ["M+l] (9) can therefore be wrl.tten as xk (M) = 1 /\ok xk (~). The eCflivalent of a "time constant" in this exponential convergence is 1/ tn (111 Ak I). The speed of convergence is thus dictated by the spectral radius of .!4. As we have shown19 later, adding neurons in a hidden layer in an APNN can significiantly reduce this spectral radius and thus improve the convergence rate. Relaxation: Both the projection and clamping operations can be relaxed to alter the network's convergence without affecting its steady state20 - 21 • For the interconnects, we choose an appropriate value of the relaxation parameter a in the interval (0,2) and 9 redefine the interconnect matrix as T aT + (1 a)I or equivalently, = {a(tnn -l)+1 a tnrn ; n =m TO see the effect of such relaxation on convergence, we need simply exam\ne the resulting ::dgenvalues. If .!4 has eigenvalues {Ar I, then .!4 has eigenvalues Ar = 1 + a (Ar - 1). A Wl.se choice of a reduces the spectral radius of .!~ with respect to that of .!4' and thus decreases the time constant of the network's convergence. Any of the operators projecting onto convex sets can be relaxed without affecting steady state convergence19 - 20 • These include the ~ operator2 and the sigmoid-type neural operator that projects onto a box. Choice of stationary relaxation parameters without numerical andlor empirical study of each specific case, however, generally remains more of an art than a science. 539 5. LAYERED APNN' S The networks thus far considered are homogeneous in the sense that any neuron can be clamped or floating. If the partition is such that the same set of neurons always provides the network stimulus and the remainder respond, then the networks can be simplified. Clamped neurons, for example, ignore the states of the other neurons. The corresponding interconnects can then be deleted from the neural network architecture. When the neurons are so partitioned, we will refer the APNN as layered. In this section, we explore various aspects of the layered APNN and in particular, the use of a so called hidden layer of neurons to increase the storage capacity of the network. An alternate architecture for a homogeneous APNN that require only Q neurons has been reported by Marks 2 • Hidden Layers: In its generic form, the APNN cannot perform a simple exclusive or (XOR). Indeed, failure to perform this same operation was a nail in the coffin of the perceptron22 . Rumelhart et. al.1 7 -18 revived the percept ron by adding additional layers of neurons. Although doing so allowed nonlinear discrimination, the iterative training of such networks can be painfully slow. With the addition of a hidden layer, the APNN likewise generalizes. In contrast, the APNN can be trained by looking at each data vector only once1 • Although neural networks will not likely be used for performing XOR's, their use in explaining the role of hidden neurons is quite instructive. The library matrix for the XOR is f- [~ ~ ~ ~ 1 The first two rOwS of F do not form a matrix of full column rank. Our approach is to augment fp with two more rows such that the resulting matrix is full rank. Most any nonlinear combination of the first two rowS will in general increase the matrix rank. Such a procedure, for example, is used in ~-classifiers23 . possible nonlinear operations include multiplication, a logical "AND" and running a weighted sum of the clamped neural states through a memoryless nonlinearity such as a sigmoid. This latter alteration is particularly well suited to neural architectures. To illustrate with the exclusive or (XOR) , a new hidden neural state is set equal to the exponentiation of the sum of the first two rows. A second hidden neurons will be assigned a value equal to the cosine of the sum of the first two neural states multiplied by Tt/2. (The choice of nonlinearities here is arbitrary. ) The augmented library matrix is 0 0 1 1 0 1 0 1 !:.+ 1 e e e 2 1 0 0 -1 0 1 1 0 540 In either the training or look-up mode, the states of the hidden neurons are clamped indirectly as a result of clamping the input neurons. The playback architecture for this network is shown in Fig .1. The interconnect values for the dashed lines are unity. The remaining interconnects are from the projection matrix formed from !+. Geometrical Interpretation In lower dimensions, the effects of hidden neurons can be nicely illustrated geometrically. Consider the library matrix F = Clearly IP = (1/2 1) . Let the determined by the nonlineariy x 2 the first row of f. Then !+ = [ t: I t; ] 1 1/2 ] neurons where x [ 1/2 = 1i4 in the hidden layer be denotes the elements in 1;2 J The corresponding geometry is shown in Fig. 2 for x the input neuron, y the output and h the hidden neuron. The augmented library vectors are shown and a portion of the generated subspace is shown lightly shaded. The surface of h = x 2 resembles a cylindrical lens in three dimensions. Note that the linear variety corresponding to f = 1/2 intersects the cylindrical lens and subspace only at 1+. Similarly, the x = 1 plane intersects the lens and subspace at 12 • Thus, in both cases, clamping the input corresponding to the first element of one of the two library vectors uniquely determines the library vector. Convergence Improvement: Use of additional neurons in the hidden layer will improve the convergence rate of the APNN19 • Specifically, the spectral radius of the .!4 matrix is decreased as additional neurons are added. The dominant time constant controlling convergence is thus decreased. Capacity: Under the assumption that nonlinearities are chosen such that the augmented fp matrix is of full rank, the number of vectors which can be stored in the layered APNN is equal to the sum of the number of neurons in the input and hidden layers. Note, then, that interconnects between the input and output neurons are not needed if there are a sufficiently large number of neurons in the hidden layer. 6. GENERALIZATION We are assured that the APNN will converge to the desired result if a portion of a training vector is used to stimulate the network. What, however, will be the response if an initialization is used that is not in the training set or, in other words, how does the network generalize from the training set? To illustrate generalization, we return to the XOR problem. Let S5 (M) denote the state of the output neuron at the Mth (synchronous) y / , "" " / X / , , "/ "541 loyer: input 3 exp hidden Figure 1. Illustration of a layered APNN fori performing an XOR. l( Figure 2. A geometrical illustration of the use of an x 2 nonlinearity to determine the states of hidden neurons. Figure 3. Response of the elementary XOR APNN using an exponential and trignometric nonlinearity in the hidden layer. Note that, at the corners, the function is equal to the XOR of the Figure 4. The generalization of the XOR networks formed by thresholding the function in Fig . 3 at 3/4. Different hidden layer nonlinearities result in different generalizations. 542 iteration. If S1 and S2 denote the input clamped value, then S5 (m+1) =t1 5 Sl + t 25 S2 + t35 S3 + t4 5 S4 + t5 5 S5 (m) where S3 =exp (Sl +S2 ) and S4 =cos [1t (S1 + S2) /2] To reach steady state, we let m tend to infinity and solve for S5 (~) : 1 A plot of S5 (~) versus (S1,S2) is shown in Figure 3. The plot goes through 1 and zero according to the XOR of the corner coordinates. Thresholding Figure 3 at 3/4 results in the generalization perspective plot shown in Figure 4. To analyze the network's generalization when there are more than one output neuron, we use (5) of which (10) is a special case. If conditions are such that there is one step convergence, then generalization plots of the type in Figure 4 can be computed from one network iteration using (7). 7. NOTES (a) There clearly exists a great amount of freedom in the choice of the nonlinearities in the hidden layer. Their effect on the network performance is currently not well understood. One can envision, however, choosing nonlinearities to enhance some network attribute such as interconnect reduction, classification region shaping (generalization) or convergence acceleration. (b) There is a possibility that for a given set of hidden neuron nonlinearities, augmentation of the fp matrix coincidentally will result in a matrix of deficent column rank, proper convergence is then not assured. It may also result in a poorly conditioned matrix, convergence will then be quite slow. A practical solution to these problems is to pad the hidden layer with additional neurons. As we have noted, this will improve the convergence rate. (c) We have shown in section 4 that if an APNN has a single bipolar output neuron, the network converges in one step in the sense of (7). Visualize a layered APNN with a single output neuron. If there are a sufficiently large number of neurons in the hidden layer, then the input layer does not need to be connected to the output layer. Consider a second neural network identical to the first in the input and hidden layers except the hidden to output interconnects are different. Since the two networks are different only in the output interconnects, the two networks can be combined into a singlee network with two output neurons. The interconnects from the hidden layer to the output neurons are identical to those used in the single output neurons architectures. The new network will also converge in one step. This process can clearly be extended to an arbitrary number of output neurons. REFERENCES 1. R.J. Marks II, "A Class of Continuous Level Associative Memory Neural Nets," ~. Opt., vo1.26, no.10, p.200S, 1987. 543 2. K.F. Cheung et. al., "Neural Net Associative Memories Based on Convex Set Projections," Proc. IEEE 1st International Conf. on Neural Networks, San Diego, 1987. 3. R.J. Marks II et. al., "A Class of Continuous Level Neural Nets," Proc. 14th Congress of International Commission for Optics Conf., Quebec, Canada, 1987. 4. J.J. Hopfield, "Neural Networks and Physical Systems with Emergent Collective Computational Abilities," Proceedings Nat. Acad. of Sciences, USA, vol.79, p.2554, 1982. 5. J.J. Hopfield et. al., "Neural Computation of Decisions in Optimization Problem," BioI. Cyber., vol. 52, p.141, 1985. 6. ·D. W. Tank et. al., "Simple Neurel Optimization Networks: an AID Converter, Signal Decision Circuit and a Linear Programming Circuit," IEEE Trans. Cir. ~., vol. CAS-33, p.533, 1986. 7. M. Takeda et. ai, "Neural Networks for Computation: Number Representation and Programming Complexity," ~. Opt., vol. 25, no. 18, p.3033, 1986. 8. S. Geman et. al., "Stochastic Relaxation, Gibb's Distributions, and the Bayesian Restoration of Images," IEEE Trans. Pattern Recog. & Machine Intelligence., vol. PAMI-6, p.721, 1984. 9. S. Kirkpatrick et. al. ,"Optimization by Simulated Annealing," Science, vol. 220, no. 4598, p.671, 1983. 10. D.H. Ackley et. al., "A Learning Algorithm for Boltzmann Machines," Cognitive Science, vol. 9, p.147, 1985. 11. K.F. Cheung et. al., "Synchronous vs. Asynchronous Behaviour of Hopfield's CAM Neural Net," to appear in Applied Optics. 12. R.P. Lippmann, "An Introduction to Computing With Neural nets," IEEE ASSP Magazine, p.7, Apr 1987. 13. N. Farhat et. al .. , "Optical Implementation of the Hopfield Model," ~. Opt., vol. 24, pp.1469, 1985. 14. L.E. Atlas, "Auditory Coding in Higher Centers of the CNS," IEEE Eng. in Medicine and Biology Magazine, p.29, Jun 1987. 15. Y.S. Abu-Mostafa et. al., "Information Capacity of the Hopfield Model, " IEEE Trans. Inf. Theory, vol. IT-31, p.461, 1985. 16. R.J. McEliece et. al.,"The Capacity of the Hopfield Associative Memory, " IEEE Trans. Inf. Theory (submitted), 1986. 17. D.E. Rumelhart et. al., Parallel Distributed Prooessing, vol. I & II, Bradford Books, Cambridge, MA, 1986. 18. D.E. Rumelhart et. al., "Learning Representations by Back-Propagation Errors," Nature. vol. 323, no. 6088, p.533, 1986. 19. R.J. Marks II et. al.,"Alternating Projection Neural Networks," ISDL report *11587, Nov. 1987 (Submitted for publication) . 20. D.C. Youla et. al, "Image Restoration by the Method of Convex Projections: Part I-Theory," IEEE Trans. Med. Imaging, vol. MI-1, p.81, 1982. 21. M. I. Sezan and H. Stark. "Image Restoration by the Method of Convex Projections: Part II-Applications and Numerical Results," IEEE Trans. Med. Imaging, vol. MI-1, p.95, 1985. 22. M. Minsky et. al., Perceptrons, MIT Press, Cambridge, MA, 1969. 23. J. Sklansky et. al., Pattern Classifiers and Trainable Machines, Springer-Verlag, New York, 1981.
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144 SPEECH RECOGNITION EXPERIMENTS WITH PERCEPTRONS D. J. Burr Bell Communications Research Morristown, NJ 07960 ABSTRACT Artificial neural networks (ANNs) are capable of accurate recognition of simple speech vocabularies such as isolated digits [1]. This paper looks at two more difficult vocabularies, the alphabetic E-set and a set of polysyllabic words. The E-set is difficult because it contains weak discriminants and polysyllables are difficult because of timing variation. Polysyllabic word recognition is aided by a time pre-alignment technique based on dynamic programming and E-set recognition is improved by focusing attention. Recognition accuracies are better than 98% for both vocabularies when implemented with a single layer perceptron. INTRODUCTION Artificial neural networks perform well on simple pattern recognition tasks. On speaker trained spoken digits a layered network performs as accurately as a conventional nearest neighbor classifier trained on the same tokens [1]. Spoken digits are easy to recognize since they are for the most part monosyllabic and are distinguished by strong vowels. It is reasonable to ask whether artificial neural networks can also solve more difficult speech recognition problems. Polysyllabic recognition is difficult because multi-syllable words exhibit large timing variation. Another difficult vocabulary, the alphabetic E-set, consists of the words B, C, D, E, G, P, T, V, and Z. This vocabulary is hard since the distinguishing sounds are short in duration and low in energy. We show that a simple one-layer perceptron [7] can solve both problems very well if a good input representation is used and sufficient examples are given. We examine two spectral representations a smoothed FFT (fast Fourier transform) and an LPC (linear prediction coefficient) spectrum. A time stabilization technique is described which pre-aligns speech templates based on peaks in the energy contour. Finally, by focusing attention of the artificial neural network to the beginning of the word, recognition accuracy of the E-set can be consistently increased. A layered neural network, a relative of the earlier percept ron [7], can be trained by a simple gradient descent process [8]. Layered networks have been © American Institute of Physics 1988 145 applied successflJ.lly to speech recognition [1], handwriting recognition [2], and to speech synthesis [11]. A variation of a layered network [3] uses feedback to model causal constraints, which can be useful in learning speech and language. Hidden neurons within a layered network are the building blocks that are used to form solutions to specific problems. The number of hidden units required is related to the problem [1,2]. Though a single hidden layer can form any mapping [12], no more than two layers are needed for disjunctive normal form [4]. The second layer may be useful in providing more stable learning and representation in the presence of noise. Though neural nets have been shown to perform as well as conventional techniques[I,5], neural nets may do better when classes have outliers [5]. PERCEPTRONS A simple perceptron contains one input layer and one output layer of neurons directly connected to each other (no hidden neurons). This is often called a one-layer system, referring to the single layer of weights connecting input to output. Figure 1. shows a one-layer perceptron configured to sense speech patterns on a two-dimensional grid. The input consists of a 64-point spectrum at each of twenty time slices. Each of the 1280 inputs is connected to each of the output neurons, though only a sampling of connections are shown. There is one output neuron corresponding to each pattern class. Neurons have standard linear-weighted inputs with logistic activation. C(1) C(2) FR:<lBC'V .... 64 units C(N-1) C(N) Figure 1. A single layer perceptron sensing a time-frequency array of sample data. Each output neuron CU) (1 <i<N) corresponds to a pattern class and is full connected to the input array (for clarity only a few connections are shown). An input word is fit to the grid region by applying an automatic endpoint detection algorithm. The algorithm is a variation of one proposed by Rabiner and Sambur [9] which employs a double threshold successive approximation 146 method. Endpoints are determined by first detecting threshold crossings of energy and then of zero crossing rate. In practice a level crossing other than zero is used to prevent endpoints from being triggered by background sounds. INPUT REPRESENTATIONS Two different input representations were used in this study. The first is a Fourier representation smoothed in both time and frequency. Speech is sampled at 10 KHz ap.d Hamming windowed at a number of sample points. A 128-point FFT spectrum is computed to produce a template of 64 spectral samples at each of twenty time frames. The template is smoothed twice with a time window of length three and a frequency window of length eight. For comparison purposes an LPC spectrum is computed using a tenth order model on 300-sample Hamming windows. Analysis is performed using the autocorrelation method with Durbin recursion [6]. The resulting spectrum is smoothed over three time frames. Sample spectra for the utterance "neural-nets" is shown in Figure 2. Notice the relative smoothness of the LPC spectrum which directly models spectral peaks. FFT LPC Figure 2. FFT and LPC time-frequency plots for the utterance "neural nets". Time is toward the left, and frequency, toward the right. DYNAMIC TIME ALIGMv1ENT Conventional speech recognition systems often employ a time normalization technique based on dynamic programming [10]. It is used to warp the time scales of two utterances to obtain optimal alignment between their spectral frames. We employ a variation of dynamic programming which aligns energy contours rather than spectra. A reference energy template is chosen for each pattern class, and incoming patterns are warped onto it. Figure 3 shows five utterances of "neural-nets" both before and after time alignment. Notice the improved alignment of energy peaks. 147 § I § § § I >\b ~ ~ II III z W ~ ~ !I ! .. .. 10 10 . .. (a. ) TIME (b) Figure 3. (a) Superimposed energy plots of five different utterances of "neural nets". (b). Same utterances after dynamic time alignment. POLYSYLLABLE RECOGNITION Twenty polysyllabic words containing three to five syllables were chosen, and five tokens of each were recorded by a single male speaker. A variable number of tokens were used to train a simple perceptron to study the effect of training set size on performance. Two performance measures were used: classification accuracy, and an RMS error measure. Training tokens were permuted to obtain additional experimental data points. Figure 4. Output responses of a perceptron trained with one token per class (left) and four tokens per class (right). 148 Figure 4 shows two representative perspective plots of the output of a perceptron trained on one and four tokens respectively per class. Plots show network response (z-coordinate) as a function of output node (left axis) and test word index (right axis). Note that more training tokens produce a more ideal map - a map should have ones along the diagonal and zeroes everywhere else. Table 1 shows the results of these experiments for three different representations: (1) FFT, (2) LPC and (3) time aligned LPC. This table lists classification accuracy as a function of number of training tokens and input representation. The perceptron learned to classify the unseen patterns perfectly for all cases except the FFT with a single training pattern. Table 1. Polysyllabic Word Recognition AccuraclT Number Training Tokens 1 2 3 4 FFT 98.7% 100% 100% 100% LPC 100% 100% 100% 100% Time Aligned LPC 100% 100% 100% 100% Permuted Trials 400 300 200 100 A different performance measure, the RMS error, evaluates the degree to which the trained network output responses Rjk approximate the ideal targets Tjk • The measure "is evaluated over the N non-trained tokens and M output nodes of the network. Tik equals 1 for J=k and 0 for J=I=k. Figure 5 shows plots of RMS error as a function of input representation and training patterns. Note that the FFT representation produced the highest error, LPC was about 40% less, and time-aligned LPC only marginally better than non-aligned LPC. In a situation where many choices must be made (i.e. vocabularies much larger than 20 words) LPC is the preferred choice, and time alignment could be useful to disambiguate similar words. Increased number of training tokens results in improved performance in all cases. 149 o ci ,-----------------------------~ '" 0 .. FFT i "! lii I0 5 0 g W tJl ~ LPC a: '" 0 TIme Aligned LPC 0 o o ~--~----~--~----~ __ ~ ____ ~ 1.0 2.0 3.0 4.0 Number Traln'ng Tokens Figure 5. RMS error versus number of training tokens for various input representations. E-SET VOCABULARY The E-Set vocabulary consists of the nine E-words of the English alphabet B, C, D, E, G, P, T, V, Z. Twenty tokens of each of the nine classes were recorded by a single male speaker. To maximize the sizes of training and test sets, half were used for training and the other half for testing. Ten permutations produced a total of 900 separate recognition trials. Figure 6 shows typical LPC templates for the nine classes. Notice the double formant ridge due to the ''E'' sound, which is common to all tokens. Another characteristic feature is the FO ridge - the upward fold on the left of all tokens which characterizes voicing or pitched sound. 150 Figure 6. LP C time-frequency plots for representative tokens of the E-set words. Figure 7. Time-frequency plots of weight values connected to each output neuron ''E'' through "z" in a trained perceptron. 151 Figure 7 shows similar plots illustrating the weights learned by the network when trained on ten tokens of each class. These are plotted like spectra, since one weight is associated with each spectral sample. Note that the patterns have some formant structure. A recognition accuracy of 91.4% included perfect scores for classes C, E, and G. Notice that weights along the FO contour are mostly small and some are slightly negative. This is a response to the voiced ''E" sound common to all classes. The network has learned to discount "voicing" as a discriminator for this vocabulary. Notice also the strong "hilly" terrain near the beginning of most templates. This shows where the network has decided to focus much of its discriminating power. Note in particular the hill-valley pair at the beginning of ''p'' and "T". These are near to formants F2/F3 and could conceivably be formant onset detectors. Note the complicated detector pattern for the ''V'' sound. The classes that are easy to discriminate (C, E, G) produce relatively fiat and uninter~sting weight spaces. A highly convoluted weight space must therefore be correlated with difficulty in discrimination. It makes little sense however that the network should be working hard in the late time C'E" sound) portion of the utterance. Perhaps additional training might reduce this activity, since the network would eventually find little consistent difference there. A second experiment was conducted to help the network to focus attention. The first k frames of each input token were averaged to produce an average spectrum. These average spectra were then used in a simple nearest neighbor recognizer scheme. Recognition accuracy was measured as a function of k. The highest performance was for k=8, indicating that the first 40% of the word contained most of the "action". B C D E C P T V Z B 08 0 0 0 0 0 0 c 0 100 0 0 0 0 0 0 0 D 0 0 08 0 0 2 0 0 0 E 0 0 0 100 0 0 0 0 0 c 0 0 0 0 100 0 0 0 0 p 0 0 3 0 0 03 4 0 0 T 0 0 0 0 0 0 100 0 0 V 2 0 0 0 0 2 0 08 0 Z 0 0 0 0 0 0 0 09 Figure 8. Confusion matrix of the E-set focused on the first 40% of each word. 152 All words were resampled to concentrate 20 time frames into the first 40% of the word. LPC spectra were recomputed using a 16th order model and the network was trained on the new templates. Performance increased from 91.4% to 98.2%. There were only 16 classification errors out of the 900 recognition tests. The confusion matrix is shown in Figure 8. Learning times for all experiments consisted of about ten passes through the training set. When weights were primed with average spectral values rather than random values, learning time decreased slightly. CONCLUSIONS Artificial neural networks are capable of high performance in pattern recognition applications, matching or exceeding that of conventional classifiers. We have shown that for difficult speech problems such as time alignment and weak discriminability, artificial neural networks perform at high accuracy exceeding 98%. One-layer perceptrons learn these difficult tasks almost effortlessly - not in spite of their simplicity, but because of it. REFERENCES 1. D. J. Burr, "A Neural Network Digit Recognizer", Proceedings of IEEE Conference on Systems, Man, and Cybernetics, Atlanta, GA, October, 1986, pp. 1621-1625. 2. D. J. Burr, "Experiments with a Connectionist Text Reader," IEEE International Conference on Neural Networks, San Diego, CA, June, 1987. 3. M. I. Jordan, "Serial Order: A Parallel Distributed Processing Approach," ICS Report 8604, UCSD Institute for Cognitive Science, La Jolla, CA, May 1986. 4. S. J. Hanson, and D. J. Burr, 'What Connectionist Models Learn: Toward a Theory of Representation in Multi-Layered Neural Networ.ks," submitted for pu blication. 5. W. Y. Huang and R. P. Lippmann, "Comparisons Between Neural Net and Conventional Classifiers," IEEE International Conference on Neural Networks, San Diego, CA, June 21-23, 1987. 6. J. D. Markel and A. H. Gray, Jr., Linear Prediction of Speech, SpringerVerlag, New York, 1976. 7. M. L. Minsky and S. Papert, Perceptrons, MIT Press, Cambridge, Mass., 1969. 153 8. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, ''Learning Internal Representations by Error Propagation," in Parallel Distributed Processing, Vol. 1, D. E. Rumelhart and J. L. McClelland, eds., MIT Press, 1986, pp. 318362. 9. L. R. Rabiner and M. R. Sambur, "An Algorithm for Determining the Endpoints of Isolated Utterances," BSTJ, Vol. 54,297-315, Feb. 1975. 10. H. Sakoe and S. Chiba, "Dynamic Programming Optimization for Spoken Word Recognition," IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-26, No.1, 43-49, Feb. 1978. 11. T. J. Sejnowski and C. R. Rosenberg, "NETtalk: A Parallel Network that Learns to Read Aloud," Technical Report JHU/EECS-86/01, Johns Hopkins University Electrical Engineering and Computer Science, 1986. 12. A. Wieland and R. Leighton, "Geometric Analysis of Neural Network Capabilities," IEEE International Conference on Neural Networks, San Deigo, CA, June 21-24, 1987.
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ON PROPERTIES OF NETWORKS OF NEURON-LIKE ELEMENTS Pierre Baldi· and Santosh S. Venkatesht 15 December 1987 Abstract The complexity and computational capacity of multi-layered, feedforward neural networks is examined. Neural networks for special purpose (structured) functions are examined from the perspective of circuit complexity. Known results in complexity theory are applied to the special instance of neural network circuits, and in particular, classes of functions that can be implemented in shallow circuits characterised. Some conclusions are also drawn about learning complexity, and some open problems raised. The dual problem of determining the computational capacity of a class of multi-layered networks with dynamics regulated by an algebraic Hamiltonian is considered. Formal results are presented on the storage capacities of programmed higher-order structures, and a tradeoff between ease of programming and capacity is shown. A precise determination is made of the static fixed point structure of random higher-order constructs, and phase-transitions (0-1 laws) are shown. 1 INTRODUCTION In this article we consider two aspects of computation with neural networks. Firstly we consider the problem of the complexity of the network required to compute classes of specified (structured) functions. We give a brief overview of basic known complexity theorems for readers familiar with neural network models but less familiar with circuit complexity theories. We argue that there is considerable computational and physiological justification for the thesis that shallow circuits (Le., networks with relatively few layers) are computationally more efficient. We hence concentrate on structured (as opposed to random) problems that can be computed in shallow (constant depth) circuits with a relatively few number (polynomial) of elements, and demonstrate classes of structured problems that are amenable to such low cost solutions. We discuss an allied problem-the complexity of learning-and close with some open problems and a discussion of the observed limitations of the theoretical approach. We next turn to a rigourous classification of how much a network of given structure can do; i.e., the computational capacity of a given construct. (This is, in ·Department of Mathematics, University of California (San Diego), La Jolla, CA 92093 tMoore School of Electrical Engineering, University of Pennsylvania, Philadelphia, PA 19104 © American Institute of Physics 1988 41 42 a sense, the mirror image of the problem considered above, where we were seeking to design a minimal structure to perform a given task.) In this article we restrict ourselves to the analysis of higher-order neural structures obtained from polynomial threshold rules. We demonstrate that these higher-order networks are a special class of layered neural network, and present formal results on storage capacities for these constructs. Specifically, for the case of programmed interactions we demonstrate that the storage capacity is of the order of n d where d is the interaction order. For the case of random interactions, a type of phase transition is observed in the distribution of fixed points as a function of attraction depth. 2 COMPLEXITY There exist two broad classes of constraints on compl,ltations. 1. Physical constraints: These are related to the hardware in which the computation is embedded, and include among others time constants, energy limitations, volumes and geometrical relations in 3D space, and bandwidth capacities. 2. Logical constraints: These can be further subdivided into • Computability constraints-for instance, there exist unsolvable problems, i.e., functions such as the halting problem which are not computable in an absolute sense . • Complexity constraints-usually giving upper and/or lower bounds on the amount of resources such as the time, or the number of gates required to compute a given function. As an instance, the assertion "There exists an exponential time algorithm for the Traveling Salesman Problem," provides a computational upper bound. If we view brains as computational devices, it is not unreasonable to think that in the course of the evolutionary process, nature may have been faced several times by problems related to physical and perhaps to a minor degree logical constraints on computations. If this is the case, then complexity theory in a broad sense could contribute in the future to our understanding of parallel computations and architectural issues both in natural and synthetic neural systems. A simple theory of parallel processing at the macro level (where the elements are processors) can be developed based on the ratio of the time spent on communications between processors [7] for different classes of problems and different processor architecture and interconnections. However, this approach does not seem to work for parallel processing at the level of circuits, especially if calculations and communications are intricately entangled. Recent neural or connectionist models are based on a common structure, that of highly interconnected networks of linear (or polynomial) threshold (or with sigmoid input-output function) units with adjustable interconnection weights. We shall therefore review the complexity theory of such circuits. In doing so, it will be sometimes helpful to contrast it with the similar theory based on Boolean (AND, OR, NOT) gates. The presentation will be rather informal and technical complements can easily be found in the references. Consider a circuit as being on a cyclic oriented graph connecting n Boolean inputs to one Boolean output. The nodes of the graph correspond to the gates (the n input units, the "hidden" units, and the output unit) of the circuit. The size of the circuit is the total number of gates and the depth is the length of the longest path connecting one input to the output. For a layered, feed-forward circuit, the width is the average number of computational units in the hidden (or interior) layers of elements. The first obvious thing when comparing Boolean and threshold logic is that they are equivalent in the sense that any Boolean function can be implemented using either logic. In fact, any such function can be computed in a circuit of depth two and exponential size. Simple counting arguments show that the fraction of functions requiring a circuit of exponential size approaches one as n -+ 00 in both cases, i.e., a random function will in general require an exponential size circuit. (Paradoxically, it is very difficult to construct a family of functions for which we can prove that an exponential circuit is necessary.) Yet, threshold logic is more powerful than Boolean logic. A Boolean gate can compute only one function whereas a threshold gate can compute to the order of 2on2 functions by varying the weights with 1/2 ~ a ~ 1 (see [19] for the lower bound; the upper bound is a classical hyperplane counting argument, see for instance [20,30)). It would hence appear plausible that there exist wide classes of problems which can be computed by threshold logic with circuits substantially smaller than those required by Boolean logic. An important result which separates threshold and Boolean logic from this point of view has been demonstrated by Yao [31] (see [10,24] for an elegant proof). The result is that in order to compute a function such as parity in a circuit of constant depth k, at least exp(cnl/2k) Boolean gates with unbounded fanin are required. As we shall demonstrate shortly, a circuit of depth two and linear size is sufficient for the computation of such functions using threshold logic. It is not unusual to hear discussions about the tradeoffs between the depth and the width of a circuit. We believe that one of the main constributions of complexity analysis is to show that this tradeoff is in some sense minimal and that in fact there exists a very strong bias in favor of shallow (Le., constant depth) circuits. There are multiple reasons for this. In general, for a fixed size, the number of different functions computable by a circuit of small depth exceeds the number of those computable by a deeper circuit. That is, if one had no a priori knowledge regarding the function to be computed and was given hidden units, then the optimal strategy would be to choose a circuit of depth two with the m units in a single layer. In addition, if we view computations as propagating in a feedforward mode from the inputs to the output unit, then shallow circuits compute faster. And the deeper a circuit, the more difficult become the issues of time delays, synchronisation, and precision on the computations. Finally, it should be noticed that given overall responses of a few hundred milliseconds and given the known time scales for synaptic integration, biological circuitry must be shallow, at least within a "module" and this is corroborated by anatomical data. The relative slowness of neurons and their shallow circuit architecture are to be taken together with the "analog factor" and "entropy factor" [1] to understand the necessary high-connectivity requirements of neural systems. 43 44 From the previous analysis emerges an important class of circuits in threshold logic characterised by polynomial size and shallow depth. We have seen that, in general, a random function cannot be computed by such circuits. However, many interesting functions-the structured problems--are far from random, and it is then natural to ask what is the class of functions computable by such circuits? While a complete characterisation is probably difficult, there are several sub-classes of structural functions which are known to be computable in shallow poly-size circuits. The symmetric functions, i.e., functions which are invariant under any permutation of the n input variables, are an important class of structured problems that can be implemented in shallow polynomial size circuits. In fact, any symmetric function can be computed by a threshold circuit of depth two and linear size; (n hidden units and one output unit are always sufficient). We demonstrate the validity of this assertion by the following instructive construction. We consider n binary inputs, each taking on values -1 and 1 only, and threshold gates as units. Now array the 2n possible inputs in n + 1 rows with the elements in each row being permuted versions of each other (i.e., n-tuples in a row all have the same number of +1's) and with the rows going monotonically from zero +1's to n +l's. Any given symmetric Boolean function clearly assumes the same value for all elements (Boolean n-tuples) in a row, so that contiguous rows where the function assumes the value +1 form bands. (There are at most n/2 bands-the worst case occuring for the parity function.) The symmetric function can now be computed with 2B threshold gates in a single hidden layer with the topmost "neuron" being activated only if the number of +1's in the input exceeds the number of +1's in the lower edge of the lowest band, and proceeding systematically, the lowest "neuron" being activated only if the number of +1's in the input exceeds the number of +1's in the upper edge of the highest band. An input string will be within a band if and only if an odd number of hidden neurons are activated startbg contiguously from the top of the hidden layer, and conversely. Hence, a single output unit can compute the given symmetric function. It is easy to see that arithmetic operations on binary strings can be performed with polysize small depth circuits. Reif [23] has shown that for a fixed degree of precision, any analytic function such as polynomials, exponentials, and trigonometric functions can be approximated with small and shallow threshold circuits. Finally, in many situations one is interested in the value of a function only for a vanishingly small (Le., polynomial) fraction of the total number of possible inputs 2n. These functions can be implemented by polysize shallow circuits and one can relate the size and depths of the circuit to the cardinal of the interesting inputs. So far we only have been concerned with the complexity of threshold circuits. We now turn to the complexity of learning, i.e., the problem of finding the weights required to implement a given function. Consider the problem of repeating m points in 1R l coloured in two colours, using k hyperplanes so that any region contains only monochromatic points. If i and k are fixed the problem can be solved in polynomial time. If either i or k goes to infinity, the problem becomes NP-complete [1]. As a result, it is not difficult to see that the general learning problem is NP-complete (see also [12] for a different proof and [21] for a proof of the fact it is already NP-complete in the case of one single threshold gate). Some remarks on the limitations of the complexity approach are a pro]XJs at this juncture: 1. While a variety of structured Boolean functions can be implemented at relatively low cost with networks of linear threshold gates (McCulloch-Pitts neurons), the extension to different input-output functions and the continuous domain is not always straightforward. 2. Even restricting ourselves to networks of relatively simple Boolean devices such as the linear threshold gate, in many instances, only relatively weak bounds are available for computational cost and complexity. 3. Time is probably the single most important ingredient which is completely absent from these threshold units and their interconnections [17,14]; there are, in addition, non-biological aspects of connectionist models [8]. 4. Finally, complexity results (where available) are often asymptotic in nature and may not be meaningful in the range corresponding to a particular application. We shall end this section with a few open questions and speculations. One problem has to do with the time it takes to learn. Learning is often seen as a very slow process both in artificial models (cf. back propagation, for instance) and biological systems (cf. human acquisition of complex skills). However, if we follow the standards of complexity theory, in order to be effective over a wide variety of scales, a single learning algorithm should be polynomial time. We can therefore ask what is learnable by examples in polynomial time by polynomial size shallow threshold circuits? The status of back propagation type of algorithms with respect to this question is not very clear. The existence of many tasks which are easily executed by biological organisms and for which no satisfactory computer program has been found so far leads to the question of the specificity of learning algorithms, i.e., whether there exists a complexity class of problems or functions for which a "program" can be found only by learning from examples as opposed to by traditional programming. There is some circumstantial evidence against such conjecture. As pointed out by Valiant [25], cryptography can be seen in some sense as the opposite of learning. The conjectures existence of one way function, i.e., functions which can be constructed in polynomial time but cannot be invested (from examples) in polynomial time suggests that learning algorithms may have strict limitations. In addition, for most of the artificial applications seen so far, the programs obtained through learning do not outperform the best already known software, though there may be many other reasons for that. However, even if such a complexity class does not exist, learning algorithm may still be very important because of their inexpensiveness and generality. The work of Valiant [26,13] on polynomial time learning of Boolean formulas in his "distribution free model" explores some additional limitations of what can be learned by examples without including any additional knowledge. Learning may therefore turn out to be a powerful, inexpensive but limited family of algorithms that need to be incorporated as "sub-routines" of more global 45 46 programs, the structure of which may be -harder to find. Should evolution be regarded as an "exponential" time learning process complemented by the "polynomial" time type of learning occurring in the lifetime of organisms? 3 CAPACITY In the previous section the focus of our investigation was on the structure and cost of minimal networks that would compute specified Boolean functions. We now consider the dual question: What is the computational capacity of a threshold network of given structure? As with the issues on complexity, it turns out that for fairly general networks, the capacity results favour shallow (but perhaps broad) circuits [29]. In this discourse, however, we shall restrict ourselves to a specified class of higher-order networks, and to problems of associative memory. We will just quote the principal rigourous results here, and present the involved proofs elsewhere [4]. We consider systems of n densely interacting threshold units each of which yields an instantaneous state -1 or +1. (This corresponds in the literature to a system of n Ising spins, or alternatively, a system of n neural states.) The state space is hence the set of vertices of the hypercube. We will in this discussion also restrict our attention throughout to symmetric interaction systems wherein the interconnections between threshold elements is bidirectional. Let Id be the family of all subsets of cardinality d + 1 of the set {1, 2, ... , n}. n Clearly IIdl = ( d + 1)· For any subset I of {1, 2, ... , n}, and for every state deC U = {Ul,U2, ... ,un }E lBn = {-1,l}n, set UI = fIiEIui. Definition 1 A homogeneous algebraic threshold network of degree d is a network of n threshold elements with interactions specified by a set of ( d: 1 ) real coefficients WI indexed by I in I d, and the evolution rule ut = sgn ( L WIUI\{i}) IeId :ieI (1) These systems can be readily seen to be natural generalisations to higherorder of the familiar case d = 1 of linear threshold networks. The added degrees of freedom in the interaction coefficients can potentially result in enhanced flexibility and programming capability over the linear case as has been noted independently by several authors recently [2,3,4,5,22,27]. Note that each d-wise product uI\i is just the parity of the corresponding d inputs, and by our earlier discussion, this can be computed with d hidden units in one layer followed by a single threshold unit. Thus the higher-order network can be realised by a network of depth three, where the first hidden layer has d( ~ ) units, the second hidden layer has ( ~ ) units, and there are n output units which feedback into the n input units. Note that the weights from the input to the first hidden layer, and the first hidden layer to the second are fixed (computing the various d-wise products), and the weights from the second hidden layer to the output are the coefficients WI which are free parameters. These systems can be identified either with long range interactions for higherorder spin glasses at zero temperature, or higher-order neural networks. Starting from an arbitrary configuration or state, the system evolves asynchronously by a sequence of single "spin" flips involving spins which are misaligned with the instantaneous "molecular field." The dynamics of these symmetric higher-order systems are regulated analogous to the linear system by higher-order extensions of the classical quadratic Hamiltonian. We define the homogeneous algebraic Hamiltonian of degree d by Hd(u) = - E WI'UI· IeId (2) The algebraic Hamiltonians are functionals akin in behaviour to the classical quadratic Hamiltonian as has been previously demonstrated [5]. Proposition 1 The functional H d is non-increasing under the evolution rule 1. In the terminology of spin glasses, the state trajectories of these higher-order networks can be seen to be following essentially a zero-temperature Monte Carlo (or Glauber) dynamics. Because of the monotonicity of the algebraic Hamiltonians given by equation 2 under the asynchronous evolution rule 1, the system always reaches a stable state (fixed point) where the relation 1 is satisfied for each of the n spins or neural states. The fixed points are hence the arbiters of system dynamics, and determine the computational capacity of the system. System behaviour and applications are somewhat different depending on whether the interactions are random or programmed. The case of random interactions lends itself to natural extensions of spin glass formulations, while programmed interactions yield applications of higher-order extensions of neural network models. We consider the two cases in turn. 3.1 PROGRAMMED INTERACTIONS Here we query whether given sets of binary n-vectors can be stored as fixed points by a suitable selection of interaction coefficients. If such sets of prescribed vectors can be stored as stable states for some suitable choice of interaction coefficients, then proposition 1 will ensure that the chosen vectors are at the bottom of "energy wells" in the state space with each vector exercising a region of attraction around it-all characterestics of a physical associative memory. In such a situation the dynamical evolution of the network can be interpreted in terms of computations: error-correction, nearest neighbour search and associative memory. Of importance here is the maximum number of states that can be stored as fixed points for an appropriate choice of algebraic threshold network. This represents the maximal information storage capacity of such higher-order neural networks. Let d represent the degree ofthe algebraic threshold network. Let u(l), ... , u(m) be the m-set of vectors which we require to store as fixed points in a suitable algebraic threshold network. We will henceforth refer to these prescribed vectors as 47 48 memories. We define the storage capacity of an algebraic threshold network of degree d to be the maximal number m of arbitrarily chosen memories which can be stored with high probability for appropriate choices of coefficients in the network. Theorem 1 The maximal (algorithm independent) storage capacity of a homogeneous algebraic threshold network of degree d is less than or equal to 2 ( ~ ). Generalised Sum of Outer-Products Rule: The classical Reb bian rule for the linear case d = 1 (cf. [11] and quoted references) can be naturally extended to networks of higher-order. The coefficients WI, IE Id are constructed as the sum of generalised Kronecker outer-products, m WI = L u~a). a=l Theorem 2 The storage capacity of the outer-product algorithm applied to a homogeneous algebraic threshold network of degree d is less than or equal to nd /2(d + l)logn (also cf. [15,27]). Generalised Spectral Rule: For d = 1 the spectral rule amounts to iteratively projecting states orthogonally onto the linear space generated by u(1), ... , u(m), and then taking the closest point on the hypercube to this projection (cf. [27,28]). This approach can be extended to higher-orders as we now describe. Let W denote the n X N(n,d) matrix of coefficients WI arranged lexicographically; i.e., Wl,l,2, ... ,d-l,d Wl,2,3, ... ,d,d+l Wl,n-d+l,n-d+2, ... ,n-l,n W= W2,l,2, ... ,d-l,d W2,2,3, ... ,d,d+l W2,n-d+l,n-d+2, ... ,n-l,n Wn ,l,2, ... ,d-l,d Wn ,2,3, ... ,d,d+l W n ,n-d+l,n-d+2, ... ,n-l,n Note that the symmetry and the "zero-diagonal" nature of the interactions have been relaxed to increase capacity. Let U be the n X m matrix of memories. Form the extended N(n,d) X m binary matrix 1 U = [lu(l) ... lu(m)], where u(a) l,2,. .. ,d-l,d (a) u1,2, ... ,d-l,d+l (a) un _ d+ l,n- d+2, ... ,n-l,n Let A = dgP'<l) ... ),(m)] be a m X m diagonal matrix with positive diagonal terms. A generalisation of the spectral algorithm for choosing coefficients yields W = UA1Ut where 1 ut is the pseudo-inverse of 1 U. Theorem 3 The storage capacity of the generalised spectral algorithm is at best n ( d ). 3.2 RANDOM INTERACTIONS We consider homogeneous algebraic threshold networks whose weights WI are Li.d., N(O, 1) random variables. This is a natural generalisation to higher-order of Ising spin glasses with Gaussian interactions. We will show an asymptotic estimate for the number of fixed points of the structure. Asymptotic results for the usual case d = 1 of linear threshold networks with Gaussian interactions have been reported in the literature [6,9,16]. For i = 1, ... , n set s~ = Ui L: WIUI\i . IeId:ieI For each n the random variables S~, i = 1, ... , n are identically distributed, jointly Gaussian variables with zero mean, and variance O'~ = ( n ~ 1 ). Definition 2 For any given f3 ~ 0, a state u E run is f3-strongly stable iff S~ ~ f3O'n, for each i = 1, ... , n. The case f3 = 0 reverts to the usual case of fixed points. The parameter f3 is essentially a measure of how deep the well of attraction surrounding the fixed point is. The following proposition asserts that a 0-1 law ("phase transition") governs the expected number of fixed points which have wells of attraction above a certain depth. Let Fd(f3) be the expected number of f3-strongly stable states. Theorem 4 Corresponding to each fixed interaction order d there exists a positive constant f3d such that as n --+ 00, if f3 < f3d if f3 = f3d if f3 > f3d , where kd(f3) > 0, and 0 ~ Cd(f3) < 1 are parameters depending solely on f3 and the interaction order d. 4 CONCLUSION In fine, it appears possible to design shallow, polynomial size threshold circuits to compute a wide class of structured problems. The thesis that shallow circuits compute more efficiently than deep circuits is borne out. For the particular case of 49 50 higher-order networks, all the garnered results appear to point in the same direction: For neural networks of fixed degree d, the maximal number of programmable states is essentially of the order of nd• The total number of fixed points, however, appear to be exponential in number (at least for the random interaction case) though almost all of them have constant attraction depths. References [1] Y. S. Abu-Mostafa, "Number of synapses per neuron," in Analog VLSI and Neural Systems, ed. C. Mead, Addison Wesley, 1987. [2] P. Baldi, II. Some Contributions to the Theory of Neural Networks. Ph.D. Thesis, California Insitute of Technology, June 1986. [3] P. Baldi and S. S. Venkatesh, "Number of stable points for spin glasses and neural networks of higher orders," Phys. Rev. Lett., vol. 58, pp. 913-916, 1987. [4] P. Baldi and S. S. Venkatesh, "Fixed points of algebraic threshold networks," in preparation. [5] H. H. Chen, et al, "Higher order correlation model of associative memory," in Neural Networks for Computing. New York: AlP Conf. Proc., vol. 151, 1986. [6] S. F. Edwards and F. Tanaka, "Analytical theory of the ground state properties of a spin glass: I. ising spin glass," Jnl. Phys. F, vol. 10, pp. 2769-2778, 1980. [7] G. C. Fox and S. W. Otto, "Concurrent Computations and the Theory of Complex Systems," Caltech Concurrent Computation Program, March 1986. [8] F. H. Grick and C. Asanuma, ~'Certain aspects of the anatomy and physiology of the cerebral cortex," in Parallel Distributed Processing, vol. 2, eds. D. E. Rumelhart and J. L. McCelland, pp. 333-371, MIT Press, 1986. [9] D. J. Gross and M. Mezard, "The simplest spin glass," Nucl. Phys., vol. B240, pp. 431-452, 1984. [10] J. Hasted, "Almost optimal lower bounds for small depth circuits," Proc. 18-th ACM STOC, pp. 6-20, 1986. [11] J. J. Hopfield, "Neural networks and physical sytems with emergent collective computational abilities," Proc. Natl. Acad. Sci. USA, vol. 79, pp. 25.54-2558, 1982. [12] J. S. Judd, "Complexity of connectionist learning with various node functions," Dept. of Computer and Information Science Technical Report, vol. 87-60, Univ. of Massachussetts, Amherst, 1987. [13] M. Kearns, M. Li, 1. Pitt, and L. Valiant, "On the learnability of Boolean formulae," Proc. 19-th ACM STOC, 1987. [14] C. Koch, T. Poggio, and V. Torre, "Retinal ganglion cells: A functional interpretation of dendritic morphology," Phil. Trans. R. Soc. London, vol. B 288, pp. 227-264, 1982. [15] R. J. McEliece, E. C. Posner, E. R. Rodemich, and S. S. Venkatesh, "The capacity of the Hopfield associative memory," IEEE Trans. Inform. Theory, vol. IT-33, pp. 461-482, 1987. [16] R. J. McEliece and E. C. Posner, "The number of stable points of an infiniterange spin glass memory," JPL Telecomm. and Data Acquisition Progress Report, vol. 42-83, pp. 209-215, 1985. [17] C. A. Mead (ed.), Analog VLSI and Neural Systems, Addison Wesley, 1987. [18] N. Megiddo, "On the complexity of polyhedral separability," to appear in Jnl. Discrete and Computational Geometry, 1987. [19] S. Muroga, "Lower bounds on the number of threshold functions," IEEE Trans. Elec. Comp., vol. 15, pp. 805-806, 1966. [20] S. Muroga, Threshold Logic and its Applications, Wiley Interscience, 1971. [21] V. N. Peled and B. Simeone, "Polynomial-time algorithms for regular setcovering and threshold synthesis," Discr. Appl. Math., vol. 12, pp. 57-69, 1985. [22] D. Psaltis and C. H. Park, "Nonlinear discriminant functions and associative memories," in Neural Networks for Computing. New York: AlP Conf. Proc., vol. 151, 1986. [23] J. Reif, "On threshold circuits and polynomial computation," preprint. [24] R. Smolenski, "Algebraic methods in the theory of lower bounds for Boolean circuit complexity," Proc. J9-th ACM STOC, 1987. [25] L. G. Valiant, "A theory of the learnable," Comm. ACM, vol. 27, pp. 1134-1142, 1984. [26] L. G. Valiant, "Deductive learning," Phil. Trans. R. Soc. London, vol. A 312, pp. 441-446, 1984. [27] S. S. Venkatesh, Linear Maps with Point Rules: Applications to Pattern Classification and Associativ~ Memory. Ph.D. Thesis, California Institute of Technology, Aug. 1986. [28] S. S. Venkatesh and D. Psaltis, "Linear and logarithmic capacities in associative neural networks," to appear IEEE Trans. Inform. Theory. [29] S. S. Venkatesh, D. Psaltis, and J. Yu, private communication. [30] R. O. Winder, "Bounds on threshold gate realisability," IRE Trans. Elec. Comp., vol. EC-12, pp. 561-564, 1963. [31] A. C. C. Yaa, "Separating the poly-time hierarchy by oracles," Proc. 26-th IEEE FOCS, pp. 1-10, 1985. 51
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'Ensemble' Boltzmann Units have Collective Computational Properties like those of Hopfield and Tank Neurons Mark Derthick and Joe Tebelskis Department of Computer Science Carnegie-Mellon University 1 Introduction 223 There are three existing connection::;t models in which network states are assigned a computational energy. These models-Hopfield nets, Hopfield and Tank nets, and Boltzmann Machines-search for states with minimal energy. Every link in the network can be thought of as imposing a constraint on acceptable states, and each violation adds to the total energy. This is convenient for the designer because constraint satisfaction problems can be mapped easily onto a network. Multiple constraints can be superposed, and those states satisfying the most constraints will have the lowest energy. Of course there is no free lunch. Constraint satisfaction problems are generally combinatorial and remain so even with a parallel implementation. Indeed, Merrick Furst (personal communication) has shown that an NP-complete problem, graph coloring, can be reduced to deciding whether a connectionist network has a state with an energy of zero (or below). Therefore designing a practical network for solving a problem requires more than simply putting the energy minima in the right places. The topography of the energy space affects the ease with which a network can find good solutions. If the problem has highly interacting constraints, there will be many local minima separated by energy barriers. There are two principal approaches to searching these spaces: monotonic gradient descent, introduced by Hopfield [1] and refined by Hopfield and Tank [2]; and stochastic gradient descent, used by the Boltzmann Machine [3]. While the monotonic methods are not guaranteed to find the optimal solution, they generally find good solutions much faster than the Boltzmann Machine. This paper adds a refinement to the Boltzmann Machine search algorithm analogous to the Hopfield and Tank technique, allowing the user to trade off the speed of search for the quality of the solution. © American Institute of Physics 1988 224 2 Hopfield nets A Hopfield net [1] consists of binary-valued units connected by symmetric weighted links. The global energy of the network is defined to be 1 E = -2 ~ ~ WijSjSj ~ljsj I Jr' I where Sj is the state of unit i, and Wjj is the weight on the link between units i and j. The search algorithm is: randomly select a unit and probe it until quiescence. During a probe, a unit decides whether to be on or off, detennined by the states of its neighbors. When a unit is probed, there are two possible resulting global states. The difference in energy between these states is called the unit's energy gap: The decision rule is s, = { o iL1i < 0 1 otherwise This rule chooses the state with lower energy. With time, the global energy of the network monotonically decreases. Since there are only a finite number of states, the network must eventually reach quiescence. 3 Boltzmann Machines A Boltzmann Machine [3] also has binary units and weighted links, and the same energy function is used. Boltzmann Machines also have a learning rule for updating weights, but it is not used in this paper. Here the important difference is in the decision rule, which is stochastic. As in probing a Hopfield unit, the energy gap is detennined. It is used to detennine a probability of adopting the on state: 1 P(Sj = 1) = 1 + e-tl;jT where T is the computational temperature. With this rule, energy does not decrease monotonically. The network is more likely to adopt low energy states, but it sometimes goes uphill. The idea is that it can search a number of minima, but spends more time in deeper ones. At low temperatures, the ratio of time spent in the deepest minima is so large that the chances of not being in the global minimum are negligible. It has been proven [4] that after searching long enough, the probabilities of the states are given by the Boltzmann distribution, which is strictly a function of energy and temperature, and is independent of topography: P ex _ -CEa-E,,)jT - e .P{3 (1) 225 The approach to equilibrium, where equation 1 holds, is speeded by initially searching at a high temperature and gradually decreasing it. Unfortunately, reaching equilibrium stills takes exponential time. While the Hopfield net settles quickly and is not guaranteed to find the best solution, a Boltzmann Machine can theoretically be run long enough to guarantee that the global optimum is found Most of the time the _ uphill moves which allow the network to escape local minima are a waste of time, however. It is a direct consequence of the guaranteed ability to find the best solution that makes finding even approximate solutions slow. 4 Hopfield and Tank networks In Hopfield and Tank nets [2], the units take on continuous values between zero and one, so the search takes place in the interior of a hypercube rather than only on its vertices. The search algorithm is deterministic gradient descent. By beginning near the center of the space and searching in the direction of steepest descent, it seems likely that the deepest minimum will be found. There is still no guarantee, but good results have been reported for many problems. The modified energy equation is 1 ~~ ~ 1 r; I ~ E = -2 ~ ~ WjjSjSj + ~ Rj 10 g- (s)ds - ~ [jSj I l ' I (2) Rj is'the input resistance to unit i, and g(u) is the sigmoidal unit transfer function 1+~2X.. The second term is zero for extreme values of Sj, and is minimized at Sj = t. The Hopfield and Tank model is continuous in time as well as value. Instead of proceeding by discrete probes, the system is described by simultaneous differential equations, one for each unit. Hopfield and Tank show that the following equation of m<?tion results in a monotonic decrease in the value of the energy function: duo _I = -u./r + ~ Woos' + [. dt I ~ IlJ I J where r = RC, C is a constant determining the speed of convergence, Uj = g-I(Sj), and the gain, .A, is analgous to (the inverse of) temperature in a Boltzmann Machine . .A determines how important it is to satisfy the constraints imposed by the links to other units. When .A is low, these constraints are largely ignored and the second term dominates, tending to keep the system near the center of the search space, where there is a single global minimum. At high gains, the minima lie at the corners of the search space, in the same locations as for the Hopfield model and the Boltzmann model. If the system is run at high gain, but the initial state is near the center of the space, the search gradually moves out towards the corners, on the way encountering "continental divides" between watersheds leading to all the various local minima. The initial steepness of the watersheds serves as a heuristic for choosing which minima is 226 likely to be lower. This search heuristic emerges automatically from the architecture, making network design simple. For many problems this single automatic heuristic results in a system comparable to the best knowledge intensive algorithms in which many domain specific heuristics are laboriously hand programmed. For many problems, Hopfield and Tank nets seem quite sufficient [5,6]. However for one network we have been using [7] the Hopfield and Tank model invariably settles into poor local minima. The solution has been to use a new model combining the advantages of Boltzmann Machines and Hopfield and Tank networks. 5 'Ensemble' Boltzmann Machines It seems the Hopfield and Tank model gets its advantage by measuring the actual gradient, giving the steepest direction to move. This is much more informative than picking a random direction and deciding which of the two corners of the space to try, as models using binary units must do. Peter Brown (personal communication) has investigated continuous Boltzmann Machines, in which units stochastically adopt a state between zero and one. The scheme presented here has a similar effect, but the units actually take on discrete states between zero and one. Each ensemble unit can be thought of as an ensemble of identically connected conventional Boltzmann units. To probe the ensemble unit, each of its constituents is probed, and the state of the ensemble unit is the average of its constituents' states. Because this average is over a number of identical independent binary random variables, the ensemble unit's state is binomially distributed. Figure 1 shows an ensemble unit with three constituents. At infinite temperature, all unit states tend toward -t, and at zero temperature the states go to zero or one unless the energy gap is exactly zero. This is similar to the behavior of a Hopfield and Tank network at low and high gain, respectively. In Ensemble Boltzmann Machines (EBMs) the tendency towards! in the absence of constraints from other units results from the shape of the binomial distribution. In contrast, the second term in the energy equation is responsible for this effect in the Hopfield and Tank model. Although an EBM proceeds in discrete time using probes, over a large number of probes the search tends to proceed in the direction of the gradient. Every time a unit is probed, a move is made along one axis whose length depends on the magnitude of the gradient in that direction. Because probing still contains a degree of stochasticity, EBMs can escape from local minima, and if run long enough are guaranteed to find the global minimum. By varying n, the number of components of each ensemble unit, the system can exhibit any intermediate behavior in the tradeoff between the speed of convergence of Hopfield and Tank networks, and the ability to escape local minima of Boltzmann Machines. Clearly when n = 1 the performance is identical to a conventional Boltzmann Machine, because each unit consists of a single Boltzmann unit. As n -+ 00 the 227 s=1/3 s=2/3 Figure 1: The heavy lines depict an 'Ensemble' Boltzmann Machine with two units. With an ensemble size of three, this network behaves like a conventional Boltzmann Machine consisting of six units (light lines). The state of the ensemble units is the average of the states of its components. value a unit takes on after probing becomes deterministic. The stable points of the system are then identical to the ones of the Hopfield and Tank model. To prove this, it suffices to show that at each probe the ensemble Boltzmann unit takes on the state which gives rise to the lowest (Hopfield and Tank) energy. Therefore the energy must monotonically decrease. Further, if the system is not at a global (Hopfield and Tank) energy minimum, there is some unit which can be probed so as to lower the energy. To show that the state resulting from a probe is the minimum possible, we show first that the derivative of the energy with resepect to the unit's state is zero at the resulting state, and second that the second derivative is positive over the entire range of possible states, zero to one. so Taking the derivative of equation 2 gives Now 1 g(u) = 1 + e-2>'w lIs g- (u) = -1n-2,\ 1 - s Let T = 2lR' The EBM update rule is 1 Sk=---= 1 + e-ilk/ T • 228 Therefore dEl dslc SI; 1 1+. Lll;/T = - Ll + Tin [l+e lLltlT 1 Ic e-Llt/T 1+e-at1t = -Lllc + Tin eLlI;/T = - Lllc + T(LlIc/D = 0 and = _1_. 1 Sic • [(1 Sic) (-Sic)] 2>"R Sic (I SIc)2 1 = 2>..Rslc(l Sic) > 0 on 0 < Sic < 1 In writing a program to simulate an EBMt it would be wasteful to explicitly represent the components of each ensemble unit. Since each component has an identical energy gapt the average of their values is given by the binomial distribution b(ntp) where n is the ensemble sizet and p is l+e 1 LlIT. There are numerical methods for sampling from this distribution in time independent of n [8]. When n is infinitet there is no need to bother with the distribution because the result is just p. Hopfield and Tank suggest [2] that the Hopfield and Tank. model is a mean field approximation to the original Hopfield model. In a mean field approximationt the average value of a variable is used to calculate its effect on other variablest rather than calculating all the individual interactions. Consider a large ensemble of Hopfield nets with two unitst A and B. To find the distribution of final states exactlYt each B unit must be updated based on the A unit in the same network. The calculation must be repeated for every network in the ensemble. Using a mean field approximationt the average value of all the B units is calculated based on the average value of all the A units. This calculation is no harder than that of the state of a single Hopfield network, yet is potentially more informative since it approximates an average property of a whole ensemble of Hopfield networks. The states of Hopfield and Tank. units can be viewed as representing the ensemble average of the states of Hopfield units in this way. Peterson and Anderson [9] demonstrate rigorously that the behavior is a mean field approximation. In the EBM, it is intuitively clear that a mean field approximation is being made. The network can be thought of as a real ensemble of Boltzmann networkst except with additional connections between the networks so that each Boltzmann unit sees not only its neighbors in the same nett but also sees the average state of the neighboring units in all the nets (see figure 1). 229 6 Traveling Salesman Problem The traveling salesman problem illustrates the use of energy-based connectionist networks, and the ease with which they may be designed. Given a list of city locations, the task is to find a tour of minimum length through all the cities and returning to the starting city. To represent a solution to an n city problem in a network, it is convenient to use n columns of n rows of units [2]. If a unit at coordinates (i, J) is on, it indicates that the ith city is the jth to be visited. A valid solution will have n units on, one in every column and one in every row. The requirements can be divided into four constraints: there can be no more than one unit on in a row, no more that one unit on in a column, there must be n units on, and the distances between cities must be minimized. Hopfield and Tank use the following energy function to effect these constraints: X i Hi B/2 L L L SXiSYi + i x Y:IX C/2 (;;~>Xi -nr + D/2 L L L dxrsXi(sY,i+l + SY,i-l) x Y:IX i (3) Here units are given two subscripts to indicate their row and column, and the subscripts "wrap around" when outside the range 1 < i < n. The first tenn is imple-mented with inhibitory links between every pair of units in a row, and is zero only if no two are on. The second term is inhibition within columns. In the third term, n is the number of cities in the tour. When the system reaches a vertex of the search space, this term is zero only if exactly n units are on. This constraint is implemented with inhibitory links between all n4 pairs of units plus an excitatory input current to all units. In the last term dxr is the distance between cities X and Y. At points in the search space representing valid tours, the summation is numerically equal to the length of the tour. I As long as the constraints ensuring that the solution is a valid tour are stronger than those minimizing distance, the global energy minimum will represent the shortest tour. However every valid tour will be a local energy minimum. Which tour is chosen will depend on the random initial starting state, and on the random probing order. 7 Empirical Results The evidence that convinced me EBMs offer improved performance over Hopfield and Tank networks was the ease of tuning them for the Ted Turner problem reported 230 in [7]. However this evidence is entirely subjective; it is impossible to show that no set of parameters exist which would make the Hopfield and Tank model perform well. Instead we have chosen to repeat the traveling salesman problem experiments reported by Hopfield and Tank [2], using the same cities and the same values for the constants in equation 3. The tour involves 10 cities, and the shortest tour is of length 2.72. An average tour has length 4.55. Hopfield and Tank report finding a valid tour in 16 of 20 settlings, and that half of these are one of the two shortest tours. One advantage of Hopfield and Tank nets over Boltzmann Machines is that they move continuously in the direction of the gradient. EBMs move in discrete jumps whose size is the value of the gradient along a given axis. When the system is far from equilibrium these jumps can be quite large, and the search is inefficient. Although Hopfield and Tank nets can do a whole search at high gain, Boltzmann Machines usually vary the temperature so the system can remain close to equilibrium as the low temperature eqUilibrium is approached. For this reason our model was more sensitive to the gain parameter than the Hopfield and Tank model, and we used temperatures much higher than 2lR' As expected, when n is infinite, an EBM produces results similar to those reported by Hopfield and Tank. 85 out of 100 settlings resulted in valid tours, and the average length was 2.73. Table 1 shows how n affects the number of valid tours and the average tour length. As n decreases from infinity, both the average tour length and the number of valid tours increases. (We have no explanation for the anomalously low number of valid tours for n = 40.) Both of these effects result from the increased sampling noise in determining the ensemble unit states for lower n. With more noise, the system has an easier time escaping local minima which do not represent valid tours. Yet at the same time the discriminability between the very best tours and moderately good tours decreases, because these smaller energy differences are swamped by the noise. Rather than stop trials when the network was observed to converge, a constant number of probes, 200 per unit, was made. However we noted that convergence was generally faster for larger values of n. Thus for the traveling salesman problem, large n give faster and better solutions, but a smaller values gives the highest reliability. Depending on the application, a value of either infinity or 50 seems best. 8 Conclusion 'Ensemble' Boltzmann Machines are completely upward compatible with conventional Boltzmann Machines. The above experiment can be taken to show that they perform better at the traveling salesman problem. In addition, at the limit of infinite ensemble size they perform similarly to Hopfield and Tank nets. For TSP and perhaps many other problems, the latter model seems an equally good choice. Perhaps due to the extreme regularity of the architecture, the energy space must be nicely behaved 231 Ensemble Size Percent Valid Average Tour Length I 93 3.32 40 84 2.92 50 95 2.79 100 89 2.79 1000 90 2.80 infinity 85 2.73 Table 1: Number of valid tours out of 100 trials and average tour length, as a function of ensemble size. An ensemble size of one corresponds to a Boltzmann Machine. Infinity loosely corresponds to a Hopfield and Tank network. in that the ravine steepness near the center of the space is a good indication of its eventual depth. In this case the ability to escape local minima is not required for good perfonnance. For the Ted Turner problem, which has a very irregular architecture and many more constraint types, the ability to escape local minima seems essential. Conventional Boltzmann Machines are too noisy, both for efficient search and for debugging. EBMs allow the designer the flexibility to add only as much noise as is necessary. In addition, lower noise can be used for debugging. Even though this may give poorer perfonnance, a more detenninistic search is easier for the debugger to understand, allowing the proper fix to be made. Acknowledgements We appreciate receiving data and explanations from David Tank, Paul Smolensky, and Erik Sobel. This research has been supported by an ONR Graduate Fellowship, by NSF grant EET-8716324, and by the Defense Advanced Research Projects Agency (DOD), ARPA Order No. 4976 under contract F33615-87-C-1499 and monitored by the: Avionics Laboratory Air Force Wright Aeronautical Laboratories Aeronautical Systems Division (AFSC) Wright-Patterson AFB, OB 45433-6543 This research was also sponsored by the same agency under contract N00039-87C-0251 and monitored by the Space and Naval Warfare Systems Command. 232 References [1] J. J. Hopfield, "Neural networks and physical systems with emergent collective computational abilities," Proceedings of the National Academy of Sciences U.SA., vol. 79, pp. 2554-2558, April 1982. [2] J. Hopfield and D. Tank, '''Neural' computation of decisions in optimization problems,»' Biological Cybernetics, vol. 52, pp. 141-152, 1985. [3] G. E. Hinton and T. J. Sejnowski, "Learning and relearning in Boltzmann Machines," in Parallel distributed processing: Explorations in the microstructure of cognition, Cambridge, MA: Bradford Books, 1986. [4] S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-6, pp. 721-741, 1984. [5] J. L. Marroquin, Probabilistic Solution of Inverse Problems. PhD thesis, MIT, September 1985. [6] J. Hopfield and D. Tank, "Simple 'Neural' optimization networks: an aid converter, signal decision circuit and a linear programming circuit," IEEE Transactions on Circuits and Systems, vol. 33, pp. 533--541, 1986. [7] M. Derthick, "Counterfactual reasoning with direct models," in AAA/-87, Morgan Kaufmann, July 1987. [8] D. E. Knuth, The Art of Computer Programming. Second Edition. Vol. 2, Addison-Wesley, 1981. [9] C. Peterson and J. R. Anderson, "A mean field theory learning algorithm for neural networks," Tech. Rep. EI-259-87, MCC, August 1987.
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262 ON TROPISTIC PROCESSING AND ITS APPLICATIONS Manuel F. Fernandez General Electric Advanced Technology Laboratories Syracuse, New York 13221 ABSTRACT The interaction of a set of tropisms is sufficient in many cases to explain the seemingly complex behavioral responses exhibited by varied classes of biological systems to combinations of stimuli. It can be shown that a straightforward generalization of the tropism phenomenon allows the efficient implementation of effective algorithms which appear to respond "intelligently" to changing environmental conditions. Examples of the utilization of tropistic processing techniques will be presented in this paper in applications entailing simulated behavior synthesis, path-planning, pattern analysis (clustering), and engineering design optimization. INTRODUCTION The goal of this paper is to present an intuitive overview of a general unsupervised procedure for addressing a variety of system control and cost minimization problems. This procedure is hased on the idea of utilizing "stimuli" produced by the environment in which the systems are designed to operate as basis for dynamically providing the necessary system parameter updates. This is by no means a new idea: countless examples of this approach abound in nature, where innate reactions to specific stimuli ("tropisms" or "taxis" --not to be confused with "instincts") provide organisms with built-in first-order control laws for triggering varied responses [8]. (It is hypothesized that "knowledge" obtained through evolution/adaptation or through learning then refines or suppresses most of these primal reactions). Several examples of the implicit utilization of this approach can also be found in the literature, in applications ranging from behavior modeling to pattern analysis. Ve very briefly depict some these applications, underlining a common pattern in their formulation and generalizing it through the use of basic field theory concepts and representations. A more rigorous and detailed exposition --regarding both mathematic and application/implementation aspects-- is presently under preparation and should be ready for publication sometime next year ([6]). TROPISMS Tropisms can be defined in general as class-invariant systemic responses to specific sets of stimuli [6]. All time-invariant systems can thus be viewed as tropistic provided that we allow all possible stimuli to form part of our set of inputs. In most tropistic systems, however, response- (or time-) invariance applies only to specific inputs: green plants, for example, twist and grow in the direction of light (phototropism), some birds' flight patterns follow changes in the Earth's magnetic field (magnetotropism), various organisms react to gravitational field © American Institute of Physics 1988 263 variations (geotropism), etc. Tropism/stimuli interactions can be portrayed in term~ of the superposition of scalar (e.g., potential) or vector (e.g., force) fields exhibiting properties paralleling those of the suitably constrained "reactions" we wish to model [1J,[6J. The resulting field can then be used as a basis for assessing the intrinsic cost of pursuing any given path of action, and standard techniques (e.g., gradient-following in the case of scalar fields or divergence computation in the case of vector fields) utilized in determining a response*. In addition, the global view of the situation provided by field representations suggest that a basic theory of tropistic behavior can also be formulated in terms of energy expenditure minimization (Euler-Lagrange equations). This formulation would yield integral-based representations (Feynman path integrals [4],[11]) satisfying the observation that tropistic processes typically obey the principle of least action. Alternatively, fields may also be collapsed into "attractors" (points of a given "mass" or "charge" in cost space) through laws defining the relationships that are to exist among these "at tractors" and the other particles traveling through the space. This provides the simplification that when updating dynamically changing situations only the effects caused by the interaction of the attractors with the particles of interest --rather than the whole cost field-- may have to be recalculated. For example, appropriately positioned point charges exerting on each other an electrostatic force inversely proportional to the square of their distance can be used to represent the effects of a coulombic-type cost potential field. A particle traveling through this field would now be affected by the combination of forces ensuing from the interaction of the attractors' charges with its own. If this particle were then to passively follow the composite of the effects of these forces it would be following the gradient of the cost field (i.e., the vector resulting from the superposition of the forces acting on the particle would point in the direction of steepest change in potential). Finally, other representations of tropism/stimuli interactions (e.g., Value-Driven Decision Theory approaches) entail associating "profit" functions (usually sigmoidal) with each tropism, modeling the relative desirability of triggering a reaction as a function of the time since it was last activated [9]. These representations are * In order to bring extra insight into tropism/stimuli interactions and simplify their formulation, one may exchange vector and scalar field representations through the utilization of appropriately selected mappings. Some of the most important of such mappings are the gradient operator (particularly so because the gradient of a scalar --potential-field is proportional to a "force" --vector-- field), the divergence (which may be thought of as performing in vector fields a function analogous to that performed in scalar fields by the gradient), and their combinations (e.g., the Laplacian, a scalar-to-scalar mapping which can be visualized as performing on potential fields the equivalent of a second derivative operation. 264 /.~ .' .• Model fly as a positive geotropislic point of mass M. • Model fence slakes as negalive geotropislic poinls with masses m 1 , m z • •••• mit· • At each update time compute sum of forces acting on frog: F • k H d 2 " • Compute frog's heading and acceleration based on the ensuing force; then update frog's position. Figure 1: Attractor-based representation of a frog-fenee-fly scenario (see [1) for a vector-field representation). The objective is to model a frog's path-planning decision-making process when approaching a fly in the presence of obstacles. (The picket fence is represented by the elliptical outline with an opening in the back, the fly --inside the fenced space-- is represented by a "+~ sign, and arrows are used to indicate the direction of a frog's trajectory into and out of fenced area). 265 particularly amenable to neural-net implementations [6J. TROPISTIC PROCESSING Tropistic processing entails building into systems tropisms appropriate for the environment in which these systems are expected to operate. This allows taking advantage of environment-produced "stimuli" for providing the required control for the systems' behavior. The idea of tropistic processing has been utilized with good results in a variety of applications. Arbib et.al., for example, have implicitly utilized tropistic processing to describe a batrachian's reaction to its environment in terms of what may be visualized as magnetic (vector) fields' interactions [1]. Vatanabe (12) devised for pattern analysis purposes an interaction of tropisms ("geotropisms") in which pattern "atoms" are attracted to each other, and hence "clustered", subject to a squared-inverse-distance ("feature distance") law similiar to that from gravitational mechanics. It can be seen that if each pattern atom were considered an "organism", its behavior would not be conceptually different from that exhibited by Arbibian frogs: in both cases organisms passively follow the force vectors resulting from the interaction of the environmental stimuli with the organisms' tropisms. It is interesting, though, to note that the "organisms'" behavior will nonetheless appear "intelligent" to the casual observer. The ability of tropistic processes to emulate seemingly rational behavior is now begining to be explored and utilized in the development of synthetic-psychological models and experiments. Braitenberg, for example, has placed tropisms as the primal building block from which his models for cognition, reason, and emotions evolve [3]**; Barto [2] has suggested the possibility of combining tropisms and associative (reinforced) learning, with aims at enabling the automatic triggering of behavioral responses by previously experienced situations; and Fernandez [6] has used CROBOTS [10], a virtual multiprocessor emulator, as laboratory for evaluating the effects of modifying tropistic responses on the basis of their projected future consequences. Other applications of tropistic processing presently being investigated include path-planning and engineering design optimization [6]. For example, consider an air-reconnaissance mission deep behind enemy lines; as the mission progresses and unexpected SAM sites are discovered, contingency flight paths may be developed in real time simply by modeling each SAM or interdiction site as a mass point towards which the aircraft exhibits negative geotropistic tendencies (i.e., gravitational forces repel it), and modeling the objective as a positive geotropistic point. A path to ** Of particular interest within the sole context of Tropistic Processing is Dewdney's [5] commented version of the first chapters of Braitenberg's book [3J, in which the "behavior" of mechanically very simple cars, provided with "~yes" and phototropism-supporting connections (including Ledley-type "neurons" [4J), is "analyzed". 266 • • • •• . "":,~ • ,. ill' ",:" • • • ••• -. -• • • • • • • • • • • A •• •• • • • • • • • • • • e ,,• • • • • • " • , • 8 .. *' ~!::. ~::: • -• • • • • • • • • e • • .. .~~ Figure ~ (Geotropistic clustering ~2]): The problem being portrayed here is that of clustering dots distributed in [x,y]-space as shown and uniformly in color ([red,blue,green]). The approach followed is that outlined in Figure 1, with the differences that normalized (Hahalanobis) distances are used and when merges occur, conservation of momentum is observed. Tags are also kept --specifying with which dots and in what order merges occur-- to allow drawing cluster boundaries in the original data set. (Efficient implementation of this clustering technique entails using a ring of processors, each of which is assigned the "features" of one or more "dots" and the task of carrying out computations with respect to these features. If the features of each dot are then transmitted through the ring, all the forces imposed on it by the rest will have been determined upon completion of the circuit). the target will then be automatically drawn by the interaction of the tropisms with the gravitational forces. (Once the mission has been completed, the target and its effects can be eliminated, leaving active only the repulsive forces, which will then "guide" the airplane out of the danger zone). 267 In engineering design applications such as lens modeling and design, lenses (gradient-index type, for example) can be modeled in terms of photons attempting to reach an objective plane through a three-dimensional scalar field of refraction indices; modeling the process tropistically (in a manner analogous to that of the air-reconnaissance example above) would yield the least-action paths that the individual photons would follow. Similarly, in "surface-of-revolution" fuselage design ("Newton's Problem"), the characteristics of the interaction of forces acting within a sheet of metal foil when external forces (collisions with a fluid's molecules) are applied can be modeled in terms of tropistic reactions which will tend to reconfigure the sheet so as to make it present the least resistance to friction when traversing a fluid. Additional applications of tropistic processing include target tracking and multisensor fusion (both can be considered instances of "clustering") [6], resource allocation and game theory (both closely related to path-planning) [9], and an assortment of other cost-minimization functions. Overall, however, one of the most important applications of tropistic processing may be in the modeling and understanding of analog processes [6], the imitation of which may in turn lead to the development of effective strategies OBSERVATlONS PAST EXPERIENCE (e.g. MEMORY MAPS) M BASIC mOPISM FUNCTION RESPONSE FUNCTION TROPISM-BASED SYSTEM PREDICTED (i.e. MODELLED) OUTCOUE p RESPONSE Figure 3: The combination of tropisms and associative (reinforced) learning can be used to enable the automatic triggering of behavioral responses by previously experienced situations [2]. Also, the modeled projection of the future consequences of a tropistic decision can be utilized in the modification of such decision (6J. (Note analogy to filtering problem in which past history and predicted behavior are used to smooth present observations). 268 -5000.0 -33».3 -I i i , i , oD 01 "' to , • • ''''.7 .lll3.J 5000.' -5000.0 -3l».l -, • • ''''.7 lJl3.J 5000.0 i , 3lJ3.J 5000.0 i , Figure 4: Simplified representation of air-reconnaissance mission example (see text): objective is at center of coordinate axis, thick dots represent SAM sites, and arrows denote airplane's direction of flight (airplane's maximum attainable speed and acceleration are constrained). All portrayed scenarios are identical except for tropistic control-law parameters (mainly objective to SAM-sites mass ratios in the first three scenarios). Varying the masses of the objective and SAM sites can be interpreted as trading off the relative importance of the mission vs. the aircraft's safety, and can produce dramatically differing flight paths, induce chaotic behavior (bottom-left scenario), or render the system unstable. The bottom-right scenario portrays the situation in which a tropistic decision is projected into the future and, if not meeting some criterion, modified (altering the direction of flight --e.g., following an isokline--, re-evaluating the mission's relative importance --revising masses--, changing the update rate, etc.). 269 for taking full advantage of parallel architectures [11]***. It is thus expected that the flexibility of tropistic processes to adapt to changing environmental conditions will prove highly valuable to the advancement of areas such as robotics, parallel processing and artificial intelligence, where at the very least they will provide some decision-making capabilities whenever unforeseen circumstances are encountered. ACKNOVLEDGEMENTS Special thanks to D. P. Bray for the ideas provided in our many discussions and for the development of the finely detailed simulations that have enabled the visualization of unexpected aspects of our work. REFERENCES [1] Arbib, M.A. and House, D.H.: "Depth and Detours: Decision Making in Parallel Systems". IEEE Vorkshop on Languages for Automation: Cognitive Aspects in Information Processing; pp. 172-180 (1985). [2] Barto, A.G. (Editor): "Simulation Experiments with Goal-Seeking Adaptive Elements". Avionics Laboratory, Vright-Patterson Air Force Base, OH. Report # AFVAL-TR-84-1022. (1984). [3] Braitenberg, V.: Vehicles: Experiments in Synthetic Psychology. The MIT Press. (1984). [4] Cheng, G.C.; Ledley, R.S.; and Ouyang, B.: "Pattern Recognition with Time Interval Modulation Information Coding". IEEE Transactions on Aerospace and Electronic Systems. AES-6, No.2; pp. 221-227 (1970). [5] Dewdney, A.K.: "Computer Recreations". Scientific American. Vol.256, No.3; pp. 16-26 (1987). [6] Fern6ndez, M.F.: "Tropistic Processing". To be published (1988). [7J Feynman, R.P.: Statistical Mechanics: A Set of Lectures. Frontiers in Physics Lecture Note Series-zI982). [8] Hirsch, J.: "Nonadaptive Tropisms and the Evolution of Behavior". Annals of the New York Academy of Sciences. Vol.223; pp. 84-88 (1973). [9] Lucas, G. and Pugh, G.: "Applications of Value-Driven Automation Methodology for the Control and Coordination of Netted Sensors in Advanced C**3". Report # RADC-TR-80-223. Rome Air Development Center, NY. (1980). [10] Poindexter, T.: "CROBOTS". Manual, programs, and files (1985). 2903 Vinchester Dr., Bloomington, IL., 61701. [11J Vallqvist, A.; Berne, B.J.; and Pangali, C.: "Exploiting Physical Parallelism Using Supercomputers: Two Examples from Chemical Physics". Computer. Vol.20, No.5; pp. 9-21 (1987). [12] Vatanabe, S.: Pattern Recognition: Human and Mechanical. John Viley & Sons; pp. 160-168 (1985). *** Optical Fourier transform operations, for instance, can be modeled in high-granularity machines through a procedure analogous to the gradient-index lens simulation example, with processors representing diffraction-grating "atoms" [6].
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814 NEUROMORPHIC NETWORKS BASED ON SPARSE OPTICAL ORTHOGONAL CODES Mario P. Vecchi and Jawad A. Salehi Bell Communications Research 435 South Street Morristown, NJ 07960-1961 Abstrad A family of neuromorphic networks specifically designed for communications and optical signal processing applications is presented. The information is encoded utilizing sparse Optical Orthogonal Code sequences on the basis of unipolar, binary (0,1) signals. The generalized synaptic connectivity matrix is also unipolar, and clipped to binary (0,1) values. In addition to high-capacity associative memory, the resulting neural networks can be used to implement general functions, such as code filtering, code mapping, code joining, code shifting and code projecting. 1 Introduction Synthetic neural nets[1,2] represent an active and growing research field. Fundamental issues, as well as practical implementations with electronic and optical devices are being studied. In addition, several learning algorithms have been studied, for example stochastically adaptive systems[3] based on many-body physics optimization concepts[4,5]. Signal processing in the optical domain has also been an active field of research. A wide variety of non-linear all-optical devices are being studied, directed towards applications both in optical computating and in optical switching. In particular, the development of Optical Orthogonal Codes (OOC)[6] is specifically interesting to optical communications applications, as it has been demonstrated in the context of Code Division Multiple Access (CDMA)[7] . In this paper we present a new class of neuromorphic networks, specifically designed for optical signal processing and communications, that encode the information in sparse OOC's. In Section 2 we review some basic concepts. The new neuromorphic networks are defined in Section 3, and their associative memory properties are presented in Section 4. In Section 5 other general network functions are discussed. Concluding remarks are given in Section 6. 2 Neural Networks and Optical Orthogonal Codes 2.1 Neural Network Model Neural network are generally based on multiply-threshold-feedback cycles. In the Hopfield model[2], for instance, a connectivity T matrix stores the M different memory elements, labeled m, by the sum of outer products, M Tij=Lu'iuj; i,j=1,2 ... N (1) m © American Institute of Physics 1988 815 where the state vectors ym represent the memory elements in the bipolar (-1,1) basis. The diagonal matrix elements in the Hopfield model are set to zero, Tii = O. For a typical memory recall cycle, an input vector .!lin, which is close to a particular memory element m = k, multiplies the T matrix, such that the output vector .!lout is given by N • out ~T. in Vi = L.J ijVj i,j = l,2 ... N (2) j=l and can be seen to reduce to vit ~ (N l)u~ + J(N - l)(M - 1) (3) for large N and in the case of randomly coded memory elements ym. In the Hopfield model, each output ~out is passed through a thresholding stage around zero. The thresholded output signals are then fed back, and the multiply and threshold cycle is repeated until a final stable output .!lout is obtained. IT the input .!lin is sufficiently close to y1c, and the number of state vectors is small (Le. M ~ N), the final output will converge to memory element m = k, that is, .!lout -+ y1c. The associative memory property of the network is thus established. 2.2 Optical Orthogonal Codes The OOC sequences have been developed[6,7] for optical CDMA systems. Their properties have been specifically designed for this purpose, based on the following two conditions: each sequence can be easily distinguished from a shifted version of itself, and each sequence can be easily distinguished from any other shifted or unshifted sequence in the set. Mathematically, the above two conditions are expressed in terms of autoand crosscorrelation functions. Because of the non-negative nature of optical signals 1 , OOC are based on unipolar (0,1) signals[7]. In general, a family of OOC is defined by the following parameters: - F, the length of the code, - K, the weight of the code, that is, the number of l's in the sequence, - >.a, the auto-correlation value for all possible shifts, other than the zero shift, - Ac , the cross-correlation value for all possible shifts, including the zero shift. For a given code length F, the maximum number of distinct sequences in a family of OOC depends on the chosen parameters, that is, the weight of the code K and the allowed overlap AaandAc. In this paper we will consider OOC belonging to the minimum overlap class, Aa = Ac = 1. lWe refer to optical inten6ity signals, and not to detection systems sensitive to phase information. 816 3 Neuromorphic Optical Networks Our neuromorphic networks are designed to take full advantage of the properties of the ~OC. The connectivity matrix T is defined as a sum of outer products, by analogy with (1), but with the following important modifications: 1. The memory vectors are defined by the sequences of a given family of OOC, with a basis given by the unipolar, binary pair (0,1). The dimension of the sparse vectors is given by the length of the code F, and the maximum number of available items depends on the chosen family of ~OC. 2. All ofthe matrix elements Ti; are clipped to unipolar, binary (0,1) values, resulting in a sparse and simplified connectivity matrix, without any loss in the functional properties defined by our neuromorphic networks. 3. The diagonal matrix elements Tii are not set to zero, as they reflect important information implicit in the OOC sequences. 4. The threshold value is not zero, but it is chosen to be equal to K, the weight of the ~OC. 5. The connectivity matrix T is generalized to allow for the possibility of a variety of outer product options: self-outer products, as in (1), for associative memory, but also cross-outer products of different forms to implement various other system functions. A simplified schematic diagram of a possible optical neuromorphic processor is shown in Figure 1. This implementation is equivalent to an incoherent optical matrix-vector multiplier[8], with the addition of nonlinear functions. The input vector is clipped using an optical hard-limiter with a threshold setting at 1, and then it is anamorphic ally imaged onto the connectivity mask for T. In this way, the ith pixel of the input vector is imaged onto the ith column of the T mask. The light passing through the mask is then anamorphically imaged onto a line of optical threshold elements with a threshold setting equal to K, such that the jth row is imaged onto the lh threshold element. 4 Associative Memory The associative memory function is defined by a connectivity matrix TMEM given by: (4) where each memory element ~m corresponds to a given sequence of the OOC family, with code length F. The matrix elements of TMEM are all clipped, unipolar values, as indicated by the function gn, such that, g{ (} = { 1 if ( ~ 1 ° if ( < 1 (5) 817 We will now show that an input vector ~Ie, which corresponds to memory element m = k, will produce a stable output (equal to the wanted memory vector) in a single pass of the multiply and threshold process. The multiplication can be written as: (6) We remember that the non-linear clipping function an is to be applied first to obtain -MEM T . Hence, v~t = ~:z:'! a {:z:'!:z:'! + ~ :z:~:z:'!'} , L.JJ 'J L.J'J j m#;1e (7) For :z:~ = 0, only the second term in (7) contributes, and the pseudo-orthogonality properties of the OOC allow us to write: (8) where the cross-correlation value is Ac < K. For :z:~ = 1, we again consider the properties of the OOC to obtain for the first term of (7): (9) where K is the weight of the OOC. Therefore, the result of the multiplication operation given by (7) can be written as: A out K Ie [value strictly 1 Vi = :Z:i + less than K (10) The thresholding operation follows, around the value K as explained in Section 3. That is, (10) is thresholded such that: { 1 if v~t > K vit = ,o ifv~t < K , , (11) hence, the final output at the end of a single pass will be given by: v:ut = :z:~. The result just obtained can be extended to demonstrate the single pass convergence when the input vector is close, but not necessarily equal, to a stored memory element. We can draw the following conclusions regarding the properties of our neuromorphic networks based on OOC: • For any given input vector ~in, the single pass output will correspond to the memory vector ~m which has the smallest Hamming distance to the input . • If the input vector ~in is missing a single 1-element from the K l's of an OOC, the single pass output will be the null or zero vector. 818 • If the input vector !lin has the same Hanuning distance to two (or more) memory vectors ~m , the single pass output will be the logical sum of those memory vectors. The ideas just discussed were tested with a computer simulation. An example of associative memory is shown in Table 1, corresponding to the OOC class of length F = 21 and weight K = 2. For this case, the maximum number of independent sequences is M = 10. The connectivity matrix TMEM is seen in Table 1, where one can clearly appreciate the simplifying features of our model, both in terms of the sparsity and of the unipolar, clipped values of the matrix elements. The computer simulations for this example are shown in Table 2. The input vectors ~ and Q show the error-correcting memory recovery properties. The input vector ~ is equally distant to memory vectors e3 and ~8, resulting in an output which is the sum (e3 EB e8 ). And finally, input vector d is closest to ~\ but one 1 is missing, and the output is the zero vector. The mask in Figure 1 shows the optical realization of the Table 1, where the transparent pixels correspond to the l's and the opaque pixels to the O's ofthe connectivity matrix TMEM. It should be pointed out that the capacity of our network is significant. From the previous example, the capacity is seen to be ::::: F /2 for single pass memory recovery. This result compares favorably with the capacity of a Hopfield model[9], of ~ F / 41n F. 5 General Network Functions Our neuromorphic networks, based on OOC, can be generalized to perform functions other than associative memory storage by constructing non-symmetrical connectivity matrices. The single pass convergence of our networks avoids the possibility of limitcycle oscillations. We can write in general: Tii = g{t Yf'Zj} , m=l (12) where each pair defined by m includes two vectors ym and em, which are not necessarily equal. The clipping function 9 {} insures that all m;:trix elements are binary (0,1) values. The possible choice of vector pairs is not completely arbitrary, but there is a wide variety of functions that can be implemented for each family of OOC. We will now discuss some of the applications that are of particular interest in optical communication systems. S.l Code Filtering (CDMA) Figure 2 shows an optical CDMA network in a star configuration. M nodes are interconnected with optical fibers to a passive MxM star coupler that broadcasts the optical signals. At each node there is a data encoder that maps each bit of information to the OOC sequence corresponding to the user for which the transmission is intended. In addition, each node has a filter and decoder that recognizes its specific OOC sequence. The optical transmission rate has been expanded by a factor F corresponding to the length of the OOC sequence. Within the context of a CDMA communication system[7], the filter or decoder must perform the function of recognizing a specific OOC sequence in the presence of other interfering codes sent on the common transmission medium. 819 We can think, then, of one of our neuromorphic networks as a filter, placed at a given receiver node, that will recognize the specific code that it was programmed for. We define for this purpose a connectivity matrix as T CDMA Ie Ie •• 1 2 F ij =ziZj; 1.,}= , ... , (13) where only one vector ~Ie is stored at each node. This symmetric, clipped connectivity matrix will give an output equal to ~Ie whenever the input contains this vector, and a null or zero output vector otherwise. It is clear by comparing (13) with (4) that the CDMA filtering matrix is equivalent to an associative memory matrix with only one item imprinted in the memory. Hence the discussion of Section 4 directly applies to the understanding of the behaviour of T CDMA In order to evaluate the performance of our neuromorphic network as a CDMA filter, computer simulations were performed. Table 3 presents the T CDM A matrix for a particular node defined by ~Ie of a CDMA system based on the OOC family F = 21, K = 2. The total number of distinct codes for this OOC family is M = 10, hence there are 9 additional OOC sequences that interfere with ~Ie, labeled in Table 3 ~l to ~9. The performance was simulated by generating random composite sequences from the set of codes ~l to ~9 arbitrarily shifted. All inputs are unipolar and clipped (0,1) signals. The results presented in Table 4 give examples of our simulation for the T CDMA matrix shown in Table 3. The input Q is the (logical) sum of a I-bit (vector ~Ie), plus interfering signals from arbitrarily shifted sequences of ~2, ~3, ~4, ~6 and ~9. The output of the neuromorphic network is seen to recover accurately the desired vector ~Ie. The input vector Q contains a O-bit (null vector), plus the shifted sequences of ~l, ~2, ~3, ~6, ~7 and ~8, and we see that the output correctly recovers a O-bit. As discussed in Section 4, our neuromorphic network will always correctly recognize a I-bit (vector ~Ie) presented to its input. On the other hand 2, there is the possibility of making an error when a O-bit is sent, and the interfering signals from other nodes happen to generate the chip positions of ~Ie. This case is shown by input vector ~ of Table 4, which contains a O-bit (null vector), plus shifted sequences of ~2, ~3, ~4, ~6, ~6, ~7 and ~8 in such a way that the output is erroneously given as a I-bit. The properties of the OOC sequences are specifically chosen to minimize these errors(7], and the statistical results of our simulation are also shown in Table 4. It is seen that, as expected, when a I-bit is sent it is always correctly recognized. On the other hand, when O-bits are sent, occasional errors occur. Our simulation, yields an overall bit error rate (BER) of BER.im = 5.88%, as shown in Table 4. These results can be compared with theoretical calculations[7] which yield an estimate for the BER for the CDMA system described: K-l ER 1 IT [ M-l-le] B calc~1-q , 2 Ie=O (14) where q = 1 ~. For the example of the OOC family F = 21, K = 2, with M = 10, the above expression yields BERcalc :::::: 5.74%. 20ur channel can be described, then, as a binary Z-channel between each two nodes dynamically establishing a communication path 820 It is seen, therefore, that our neuromorphic network approaches the minimum possible BER for a given family of OOC. In fact, the results obtained usin~ our T CDMA are equivalent CDMA detection scheme based on "optical-AND-gates,,[1 1, which corresponds to the limiting BER determined by the properties of the OOC themselves3 . The optical mask corresponding to the code filtering function is shown in Figure 3. 5.2 Other Functions As a first example of a non-symmetric T matrix, let us consider the function of mapping an input code to a corresponding different output code. We define our mapping matrix as: T;fAP = g {~Y'Zj} ; i,i = l,2 ... F, (15) where an input vector ~m will produce a different output vector code llm. The function of code joining is defined by a transfer function that takes a given input code and produces at the output a chosen combination of two or more codes. This function is performed by expressing the general matrix given by 12 as follows: T J01N r! {"( m m ) m} . . . - 1 2 F ij '!:I ~ Yi + wi + .. , Zj I &,1 , ... , (16) where an input vector ~m will result in an output that joins several vector codes (Ilm E9 wmffi ... ). The code shifting matrix TSHIFT will allow for the shift of a given code sequence, such that both input and output correspond to the same code, but shifted with respect to itself. That is, (17) where we have indicated an unshifted code sequence by ~(O)m, and its corresponding output pair as a shifted version of itself ~(s)m. The code projecting function corresponds to processing an input vector that contains the logical sum of several codes, and projecting at the output a selected single code Th din · T PROJ . . b sequence. e correspon g matrIx IS glven y: T PROJ r! {" m( m m )}... - 1 2 F ij -'!:I ~Zi Yj +Wj +... 1&,1-, ... , (18) where each input vector (~m ffi w m ffi ... ) will project at the output to a single code ~m. In general, the resulting output code sequence ~m could correspond to a code not necessarely contained in the input vector. The performance and error correcting properties of these, and other, general functions follow a similar behaviour as discussed in Section 4. 3The BER for the OOC family shown in this example are far too large for a useful CDMA communications system. Our choice intended to show computer simulated results within a reasonable computation time. 821 6 Conclusions The neuromorphic networks presented, based on sparse Optical Orthogonal Code (OOC) sequences, have been shown to have a number of attractive properties. The unipolar, clipped nature of the synaptic connectivity matrix simplifies the implementation. The single pass convergence further allows for general network functions that are expected to be of particular interest in communications and signal processing systems. The coding of the information, based on ~OC, has also been shown to result in high capacity associative memories. The combination of efficient associative memory properties, plus a variety of general network functions, also suggests the possible application of our neuromorphic networks in the implementation of computational functions based on optical symbolic substitution. The family of neuromorphic networks discussed here emphasizes the importance of understanding the general properties of non-negative systems based on sparse codes[lll. It is hoped that our results will stimulate further work on the fundamental relationship between coding, or representations, and the information processing properties of neural nets. Acknowledgement We thank J. Y. N. Hui and J. Alspector for many useful discussions, and C. A. Brackett for his support and encouragement of this research. References [1] S. Grossberg. In K. Schmitt, editor, Delay and Functional-Differential Equation6 and Their ApplicatioN, page 121, Academic Press, New York, NY, 1972. [2] J. J. Hopfield. Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proc. Nat. Acad. Sci. USA, 79:2254, 1982. [3] D. H. Ackley, G. E. Hinton, and T. J. Sejnowski. A Learning Algorithm for Boltzmann Machines. Cogn. Sci., 9:147, 1985. [4] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchio Optimization by Simulated Annealing. Science, 220:671, 1983. [5] M. P. Vecchi and S. Kirkpatrick. Global Wiring by Simulated Annealing. IEEE Tran6. CAD of Integrated Circuit. and Sydem6, CAD-2:215, 1983. [6] F. R. K. Chung, J. A. Salehi, and V. K. Wei. Optical Orthogonal Codes: Design, Analysis and Applications. In IEEE International Symp06ium on Information Theory, Catalog No. 86CH!374-7, 1986. Accepted for publication in IEEE Trans. on Information Theory. [7] J. A. Salehi and C. A. Brackett. Fundamental Principles of Fiber Optics Code Division Multiple Access. In IEEE International Conference on CommunicatiON, 1987. [8] N. H. Farhat, D. Psaltis, A. Prata, and E. Paek. Optical Implementation of the Hopfield Model. Appl. Opt., 24:1469, 1985. [9] R. J. McEliece, E. C. Posner, E. R. Rodemich, and S. S. Venkatesh. The Capacity of Hopfield Associative Memory. IEEE Tran6. on Information Theory, IT-33:461, 1987. [10] J. A. Salehi. Principles and Applications of Optical AND Gates in Fiber Optics Code Division Multiple Access Networks. In preparation, 1987. [11] G. Palm. Technical comments. Science, 235:1226, 1987. 822 T .... I: A_wi .. Yo..,. Eo ....... ~ ..... _ ... 0110 wI ... ... ~ .. ~IO ......... 0OCf ..... OOC J'anolq. r ., II. It = I OOC J'aaoi\r1 r I: II. It = I CodoVod_ I....,.",.'" Co4e ... 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I • • 1 • • • • • • • • • • • • • • , • • • • • , • • • I • • • • • • • • • • • • • • • • • • • .. • • • • • • • I • • • • 1 • • • • • • • • , • • I • • • • , :1 • • • • • • • • • " • • • • • • • • • • , • • • • • • I • • • • • • • • • • • • • • • , • • • • • • • 1 • , • • • • I • • • • • • • • I • • • • • • • I • • • • • , • • • • • • • • • I • • • • .' T .... I: A-o.&i .. 11-.,. ~ ~ ~ ... -_ .... ... ,..." _ria P. ;;. 'bWo I. 1IIr ............ a.-a.. ........... .....,._ r-..... 'bWo., c.. FI\orioc IC'DIIU). ~ so-IalioL ..... ___ ..... _ ... SOC"',, _ .. p_. ~ "'1. EMIl ............ .., ................. ... (' .... ~;~.--).""" .............. .. ooc ...." r • II, It .. I .... -. -'iInriIr ................... -. .... ~ .. v..... • I , • I • • • I • • • • • • • • • • .: OOC hili§! r. 1I,1t. I ............. W-_ ••• 0.1, .. V ...... • I , • • • • • • • • • • • • • • • • • • '1 .1"1""."'1 • • • • • • J L ..... V_ a" •••••••• • •••• • • • • • • l I • I • • I • I • • • • • I • • I • " ........... .u.'-"-, •• o..~v ...... r .... V_= _. ' •••• ..-••.• , .o' l I •••• '., •••• " • • I • • .. • • • • • • • I • • • • I • • • • • • • D :&CHl ••••••••••••••••••••• Ll., .. V_ C I • • I • • I • • I • I • • • I • • • I • IIIanmoI.& .. lot .... "- r:: • • 1 ............ 4Iot_"-~ ... I.loop .. v_ .. r ..... ~l.a-' .,' ••• " .-a-. I • a"! c T' •• I , • • I I , • , , • , , • • I , • Ou'_V __ Out,.. v ...... C • • • • • • • • , • • • • • I • • • • • • , ... • • • , • • I • • • • , • • • • • • • I • ~ .1····· ............. 0, s .. , ..... aI ..... no.. TogJ I I· WI ... 11 ................. , .... "-. I: I MIOJf N»I I_II Outp .. V ...... • ltol ltol UN •••••••••••••••••••• ' 0 BU" 0Il."" I ... Figure 1: Schematic diagram of an optical neuromorphic processor using sparse Optical Orthogonal Codes. Notice the absence oHeedback because ofthe single-pass convergence. The mask shown represents the realisation of the content-addressable memory of Table 1. 823 Figure 3: Optical realization oC a code filtering (CDMA) maslc o( Table 3. The l's are represented by the transpar' ent pixels, and the 0'. by the opaque pixels. Nkll PASSM STAll COUI'lEIIS Figure 2: Schematic diagram of a COMA communications system over an Optical Fiber interconnection network. Each node represents one of the M possible distinct users in the system.
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794 A 'Neural' Network that Learns to Play Backgammon G. Tesauro Center for Complex Systems Research, University of Illinois at Urbana-Champaign, 508 S. Sixth St., Champaign, IL 61820 T. J. Sejnowski Biophysics Dept., Johns Hopkins University, Baltimore, MD 21218 ABSTRACT We describe a class of connectionist networks that have learned to play backgammon at an intermediate-to-advanced level. TIle networks were trained by a supervised learning procedure on a large set of sample positions evaluated by a human expert. In actual match play against humans and conventional computer programs, the networks demonstrate substantial ability to generalize on the basis of expert knowledge. Our study touches on some of the most important issues in network learning theory, including the development of efficient coding schemes and training procedures, scaling, generalization, the use of real-valued inputs and outputs, and techniques for escaping from local minima. Practical applications in games and other domains are also discussed. INTRODUCTION A potentially quite useful testing ground for studying issues of knowledge representation and learning in networks can be found in the domain of game playing. Board games such as chess, go, backgammon, and Othello entail considerable sophistication and complexity at the advanced level, and mastery of expert concepts and strategies often takes years of intense study and practice for humans. However, the complexities in board games are embedded in relatively "clean" structured tasks with well-defined rules of play, and well-defined criteria for success and failure. This makes them amenable to automated play, and in fact most of these games have been exten')ively studied with conventional computer science techniques. Thus, direct comparisons of the results of network learning can be made with more conventional approaches. In this paper, we describe an application of network learning to the game of backgammon. Backgammon is a difficult board game which appears to be well-suited to neural networks, because the way in which moves are selected is primarily on the basis of pattern-recognition or "judgemental" reasoning, as opposed to explicit "look-ahead," or tree-search computations. This is due to the probabilistic dice rolls in backgammon, which greatly expand the branching factor at each ply in the search (to over 400 in typical positions). Our learning procedure is a supervised one 1 that requires a database of positions and moves that have been evaluated by an expert "teacher." In contrast, in an unsupervised procedure2-4 learning would be based on the consequences of a given move (e.g., whether it led to a won or lost position), and explicit teacher instructions would not be required. However, unsupervised learning procedures thus far have been much less efficient at reaching high levels of performance than supervised learning procedures. In part, this advantage of supervised learning can be traced to the higher © American Institute of Physics 1988 795 quantity and quality of information available from the teacher. Studying a problem of the scale and complexity of backgammon leads one to confront important general issues in network learning. Amongst the most important are scaling and generalization. Most of the problems that have been examined with connectionist learning algorithms are relatively small scale and it is not known how well they will perform on much larger problems. Generalization is a key issue in learning to play backgammon since it is estimated that there are 1020 possible board positions, which is far in excess of the number of examples that can be provided during training. In this respect our study is the most severe test of generalization in any connectionist network to date. We have also identified in this study a novel set of special techniques for training the network which were necessary to achieve good performance. A training set based on naturally occurring or random examples was not sufficient to bring the network to an advanced level of performance. Intelligent data-base design was necessary. Performance also improved when noise was added to the training procedure under some circumstances. Perhaps the most important factor in the success of the network was the method of encoding the input information. The best perfonnance was achieved when the raw input infonnation was encoded in a conceptually significant way, and a certain number of pre-computed features were added to the raw infonnation. These lessons may also be useful when connectionist learning algorithms are applied to other difficult large-scale problems. NElWORK AND DATA BASE SET-UP Our network is trained to select moves (i.e. to produce a real-valued score for any given move), rather than to generate them. This avoids the difficulties of having to teach the network the concept of move legality. Instead, we envision our network operating in tandem with a preprocessor which would take the board position and roll as input, and produce all legal moves as output. The network would be trained to score each move, and the system would choose the move with the highest network score. Furthermore, the network is trained to produce relative scores for each move, rather than an absolute evaluation of each final position. This approach would have greater sensitivity in distinguishing between close alternatives, and corresponds more closely to the way humans actually evaluate moves. The current data base contains a totaJ of 3202 board positions, taken from various sources5• For each position there is a dice roll and a set of legal moves of that roll from that pOSition. The moves receive commentary from a human expert in the form of a relative score in the range [100,+100), with +100 representing the best possible move and -100 representing the worst possible move. One of us (G.T.) is a strong backgammon player, and played the role of human expert in entering these scores. Most of the moves in the data base were not scored, because it is not feasible for a human expert to comment on all possible moves. (The handling of these unscored lines of data in the training procedure will be discussed in the following section.) An important result of our study is that in order to achieve the best perfonnance, the data base of examples must be intelligently designed, rather than haphazardly accumulated. If one simply accumulates positions which occur in actual game play, for example, one will find that certain principles of play will appear over and over again in these positions, while other important principles may be used only rarely. This causes problems for the network, as it tends to "overlearn" the commonly used principles, and not learn at aJl the rarely used principles. Hence it is necessary to have both an intelligent selection mechanism to reduce the number of over-represented situations, and an intelligent design mechanism to enhance the number of examples which illustrate under-represented situations. This process is described in more detail elsewhere5. We use a detenninistic, feed-forward network with an input layer, an output layer, and either one or two layers of hidden units, with full connectivity between adjacent layers. (We have tried a number of experiments with restricted receptive fields, and generally have not found them to be useful.) Since the desired output of the network is a single real value, only one output unit is required. 796 TIle coding of the input patterns is probably the most difficult and most important design issue. In its current configuration the input layer contains 459 input units. A location-based representation scheme is used, in which a certain number of input units are assigned to each of the 26 locations (24 basic plus White and Black bar) on the board. TIle input is inverted if necessary so that the network always sees a problem in which White is to play. An example of the coding scheme used until very recently is shown in Fig. I. This is essentially a unary encoding of the number of men at each board location, with a few exceptions as indicated in the diagram. This representation scheme worked fairly well, but had one peculiar problem in that after training, the network tended to prefer piling large numbers of men on certain points, in particular White's 5 point (the 20 point in the 1-24 numbering scheme). Fig. 2 illustrates an example of this peculiar behavior. In this position White is to play 5-1. Most humans would play 4-5,4-9 in this position; however, the network chose the move 4-9,19-20. This is actually a bad move, because it reduces White's chances of making further points in his inner board. The fault lies not with the data base used to train the network, but rather with the representation scheme used. In Fig. I a, notice that unit 12 is turned on whenever the final position is a point, and the number of men is different from the initial position. For the 20 point in particular, this unit will develop strong excitatory weights due to cases in which the initial position is not a point (i.e., the move makes the point). The 20 point is such a valuable point to make that the excitation produced by turning unit 12 on might overwhelm the inhibition produced by the poor distribution of builders. (0) ~-5 -4 -3 ~-2 -I o 0 000 I 2 3 4 5 I ~2 3 4 ~5 o • 0 0 0 6 7 8 9 10 o I ~2 3 4 ~5 o 0 • • 0 0 II 12 13 14 15 16 ( b) ~-5 -4 -3 5-2 -I o DOD 0 I 234 5 I ~2 3 4 >5 o • DOn 678910 o It I~ ~2t 2~ 3 4 ~5 o 000 0 .00 II 12 13 14 15 16 17 18 Figure 1-- Two schemes used to encode the raw position infonnation in the network's input. illustrated in each case is the encoding of two White men p~sent befo~ the move, and three White men p~Jent after the move. (a) An essentiaUy unary coding of the number of men at a particular board location. Units 1-10 encode the initial position, units 11-16 encode the final position if the~ has been a change from the initial position. Units are tumed on in the cases indicated on top of each unit, e.g., unit 1 is turned on if the~ are 5 or more Black men p~sent, etc .. (b) A superior coding scheme with more units u~ed to characterize the type of transition from initial to final position. An up arrow indicates an increase in the number of men. a down arrow indicates a decrease. Units 11-15 have conceptual interpretations: l1="dearing." 12="slotting," 13="b~aking," 14="making," 15="stripping" a point. 12 11 10 9 8 7 6 5 4 321 DO 13 14 15 16 17 18 19 20 21 22 23 24 Figure 2-- A sample position illustrating a defect of the coding scheme in Fig. 1a. White is to play 5-1. With coding scheme (1a). the network prefers 4-9. 19-20. With coding scheme (lb). the network prefers 4-9. 4-5. The graphic display was generated on a Sun Microsystems workstation using the Garnmontool program. 797 In conceptual tenns, humans would say that unit 12 participates in the representation of two different concepts: the concept of making a point, and the concept of changing the number of men occupying a made point. These two concepts are unrelated, and there is no point in representing them with a common input unit. A superior representation scheme in which these concepts are separated is shown in Fig. 1 b: In this representation unit 13 is turned on only for moves which make the point. Other moves which change the number of men on an already-made point do not activate unit 13, and thus do not receive any undeserved excitation. With this representation scheme the network no longer tends to pile large numbers of men on certain points, and its overall perfonnance is significantly better. In addition to this representation of the raw board position, we also utilize a number of input units to represent certain "pre-computed" features of the raw input. The principal goal of this study has been to investigate network learning, rather than simply to obtain high perfonnance, and thus we have resisted the temptation of including sophisticated hand-crafted features in the input encoding. However, we have found that a few simple features are needed in practice to obtain minimal standards of competent play. With only' 'raw" board infonnation, the order of the desired computation (as defined by Minsky and Papert6) is probably quite high, and the number of examples needed to learn such a difficult computation might be intractably large. By giving the network "hints" in the fonn of pre-computed features, this reduces the order of the computation, and thus might make more of the problem learnable in a tractable number of examples. 798 TRAINING AND TESTING PROCEDURES To train the network, we have used the standard "back-propagation" learning algorithm 7-9 for modifying the connections in a multilayer feed-forward network. (A detailed discussion of learning parameters, etc., is provided elsewheres.) However, our procedure differs from the standard procedure due to the necessity of dealing with the large number of uncommented moves in the data base. One solution would be simply to avoid presenting these moves to the network. However, this would limit the variety of input patterns presented to the network in training, and certain types of inputs probably would be eliminated completely. TIle alternative procedure which we have adopted is to skip the uncommented moves most of the time (75% for ordinary rolls and 92% for double rolls), and the remainder of the time present the pattern to the network and generate a random teacher signal with a slight negative bias. This makes sense, because if a move has not received comment by the human expert, it is more likely to be a bad move than a good move. The random teacher signal is chosen uniformly from the interval [-65,+35]. We have used the following four measures to assess the network's performance after it has been trained: (i) performance on the training data, (ii) perfonnance on a set of test data (1000 positions) which was not used to train the network, (iii) perfonnance in actual game play against a conventional computer program (the program Gammontool of Sun Microsystems Inc.), and (iv) performance in game play against a human expert (G.T.). In the first two measures, we define the performance as the fraction of positions in which the network picks the correct move, i.e., those positions for which the move scored highest by the network agrees with the choice of the human expert. In the latter two measures, the perfonnance is defined simply as the fraction of games won, without considering the complications of counting gammons or backgammons. QUANTITATIVE RESULTS A summary of our numerical results as measured by perfonnance on the training set and against Gammontool is presented in Table 1. The best network that we have produced so far appears to defeat Gammontool nearly 60% of the time. Using this as a benchmark, we find that the most serious decrease in performance occurs by removing aU pre-computed features from the input coding. This produces a network which wins at most about 41 % of the time. 'The next most important effect is the removal of noise from the training procedure; this results in a network which wins 45% of the time. Next in importance is the presence of hidden units; a network without hidden units wins about 50% of the games against Gammontool. In contrast, effects such as varying the exact number of hidden units, the number of layers, or the size of the training set, results in only a few (1-3) percentage point decrease in the number of games won. Also included in Table 1 is the result of an interesting experiment in which we removed our usual set of pre-computed features and substituted instead the individual tenns of the Gammontool evaluation function. We found that the resulting network, after being trained on our expert training set, was able to defeat the Gammontool program by a small margin of 54 to 46 percent. 'The purpose of this experiment was to provide evidence of the usefulness of network learning as an adjunct to standard AI techniques for hand-crafting evaluation functions. Given a set of features to be used in an evaluation function which have been designed, for example; by interviewing a human expert, the problem remains as to how to "tune" these features, i.e., the relative weightings to associate to each feature, and at an advanced level, the context in which each feature is relevant. Little is known in general about how to approach this problem, and often the human programmer must resort to painstaking trial-and-error tuning by hand. We claim that network learning is a powerful, generaJpurpose, automated method of approaching this problem, and has the potentiaJ to produce a tuning which is superior to those produced by humans, given a data base of sufficiently high quality, and a suitable scheme for encoding the features. The result of our experiment provides evidence to support this claim, although it is not firmly established since we do not have highly accurate statistics, and we do not know how much human effort went into the tuning of the Gammontool evaluation 799 function. More conclusive evidence would be provided if the experiment were repeated with a more sophisticated program such as Berliner's BKOlO, and similar results were obtained. Network Training Perf. on Perf. va. Comments sIZe cycles test set Ga.mmontool (a) 459-24-24-1 20 .540 .59 ± .03 (b) 459-24-1 22 .542 .57 ± .05 (c) 459-24-1 24 .518 . 58 ± .05 1600 posn. D.B . (d) 459-12-1 10 .538 .54 ± .05 (e) 410-24-12-1 16 .493 .54 ± .03 Gammontool features (f) 459-1 22 .485 .50 ± .03 No hidden units (g) 459-24-12-1 10 .499 .45 ± .03 No training noise (h) 393-24-12-1 12 .488 .41 ± .02 No features Table 1-- Summary of perfonnance statistics for various networks. (a) The best network we have produced. containing two layers of hidden units. with 24 units in each layer. (b) A network with only one layer of 24 hidden units. (c) A network with 24 hidden units in a single layer, trained on a training set half the nonnal size. (d) A network with half the number of hidden units as in (b). (e) A network with features from the Gammontool evaluation function substituted for the nonnal features. (f) A network without hidden units. (g) A network trained with no noise in the training procedure. (h) A network with only a raw board description as input. QUALITATIVE RESULTS Analysis of the weights produced by training a network is an exceedingly difficult problem, which we have only been able to approach qualitatively. In Fig. 3 we present a diagram showing the connection strengths in a network with 651 input units and no hidden units. lbe figure shows the weights from each input unit to the output unit. (For purposes of illustration, we have shown a coding scheme with more units than nonnal to explicitly represent the transition from initial to final position.) Since the weights go directly to the output, the corresponding input units can be clearly interpreted as having either an overall excitatory or inhibitory effect on the score produced by the network. A great deal of columnar structure is apparent in Fig. 3. This indicates that the network has learned that a particular number of men at a given location, or a particular type of transition at a given location, is either good or bad independent of the exact location on the board where it occurs. Furthennore, we can see the importance of each of the pre-computed features in the input coding. The most significant features seem to be the number of points made in the network's inner board, and the total blot exposure. 800 features { roll { ........ bar 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 ABC 0 E F G H I J K L M N 0 P Q R STU V W Figure 3-- A Hinton diagram for a network with 651 input units and no hidden units. Small squares indicate weights from a particular input unit to the output unit. White squares indicate positive weights, and black squares indicate negative weights. Size of square indicates magnitude of weight. rust 24 rows from bottom up indicate raw board infonnation. Letting x=number of men before the move and y=number of men after the move, the interpretations of columns are as follows: A: x<=-5; B: x=-4; C: x=-3; D: x<=-2; E: x=-I: F: x=l: G: x>=2; H: x=3: I: x=4: J: x>=5: K: x<1 & y=l; L: x<2 & y>=2: M: x<3 & y=3: N: x<4 & y=4: 0: x<y & y>=5: P: x=1 & y=O; Q: x>=2 & y=O; R: x>=2 & y=l: S: x>=3 & y=2: T: x>=4 & y=3: U: x>=5 & y=4: V: x>y & y>=5: W: prob. of a White blot at thi~ location being hit (precomputed feature). The next row encodes number of men on White and Blnck bar~. The next 3 rows encode roll infonnation. Remaining rows encode various pre-computed feature~. Much insight into the basis for the network's judgement of various moves has been gained by actually playing games against it. In fact, one of the most revealing tests of what the network has and has not learned came from a 20-game match played by G.T. against one of the latest generation of networks with 48 hidden units. (A detailed description of the match is given in Ref. II.) The surprising result of this match was that the network actually won, 11 games to 9. However, a 801 detailed analysis of the moves played by the network during the match indicates that the network was extremely lucky to have won so many games, and could not reasonably be expected to continue to do so well over a large number of games. Out of the 20 games played, there were 11 in which the network did not make any serious mistakes. The network won 6 out of these 11 games, a result which is quite reasonable. However, in 9 of the 20 games, the network made one or more serious (i.e. potentially fatal) "blunders." The seriousness of these mistakes would be equivalently to dropping a piece in chess. Such a mistake is nearly always fatal in chess against a good opponent; however in backgammon there are still chances due to the element of luck involved. In the 9 games in which the network blundered, it did manage to survive and win 5 of the games due to the element of luck. (We are assuming that the mistakes made by the human, if any, were only minor mistakes.) It is highly unlikely that this sort of result would be repeated. A much more likely result would be that the network would win only one or two of the games in which it made a serious error. This would put the network's expected perfonnance against expert or near-expert humans at about the 35-40% level. (This has also been confinned in play against other networks.) We find that the network does act as if it has picked up many of the global concepts and strategies of advanced play. The network has also learned many important tactical elements of play at the advanced level. As for the specific kinds of mistakes made by the network, we find that they are not at all random, senseless mistakes, but instead fall into clear, well-defined conceptual categories, and furthennore, one can understand the reasons why these categories of mistakes are made. We do not have space here to describe these in detail, and refer the reader instead to Ref. 5. To summarize, qualitative analysis of the network's play indicates that it has learned many important strategies and tactics of advanced backgammon. This gives the network very good overall perfonnance in typical positions. However, the network's worst case perfonnance leaves a great deal to be desired. The network is capable of making both serious, obvious, "blunders," as well more subtle mistakes, in many different types of positions. Worst case perfonnance is important, because the network must make long sequences of moves throughout the course of a game without any serious mistakes in order to have a reasonable chance of winning against a skilled opponent. The prospects for improving the network's worst case perfonnance appear to be mixed. It seems quite likely that many of the current "blunders" can be fixed with a reasonable number of handcrafted examples added to the training set. However, many of the subtle mistakes are due to a lack of very sophisticated knowledge, such as the notion of timing. It is difficult to imagine that this kind of knowledge could be imparted to the network in only a few examples. Probably what is required is either an intractably large number of examples, or a major overhaul in either the pre-computed features or the training paradigm. DISCUSSION We have seen from both quantitative and qualitative measures that the network has learned a great deal about the general principles of backgammon play, and has not simply memorized the individual positions in the training set. Quantitatively, the measure of game perfonnace provides a clear indication of the network's ability to generalize, because apart from the first couple of moves at the start of each game, the network must operate entirely on generalization. Qualitatively, one can see after playing several games against the network that there are certain characteristic kinds of positions in which it does well, and other kinds of positions in which it systematically makes welldefined types of mistakes. Due to the network's frequent "blunders," its overall level of play is only intennediate level, although it probably is somewhat better than the average intennediate-Ievel player. Against the intennediate-level program Gammontool, our best network wins almost 60% of the games. However, against a human expert the network would only win about 35-40% of the time. Thus while the network does not play at expert level, it is sufficiently good to give an expert a hard time, and with luck in its favor can actually win a match to a small number of games. Our simple supervised learning approach leaves out some very important sources of 802 infonnation which are readily available to humans. 1be network is never told that the underlying topological structure of its input space really corresponds to a one-dimensional spatial structure; all it knows is that the inputs form a 459-dimensional hypercube. It has no idea of the object of the game, nor of the sense of temporal causality, i.e. the notion that its actions have consequences, and how those consequences lead to the achievement of the objective. The teacher signal only says whether a given move is good or bad, without giving any indication as to what the teacher's reasons are for making such a judgement. Finally, the network is only capable of scoring single moves in isolation, without any idea of what other moves are available. 1bese sources of knowledge are essential to the ability of humans to play backgammon well, and it seems likely that some way of incorporating them into the network learning paradigm will be necessary in order to achieve further substantial improvements in performance. 111ere are a number of ways in which these additional sources of knowledge might be incorporated, and we shall be exploring some of them in future work. For example, knowledge of alternative moves could be introduced by defining a more sophisticated error signal which takes into account not only the network and teacher scores for the current move, but also the network and teacher scores for other moves from the same position. However, the more immediate plans involve a continuation of the existing strategies of hand-crafting examples and coding scheme modifications to eliminate the most serious errors in the network's play. If these errors can be eliminated, and we are confident that this can be achieved, then the network would become substantially better than any commercially available program, and would be a serious challenge for human experts. We would expect 65% performance against Gammontool, and 45% performance against human experts. Some of the results of our study have implications beyond backgammon to more general classes of difficult problems. One of the limitations we have found is that substantial human effort is required both in the design of the coding scheme and in the design of the training set. It is not sufficient to use a simple coding scheme and random training patterns, and let the automated network learning procedure take care of everything else. We expect this to be generally true when connectionist learning is applied to difficult problem domains. On the positive side, we foresee a potential for combining connectionist learning techniques with conventional AI techniques for hand-crafting knowledge to make significant progress in the development of intelligent systems. From the practical point of view, network learning can be viewed as an "enhancer" of traditional techniques, which might produce systems with superior perfonnance. For this particular application, the obvious way to combine the two approaches is in the use of pre-computed features in the input encoding. Any set of hand-crafted features used in a conventional evaluation function could be encoded as discrete or continuous activity levels of input units which represent the current board state along with the units representing the raw information. Given a suitable encoding scheme for these features, and a training set of sufficient size and quality (i.e., the scores in the training set should be better than those of the original evaluation function), it seems possible that the resulting network could outperform the original evaluation function, as evidenced by our experiment with the Gammontool features. Networlc learning might also hold promise as a means of achieving the long-sought goal of automated feature discovery2. Our network certainly appears to have learned a great deal of knowledge from the training set which goes far beyond the amount of knowledge that was explicitJy encoded in the input features. Some of this knowledge (primarily the lowest level components) is apparent from the weight diagram when there are no hidden units (Fig. 3). However, much of the network's knowledge remains inaccessible. What is needed now is a mean'! of disentangling the novel features discovered by the network from either the patterns of activity in the hidden units, or from the massive number of connection strengths which characterize the network. This is one our top priorities for future research, although techniques for such "reverse engineering" of parallel networlcs are only beginning to be developedl2. 803 ACKNOWLEDGEMrnNTS lhis work was inspired by a conference on "Evolution, Games and Learning" held at Los Alamos National Laboratory, May 20-24, 1985. We thank Sun Microsystems Inc. for providing the source code for their Gammontool program, Hans Berliner for providing some of the po!>itions used in the data base, Subutai Ahmad for writing the weight display graphics package, Bill Bogstad for assistance in programming the back-propagation simulator, and Bartlett Mel, Peter Frey, and Scott Kirkpatrick for critical reviews of the manuscript. G.T. was supponed in part by the National Center for Supercomputing Applications. TJ.S. was supponed by a NSF Presidential Young Investigator Award, and by grants from the Seaver Institute and the Lounsbury Foundation. REFERENCES 1. D. E. Rumelart and J. L. McClelland, eds., Parallel Distributed Processing: Explorations in the Microstructure o/Cognition, Vols. 1 and 2 (Cambridge: MIT Press, 1986). 2. A. L. Samuel, "Some studies in machine learning using the game of checkers." IBM J. Res. Dev. 3, 210--229 (1959). 3. J. H. Holland, "Escaping brittleness: the possibilities of general-purpose learning algorithms applied to parallel rule-based systems." In: R. S. Michalski et aI. (eds.), Machine learning: an artificial ;ntelligence approach. Vol. II (Los Altos CA: Morgan-Kaufman, 1986). 4. R. S. Sutton, "Learning to predict by the methods of temporal differences," GTE Labs Tech. Repon TR87-509.1 (1987). 5. G. Tesauro and T. J. Sejnowski, "A parallel network that learns to play backgammon." Univ. of Illinois at Urbana-Champaign, Center for Complex Systems Research Technical Repon (1987). 6. M. Minsky and S. Papen, Perceptrons (Cambridge: MIT Press, 1969). 7. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, "Learning representations by backpropagating errors." Nature 323,533--536 (1986). 8. Y. Le Cun, "A learning procedure for asymmetric network." Proceedings o/Cognitiva (Par;s) 85,599--604 (1985). 9. D. B. Parker, "Learning-logic." MIT Center for Computational Research in Economics and Management Science Tech. Repon TR-47 (1985). 10. H. Berliner, "Backgammon computer program beats world champion." Artificial Intelligence 14,205--220 (1980). 11. G. Tesauro, "Neural network defeats creator in backgammon match." Univ. of llIinois at Urbana-Champaign, Center for Complex Systems Research Technical Repon (1987). 12. C. R. Rosenberg, "Revealing the structure of NETtalk's internal representations." Proceedings of the Ninth Annual Conference of the Cognitive Science Society (Hillsdale, NJ: Lawrence Erlbaum Associates, 1987).
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CONNECTIVITY VERSUS ENTROPY Yaser S. Abu-Mostafa California Institute of Technology Pasadena, CA 91125 ABSTRACT 1 How does the connectivity of a neural network (number of synapses per neuron) relate to the complexity of the problems it can handle (measured by the entropy)? Switching theory would suggest no relation at all, since all Boolean functions can be implemented using a circuit with very low connectivity (e.g., using two-input NAND gates). However, for a network that learns a problem from examples using a local learning rule, we prove that the entropy of the problem becomes a lower bound for the connectivity of the network. INTRODUCTION The most distinguishing feature of neural networks is their ability to spontaneously learn the desired function from 'training' samples, i.e., their ability to program themselves. Clearly, a given neural network cannot just learn any function, there must be some restrictions on which networks can learn which functions. One obvious restriction, which is independent of the learning aspect, is that the network must be big enough to accommodate the circuit complexity of the function it will eventually simulate. Are there restrictions that arise merely from the fact that the network is expected to learn the function, rather than being purposely designed for the function? This paper reports a restriction of this kind. The result imposes a lower bound on the connectivity of the network (number of synapses per neuron). This lower bound can only be a consequence of the learning aspect, since switching theory provides purposely designed circuits of low connectivity (e.g., using only two-input NAND gates) capable of implementing any Boolean function [1,2]. It also follows that the learning mechanism must be restricted for this lower bound to hold; a powerful mechanism can be © American Institute of Physics 1988 2 designed that will find one of the low-connectivity circuits (perhaps byexhaustive search), and hence the lower bound on connectivity cannot hold in general. Indeed, we restrict the learning mechanism to be local; when a training sample is loaded into the network, each neuron has access only to those bits carried by itself and the neurons it is directly connected to. This is a strong assumption that excludes sophisticated learning mechanisms used in neural-network models, but may be more plausible from a biological point of view. The lower bound on the connectivity of the network is given in terms of the entropy of the environment that provides the training samples. Entropy is a quantitative measure of the disorder or randomness in an environment or, equivalently, the amount of information needed to specify the environment. There are many different ways to define entropy, and many technical variations of this concept [3]. In the next section, we shall introduce the formal definitions and results, but we start here with an informal exposition of the ideas involved. The environment in our model produces patterns represented by N bits x = Xl ••• X N (pixels in the picture of a visual scene if you will). Only h different patterns can be generated by a given environment, where h < 2N (the entropy is essentially log2 h). No knowledge is assumed about which patterns the environment is likely to generate, only that there are h of them. In the learning process, a huge number of sample patterns are generated at random from the environment and input to the network, one bit per neuron. The network uses this information to set its internal parameters and gradually tune itself to this particular environment. Because of the network architecture, each neuron knows only its own bit and (at best) the bits of the neurons it is directly connected to by a synapse. Hence, the learning rules are local: a neuron does not have the benefit of the entire global pattern that is being learned. After the learning process has taken place, each neuron is ready to perform a function defined by what it has learned. The collective interaction of the functions of the neurons is what defines the overall function of the network. The main result of this paper is that (roughly speaking) if the connectivity of the network is less than the entropy of the environment, the network cannot learn about the environment. The idea of the proof is to show that if the connectivity is small, the final function of each neuron is independent of the environment, and hence to conclude that the overall network has accumulated no information about the environment it is supposed to learn about. FORMAL RESULT A neural network is an undirected graph (the vertices are the neurons and the edges are the synapses). Label the neurons 1"", N and define Kn C {I"", N} to be the set of neurons connected by a synapse to neuron n, together with neuron n itself. An environment is a subset e C {O,I}N (each x E e is a sample 3 from the environment). During learning, Xl,"', xN (the bits of x) are loaded into the neurons 1"", N, respectively. Consider an arbitrary neuron nand relabel everything to make Kn become {I"", K}. Thus the neuron sees the first K coordinates of each x. Since our result is asymptotic in N, we will specify K as a function of N; K = a.N where a. = a.(N) satifies limN-+oo a.(N) = 0.0 (0 < 0.0 < 1). Since the result is also statistical, we will consider the ensemble of environments e e=e(N)={eC{O,I}N I lel=h} where h = 2~N and /3 = /3(N) satifies limN-+oo /3(N) = /30 (0 < /30 < 1). The probability distribution on e is uniform; any environment e E e is as likely to occur as any other. The neuron sees only the first K coordinates of each x generated by the environment e. For each e, we define the function n : {O,I}K -+ {O, 1,2,··.} where n(al" .aK) = I{x Eel Xle = ale for k = 1,'" ,K}I and the normalized version The function v describes the relative frequency of occurrence for each of the 2K binary vectors Xl'" XK as x = Xl ••• XN runs through all h vectors in e. In other words, v specifies the projection of e as seen by the neuron. Clearly, veal > 0 for all a E {O,l}K and LaE{O,l}K veal = 1. Corresponding to two environments el and e2, we will have two functions VI and V2. IT VI is not distinguishable from V2, the neuron cannot tell the difference between el and e2' The distinguishability between VI and V2 can be measured by 1 d(Vl,V2) = 2: IV1(a) - V2(a) I 2 aE{O,l}K The range of d(Vb V2) is 0 < d(Vl' V2) < 1, where '0' corresponds to complete indistinguishability while '1' corresponds to maximum distinguishability. We are now in a position to state the main result. Let el and e2 be independently selected environments from e according to the uniform probability distribution. d(Vl' V2) is now a random variable, and we are interested in the expected value E(d(Vl' V2))' The case where E(d(Vb V2)) = 0 corresponds to the neuron getting no information about the environment, while the case where E(d(Vb V2)) = 1 corresponds to the neuron getting maximum information. The theorem predicts, in the limit, one of these extremes depending on how the connectivity (0.0) compares to the entropy (/30)' 4 Theorem. 1. H Q o > Po , then limN ..... co E (d(VI, V2)) = 1. 2. H Q o < Po , then limN ..... co E (d(v}, V2)) = O. The proof is given in the appendix, but the idea is easy to illustrate informally. Suppose h = 2K +10 (corresponding to part 2 of the theorem). For most environments e E e, the first K bits of x E e go through all 2K possible values approximately 210 times each as x goes through all h possible values once. Therefore, the patterns seen by the neuron are drawn from the fixed ensemble of all binary vectors of length K with essentially uniform probability distribution, i.e., v is the same for most environments. This means that, statistically, the neuron will end up doing the same function regardless of the environment at hand. What about the opposite case, where h = 2K - 10 (corresponding to part lof the theorem)? Now, with only 2K - 10 patterns available from the environment, the first K bits of x can assume at most 2K- 10 values out of the possible 2K values a binary vector of length K can assume in principle. Furthermore, which values can be assumed depends on the particular environment at hand, i.e., v does depend on the environment. Therefore, although the neuron still does not have the global picture, the information it has says something about the environment. ACKNOWLEDGEMENT This work was supported by the Air Force Office of Scientific Research under Grant AFOSR-86-0296. APPENDIX In this appendix we prove the main theorem. We start by discussing some basic properties about the ensemble of environments e. Since the probability distribution on e is uniform and since Ie I = e:), we have ( 2N)-1 Pr(e) = h which is equivalent to generating e by choosing h elements x E {O,l}N with uniform probability (without replacement). It follows that h Pr(x E e) = 2N and so on. h h-l Pr(Xl E e , X2 E e) = 2N X 2N _ 1 5 The functions n and v are defined on K-bit vectors. The statistics of n(a) (a random variable for fixed a) is independent of a Pr(n(at} = m) = Pr(n(a2) = m) which follows from the symmetry with respect to each bit of a. The same holds for the statistics of v(a). The expected value E(n(a)) = h2-K (h objects going into 2K cells), hence E(v(a)) = 2-K . We now restate and prove the theorem. Theorem. 1. If ao > Po , then limN_oo E (d(vt, V2)) = 1. 2. If ao < Po , then limN_oo E (d(vt, V2)) = 0. Proof. We expand E (d(vt, V2)) as follows where nl and n2 denote nl(O. ··0) and n2(0·· ·0), respectively, and the last step follows from the fact that the statistics of nl(a) and n2(a) is independent of a. Therefore, to prove the theorem, we evaluate E(lnl - n21) for large N. 1. Assume ao > Po. Let n denote n(O··· 0), and consider Pr(n = 0). For n to be zero, all 2N - K strings x of N bits starting with K O's must not be in the environment e. Hence h h h Pr(n = 0) = (1 - -) (1 ) ... (1 ) 2N 2N - 1 2N 2N- K + 1 where the first term is the probability that 0· . ·00 f/. e, the second term is the 6 probability that O· .. 01 ~ f given that o· .. 00 ~ f, and so on. > (1- 2N _h2N_K )'N-K = (1- h2-N(1- 2-K)-1) 2N - K > (1 - 2h2-N)2N - K > 1- 2h2-N 2N - K = 1- 2h2-K Hence, Pr(nl = 0) = Pr(n2 = 0) = Pr(n = 0) > 1 - 2h2-K • However, E(nl) = E( n2) = h2-K. Therefore, " " E(lnl - n2\) = LLPr(nl = i,n2 = j)li - jl i=O;=O " " = L L Pr(nl = i)Pr(n2 = j) Ii - jl i=O;=O " > L Pr(nl = 0)Pr(n2 = j)j ;=0 " + L Pr(nl = i)Pr(n2 = O)i i=O which follows by throwing away all the terms where neither i nor j is zero (the term where both i an j are zero appears twice for convenience, but this term is zero anyway). = Pr(nl = 0)E(n2) + Pr(n2 = O)E(nl) > 2(1 - 2h2-K )h2-K Substituting this estimate in the expression for E(d(Vb V2)), we get 2K E(d(vl, V2)) = 2h E(lnl - n21) 2K > x 2(1 - 2h2-K )h2-K 2h = 1- 2h2-K = 1 - 2 X 2(,8-a)N Since ao > 130 by assumption, this lower bound goes to 1 as N goes to infinity. Since 1 is also an upper bound for d( VI, V2) (and hence an upper bound for the expected value E(d(vl, V2))) , limN_oo E(d(vl, V2)) must be 1. 2. Assume ao < Po. Consider E(lnl - n21) = E (I(nl - h2-K) - (n2 - h2-K )I) < E(\nl - h2- K \ + In2 - h2-KI) = E(\nl - h2- K I) + E(ln2 - h2-K I) = 2E(ln - h2-KI) 7 To evaluate E(ln - h2-K I), we estimate the variance of n and use the fact that E(ln - h2- K I) < ..jvar(n) (recall that h2-K = E(n»). Since var(n) = E(n2) - (E(n))2, we need an estimate for E(n2). We write n = E.E{O,l}N-K 6., where 6 , , { 1 if 0 .. ·Oa E e· • 0, otherwise. In this notation, E(n2 ) can be written as E(n2) = E (I: I: 6.6t,) .E{O,l}N-K bE{O,l}N-K I: L E(6.6t,) .E{O,l}N-K bE{O,l}N-K For the 'diagonal' terms (a = b), E(6.6.) = Pr(6. = 1) = h2-N There are 2N - K such diagonal terms, hence a total contribution of 2N - K x h2-N = h2- K to the sum. For the 'off-diagonal' terms (a '# b), E(6.6b ) = Pr( 6. = 1,6b = 1) = Pr(6. = 1)Pr(6b = 116. = 1) h h-l =-x--::-:::-2N 2N_1 There are 2N - K (2N - K -1) such off-diagonal terms, hence a total contribution of 2N - K(2N - K -1) x 2;~:N~1) < (h2-K)2 2~~1 to the sum. Putting the contributions 8 from the diagonal and off-diagonal terms together, we get 2N E(n2) < h2-K + (h2-K)2 2N _ 1 var(n) = E(n2) - (E(n))2 < (h2- K + (h2- K )' 2:: 1) - (h2- K )' 1 = h2-K + (h2- K)2----:-:-_ 2N -1 = h2-K 1 + ---:-:-( h2- K ) 2N -1 < 2h2-K The last step follows since h2-K is much smaller than 2N -1. Therefore, E(ln1 h2-KI) < vvar(n) < (2h2- K)?i. Substituting this estimate in the expression for E( d( Vb V2)), we get 2K E(d(vb V2)) = 2h E(lnl - n21) 2K < 2h x 2E(ln - h2-KI) 2K 1 < 2h x 2 x (2h2-K)?i _ ( 2K) ~ 2h = v'2 X 2~(Q-~)N Since ao < Po by assumption, this upper bound goes to 0 as N goes to infinity. Since 0 is also a lower bound for d(vb V2) (and hence a lower bound for the expected value E(d(vb V2))), limN_oo E(d(vb V2)) must be O .• REFERENCES [1] Y. Abu-Mostafa, "Neural networks for computing?," AlP Conference Proceedings # 151, Neural Networks for Computing, J. Denker (ed.), pp. 1-6, 1986. [2] Z. Kohavi, Switching and Finite Automata Theory, McGraw-Hill, 1978. [3] Y. Abu-Mostafa, "The complexity of information extraction," IEEE Trans. on Information Theory, vol. IT-32, pp. 513-525, July 1986. [4] Y. Abu-Mostafa, "Complexity in neural systems," in Analog VLSI and Neural Systems by C. Mead, Addison-Wesley, 1988.
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358 LEARNING REPRESENTATIONS BY RECIRCULATION Geoffrey E. Hinton Computer Science and Psychology Departments, University of Toronto, Toronto M5S lA4, Canada James L. McClelland Psychology and Computer Science Departments, Carnegie-Mellon University, Pittsburgh, PA 15213 ABSTRACT We describe a new learning procedure for networks that contain groups of nonlinear units arranged in a closed loop. The aim of the learning is to discover codes that allow the activity vectors in a "visible" group to be represented by activity vectors in a "hidden" group. One way to test whether a code is an accurate representation is to try to reconstruct the visible vector from the hidden vector. The difference between the original and the reconstructed visible vectors is called the reconstruction error, and the learning procedure aims to minimize this error. The learning procedure has two passes. On the fust pass, the original visible vector is passed around the loop, and on the second pass an average of the original vector and the reconstructed vector is passed around the loop. The learning procedure changes each weight by an amount proportional to the product of the "presynaptic" activity and the difference in the post-synaptic activity on the two passes. This procedure is much simpler to implement than methods like back-propagation. Simulations in simple networks show that it usually converges rapidly on a good set of codes, and analysis shows that in certain restricted cases it performs gradient descent in the squared reconstruction error. INTRODUCTION Supervised gradient-descent learning procedures such as back-propagation 1 have been shown to construct interesting internal representations in "hidden" units that are not part of the input or output of a connectionist network. One criticism of back-propagation is that it requires a teacher to specify the desired output vectors. It is possible to dispense with the teacher in the case of "encoder" networks2 in which the desired output vector is identical with the input vector (see Fig. 1). The purpose of an encoder network is to learn good "codes" in the intermediate, hidden units. If for, example, there are less hidden units than input units, an encoder network will perform data-compression3. It is also possible to introduce other kinds of constraints on the hidden units, so we can view an encoder network as a way of ensuring that the input can be reconstructed from the activity in the hidden units whilst also making nus research was supported by contract NOOOl4-86-K-00167 from the Office of Naval Research and a grant from the Canadian National Science and Engineering Research Council. Geoffrey Hinton is a fellow of the Canadian Institute for Advanced Research. We thank: Mike Franzini, Conrad Galland and Geoffrey Goodhill for helpful discussions and help with the simulations. © American Institute of Physics 1988 359 the hidden units satisfy some other constraint. A second criticism of back-propagation is that it is neurally implausible (and hard to implement in hardware) because it requires all the connections to be used backwards and it requires the units to use different input-output functions for the forward and backward passes. Recirculation is designed to overcome this second criticism in the special case of encoder networks. output units I \ hidden units / r-. input units Fig. 1. A diagram of a three layer encoder network that learns good codes using back-propagation. On the forward pass, activity flows from the input units in the bottom layer to the output units in the top layer. On the backward pass, errorderivatives flow from the top layer to the bottom layer. Instead of using a separate group of units for the input and output we use the very same group of "visible" units, so the input vector is the initial state of this group and the output vector is the state after information has passed around the loop. The difference between the activity of a visible unit before and after sending activity around the loop is the derivative of the squared reconstruction error. So, if the visible units are linear, we can perfonn gradient descent in the squared error by changing each of a visible unit's incoming weights by an amount proportional to the product of this difference and the activity of the hidden unit from which the connection emanates. So learning the weights from the hidden units to the output units is simple. The harder problem is to learn the weights on connections coming into hidden units because there is no direct specification of the desired states of these units. Back-propagation solves this problem by back-propagating error-derivatives from the output units to generate error-derivatives for the hidden units. Recirculation solves the problem in a quite different way that is easier to implement but much harder to analyse. 360 THE RECIRCULATION PROCEDURE We introduce the recirculation procedure by considering a very simple architecture in which there is just one group of hidden units. Each visible unit has a directed connection to every hidden unit, and each hidden unit has a directed connection to every visible unit. The total input received by a unit is Xj = LYiWji - 9j (1) i where Yi is the state of the ith unit, K'ji is the weight on the connection from the ith to the Jib unit and 9j is the threshold of the Jh unit. The threshold tenn can be eliminated by giving every unit an extra input connection whose activity level is fIXed at 1. The weight on this special connection is the negative of the threshold, and it can be learned in just the same way as the other weights. This method of implementing thresholds will be assumed throughout the paper. The functions relating inputs to outputs of visible and hidden units are smooth monotonic functions with bounded derivatives. For hidden units we use the logistic function: y. = <1(x.) = I J J I +e-Xj (2) Other smooth monotonic functions would serve as well. For visible units, our mathematical analysis focuses on the linear case in which the output equals the total input, though in simulations we use the logistic function. We have already given a verbal description of the learning rule for the hiddento-visible connections. The weight, Wij , from the Ih hidden unit to the itlr visible unit is changed as follows: f:t.wij = £y/I) [Yi(O)-Yi(2)] (3) where Yi(O) is the state of the ith visible unit at time 0 and Yi(2) is its state at time 2 after activity has passed around the loop once. The rule for the visible-to-hidden connections is identical: (4) where y/I) is the state of the lh hidden unit at time I (on the frrst pass around the loop) and y/3) is its state at time 3 (on the second pass around the loop). Fig. 2 shows the network exploded in time. In general, this rule for changing the visible-to-hidden connections does not perfonn steepest descent in the squared reconstruction error, so it behaves differently from back-propagation. This raises two issues: Under what conditions does it work, and under what conditions does it approximate steepest descent? time = 1 time = 3 time =0 time =2 Fig. 2. A diagram showing the states of the visible and hidden units exploded in time. The visible units are at the bottom and the hidden units are at the top. Time goes from left to right. CONDITIONS UNDER WHICH RECIRCULATION APPROXIMATES GRADIENT DESCENT 361 For the simple architecture shown in Fig. 2, the recirculation learning procedure changes the visible-to-hidden weights in the direction of steepest descent in the squared reconstruction error provided the following conditions hold: 1. The visible units are linear. 2. The weights are symmetrical (i.e. wji=wij for all i,j). 3. The visible units have high regression. "Regression" means that, after one pass around the loop, instead of setting the activity of a visible unit, i, to be equal to its current total input, xi(2), as determined by Eq 1, we set its activity to be y;(2) = AY;(O) + (I-A)x;(2) (5) where the regression, A, is close to 1. Using high regression ensures that the visible units only change state slightly so that when the new visible vector is sent around the loop again on the second pass, it has very similar effects to the first pass. In order to make the learning rule for the hidden units as similar as possible to the rule for the visible units, we also use regression in computing the activity of the hidden units on the second pass (6) For a given input vector, the squared reconstruction error, E, is For a hidden unit, j, 362 where For a visible-to ... hidden weight wj ; dE, dE = Yj(1)Yi(O)-dwj ; dYj(l) So, using Eq 7 and the assumption that Wkj=wjk for all k,j dE = y/(l) y;(O) [LYk(2) Yk'(2) Wjk - LYk(O) Yk'(2) Wjk] dw·· k k }l The assumption that the visible units are linear (with a gradient of 1) means that for all k, Yk'(2) = 1. So using Eq 1 we have dE = y.'(l) y.(O)[x.(3)-x~1)] dw .. } I ) } }l Now, with sufficiently high regression, we can assume that the states of units only change slightly with time so that and Yt(O) ::::: y;(2) So by substituting in Eq 8 we get dE 1 -aw-ji ::::: (1 _ A) y;(2) [y/3) - y/l)] (8) (9) An interesting property of Eq 9 is that it does not contain a tenn for the gradient of the input-output function of unit } so recirculation learning can be applied even when unit} uses an unknown non-linearity. To do back-propagation it is necessary to know the gradient of the non-linearity, but recirculation measures the gradient by measuring the effect of a small difference in input, so the tenn y/3)-y/l) implicitly contains the gradient. 363 A SIMULATION OF RECIRCULATION From a biological standpoint, the synunetry requirement that wij=Wji is unrealistic unless it can be shown that this synunetry of the weights can be learned. To investigate what would happen if synunetry was not enforced (and if the visible units used the same non-linearity as the hidden units), we applied the recirculation learning procedure to a network with 4 visible units and 2 hidden units. The visible vectors were 1000, 0100, 0010 and 0001, so the 2 hidden units had to learn 4 different codes to represent these four visible vectors. All the weights and biases in the network were started at small random values uniformly distributed in the range -0.5 to +0.5. We used regression in the hidden units, even though this is not strictly necessary, but we ignored the teon 1/ (1 - A) in Eq 9. Using an E of 20 and a A. of 0.75 for both the visible and the hidden units, the network learned to produce a reconstruction error of less than 0.1 on every unit in an average of 48 weight updates (with a maximum of 202 in 100 simulations). Each weight update was perfonned after trying all four training cases and the change was the sum of the four changes prescribed by Eq 3 or 4 as appropriate. The final reconstruction error was measured using a regression of 0, even though high regression was used during the learning. The learning speed is comparable with back-propagation, though a precise comparison is hard because the optimal values of E are different in the two cases. Also, the fact that we ignored the tenn 1/ (1-A.) when modifying the visible-to-hidden weights means that recirculation tends to change the visible-to-hidden weights more slowly than the hidden-to-visible weights, and this would also help back -propagation. It is not inunediately obvious why the recirculation learning procedure works when the weights are not constrained to be synunetrical, so we compared the weight changes prescribed by the recirculation procedure with the weight changes that would cause steepest descent in the sum squared reconstruction error (i.e. the weight changes prescribed by back-propagation). As expected, recirculation and backpropagation agree on the weight changes for the hidden-to-visible connections, even though the gradient of the logistic function is not taken into account in weight adjustments under recirculation. (Conrad Galland has observed that this agreement is only slightly affected by using visible units that have the non-linear input-output function shown in Eq 2 because at any stage of the learning, all the visible units tend to have similar slopes for their input-output functions, so the non-linearity scales all the weight changes by approximately the same amount.) For the visible-to-hidden connections, recirculation initially prescribes weight changes that are only randomly related to the direction of steepest descent, so these changes do not help to improve the perfonnance of the system. As the learning proceeds, however, these changes come to agree with the direction of steepest descent. The crucial observation is that this agreement occurs after the hidden-tovisible weights have changed in such a way that they are approximately aligned (symmetrical up to a constant factor) with the visible-to-hidden weights. So it appears that changing the hidden-to-visible weights in the direction of steepest descent creates the conditions that are necessary for the recirculation procedure to cause changes in the visible-to-hidden weights that follow the direction of steepest descent. It is not hard to see why this happens if we start with random, zero-mean 364 visible-to-hidden weights. If the visible-to-hidden weight wji is positive, hidden unit j will tend to have a higher than average activity level when the ith visible unit has a higher than average activity. So Yj will tend to be higher than average when the reconstructed value of Yi should be higher than average -- i.e. when the tenn [Yi(O)-Yi(2)] in Eq 3 is positive. It will also be lower than average when this tenn is negative. These relationships will be reversed if wji is negative, so w ij will grow faster when wJi is positive than it will when wji is negative. Smolensky4 presents a mathematical analysis that shows why a similar learning procedure creates symmetrical weights in a purely linear system. Williams5 also analyses a related learning rule for linear systems which he calls the "symmetric error correction" procedure and he shows that it perfonns principle components analysis. In our simulations of recirculation, the visible-to-hidden weights become aligned with the corresponding hidden-to-visible weights, though the hidden-to-visible weights are generally of larger magnitude. A PICTURE OF RECIRCULATION To gain more insight into the conditions under which recirculation learning produces the appropriate changes in the visible-to-hidden weights, we introduce the pictorial representation shown in Fig. 3. The initial visible vector, A, is mapped into the reconstructed vector, C, so the error vector is AC. Using high regression, the visible vector that is sent around the loop on the second pass is P, where the difference vector AP is a small fraction of the error vector AC. If the regression is sufficiently high and all the non-linearities in the system have bounded derivatives and the weights have bounded magnitudes, the difference vectors AP, BQ, and CR will be very small and we can assume that, to first order, the system behaves linearly in these difference vectors. If, for example, we moved P so as to double the length of AP we would also double the length of BQ and CR. Fig. 3. A diagram showing some vectors (A, P) over the visible units, their "hidden" images (B, Q) over the hidden units, and their "visible" images (C, R) over the visible lUlits. The vectors B' and C' are the hidden and visible images of A after the visible-to-hidden weights have been changed by the learning procedure. 365 Suppose we change the visible-to-hidden weights in the manner prescribed by Eq 4, using a very smaIl value of £. Let Q' be the hidden image of P (i.e. the image of P in the hidden units) after the weight changes. To first order, Q' will lie between B and Q on the line BQ. This follows from the observation that Eq 4 has the effect of moving each y/3) towards y/l) by an amount proportional to their difference. Since B is close to Q, a weight change that moves the hidden image of P from Q to Q' will move the hidden image of A from B to B', where B' lies on the extension of the line BQ as shown in Fig. 3. If the hidden-to-visible weights are not changed, the visible image of A will move from C to C', where C' lies on the extension of the line CR as shown in Fig. 3. So the visible-to-hidden weight changes will reduce the squared reconstruction error provided the vector CR is approximately parallel to the vector AP. But why should we expect the vector CR to be aligned with the vector AP? In general we should not, except when the visible-to-hidden and hidden-to-visible weights are approximately aligned. The learning in the hidden-to-visible connections has a tendency to cause this alignment. In addition, it is easy to modify the recirculation learning procedure so as to increase the tendency for the learning in the hidden-to-visible connections to cause alignment. Eq 3 has the effect of moving the visible image of A closer to A by an amount proportional to the magnitude of the error vector AC. If we apply the same rule on the next pass around the loop, we move the visible image of P closer to P by an amount proportional to the magnitude of PRo If the vector CR is anti-aligned with the vector AP, the magnitude of AC will exceed the magnitude of PR, so the result of these two movements will be to improve the alignment between AP and CR. We have not yet tested this modified procedure through simulations, however. This is only an infonnal argument and much work remains to be done in establishing the precise conditions under which the recirculation learning procedure approximates steepest descent. The infonnal argument applies equally well to systems that contain longer loops which have several groups of hidden units arranged in series. At each stage in the loop, the same learning procedure can be applied, and the weight changes will approximate gradient descent provided the difference of the two visible vectors that are sent around the loop aligns with the difference of their images. We have not yet done enough simulations to develop a clear picture of the conditions under which the changes in the hidden-to-visible weights produce the required alignment. USING A HIERARCHY OF CLOSED LOOPS Instead of using a single loop that contains many hidden layers in series, it is possible to use a more modular system. Each module consists of one "visible" group and one "hidden" group connected in a closed loop, but the visible group for one module is actually composed of the hidden groups of several lower level modules, as shown in Fig. 4. Since the same learning rule is used for both visible and hidden units, there is no problem in applying it to systems in which some units are the visible units of one module and the hidden units of another. Ballard6 has experimented with back-propagation in this kind of system, and we have run some simulations of recirculation using the architecture shown in Fig. 4. The network 366 learned to encode a set of vectors specified over the bottom layer. After learning, each of the vectors became an attractor and the network was capable of completing a partial vector, even though this involved passing information through several layers. 00 00 00 0000 0000 Fig 4. A network in which the hidden units of the bottom two modules are the visible units of the top module. CONCLUSION We have described a simple learning procedure that is capable of fonning representations in non-linear hidden units whose input-output functions have bounded derivatives. The procedure is easy to implement in hardware, even if the non-linearity is unknown. Given some strong assumptions, the procedure petforms gradient descent in the reconstruction error. If the synunetry assumption is violated, the learning procedure still works because the changes in the hidden-to-visible weights produce symmetry. H the assumption about the linearity of the visible units is violated, the procedure still works in the cases we have simulated. For the general case of a loop with many non-linear stages, we have an informal picture of a condition that must hold for the procedure to approximate gradient descent, but we do not have a fonnal analysis, and we do not have sufficient experience with simulations to give an empirical description of the general conditions under which the learning procedure works. REFERENCES 1. D. E. Rumelhart, G. E. Hinton and R.I. Williams, Nature 323, 533-536 (1986). 2. D. H. Ackley, G. E. Hinton and T. 1. Sejnowski, Cognitive Science 9,147-169 (1985). 3. G. Cottrell, 1. L. Elman and D. Zipser, Proc. Cognitive Science Society, Seattle, WA (1987). 4. P. Smolensky, Technical Report CU-CS-355-87, University of Colorado at Boulder (1986). 5. R.I. Williams, Technical Report 8501, Institute of Cognitive Science, University ofCalifomia, San Diego (1985). 6. D. H. Ballard, Proc. American Association for Artificial Intelligence, Seattle, W A (1987).
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715 A COMPUTER SIMULATION OF CEREBRAL NEOCORTEX: COMPUTATIONAL CAPABILITIES OF NONLINEAR NEURAL NETWORKS Alexander Singer* and John P. Donoghue** *Department of Biophysics, Johns Hopkins University, Baltimore, MD 21218 (to whom all correspondence should be addressed) **Center for Neural Science, Brown University, Providence, RI 02912 © American Institute of Physics 1988 716 A synthetic neural network simulation of cerebral neocortex was developed based on detailed anatomy and physiology. Processing elements possess temporal nonlinearities and connection patterns similar to those of cortical neurons. The network was able to replicate spatial and temporal integration properties found experimentally in neocortex. A certain level of randomness was found to be crucial for the robustness of at least some of the network's computational capabilities. Emphasis was placed on how synthetic simulations can be of use to the study of both artificial and biological neural networks. A variety of fields have benefited from the use of computer simulations. This is true in spite of the fact that general theories and conceptual models are lacking in many fields and contrasts with the use of simulations to explore existing theoretical structures that are extremely complex (cf. MacGregor and Lewis, 1977). When theoretical superstructures are missing, simulations can be used to synthesize empirical findings into a system which can then be studied analytically in and of itself. The vast compendium of neuroanatomical and neurophysiological data that has been collected and the concomitant absence of theories of brain function (Crick, 1979; Lewin, 1982) makes neuroscience an ideal candidate for the application of synthetic simulations. Furthennore, in keeping with the spirit of this meeting, neural network simulations which synthesize biological data can make contributions to the study of artificial neural systems as general infonnation processing machines as well as to the study of the brain. A synthetic simulation of cerebral neocortex is presented here and is intended to be an example of how traffic might flow on the two-way street which this conference is trying to build between artificial neural network modelers and neuroscientists. The fact that cerebral neocortex is involved in some of the highest fonns of information processing and the fact that a wide variety of neurophysiological and neuroanatomical data are amenable to simulation motivated the present development of a synthetic simulation of neocortex. The simulation itself is comparatively simple; nevertheless it is more realistic in tenns of its structure and elemental processing units than most artificial neural networks. The neurons from which our simulation is constructed go beyond the simple sigmoid or hard-saturation nonlinearities of most artificial neural systems. For example, 717 because inputs to actual neurons are mediated by ion currents whose driving force depends on the membrane potential of the neuron. the amplitude of a cell's response to an input. i.e. the amplitude of the post-synaptic potential (PSP). depends not only on the strength of the synapse at which the input arrives. but also on the state of the neuron at the time of the input's arrival. This aspect of classical neuron electrophysiology has been implemented in our simulation (figure lA). and leads to another important nonlinearity of neurons: namely. current shunting. Primarily effective as shunting inhibition. excitatory current can be shunted out an inhibitory synapse so that the sum of an inhibitory postsynaptic potential and an excitatory postsynaptic potential of equal amplitude does not result in mutual cancellation. Instead. interactions between the ion reversal potentials. conductance values. relative timing of inputs. and spatial locations of synapses determine the amplitude of the response in a nonlinear fashion (figure IB) (see Koch. Poggio. and Torre. 1983 for a quantitative analysis). These properties of actual neurons have been ignored by most artificial neural network designers. though detailed knowledge of them has existed for decades and in spite of the fact that they can be used to implement complex computations (e.g. Torre and Poggio. 1978; Houchin. 1975). The development of action potentials and spatial interactions within the model neurons have been simplified in our simulation. Action potentials involve preprogrammed \ fluctuations in the membrane potential of our neurons and result in an absolute and a relative refractory period. Thus. during the time a cell is firing a spike synaptic inputs are ignored. and immediately following an action potential the neuron is hyperpolarized. The modeling of spatial interactions is also limited since neurons are modeled primarily as spheres. Though the spheres can be deformed through control of a synaptic weight which modulates the amplitudes of ion conductances. detailed dendritic interactions are not simulated. Nonetheless. the fact that inhibition is generally closer to a cortical neuron's soma while excitation is more distal in a cell's dendritic tree is simulated through the use of stronger inhibitory synapses and relatively weaker excitatory synapses. The relative strengths of synapses in a neural network define its connectivity. Though initial connectivity is random in many artificial networks. brains can be thought to contain a combination of randomness and fixed structure at distinct levels (Szentagothai. 1978). From a macroscopic perspective. all of cerebral neocortex might be structured in a modular fashion analogous to the way the barrel field of mouse somatosensory cortex is structured (Woolsey and Van der Loos. 1970). Though speculative, arguments for the existence of some sort of anatomical modularity over the entire cortex are gaining ground 718 (Mountcastle, 1978; Szentagothai, 1979; Shepherd, in press). Thus, inspired by the barrels of mice and by growing interest in functional units of 50 to 100 microns with on the order of 1000 neurons, our simulation is built up of five modules (60 cells each) with more dense local interconnections and fewer intermodular contacts. Furthermore, a wide variety of neuronal classification schemes have led us to subdivide the gross structure of each module so as to contain four classes of neurons: cortico-cortical pyramids, output pyramids, spiny stellate or local excitatory cells, and GABAergic or inhibirtory cells. At this level of analysis, the impressed structure allows for control over a variety of pathways. In our simulation each class of neurons within a module is connected to every other class and intermodular connections are provided along pathways from corti co-cortical pyramids to inhibitory cells, output pyramids, and cortico-cortical pyramids in immediately adjacent modules. A general sense of how strong a pathway is can be inferred from the product of the number of synapses a neuron receives from a particular class and the strength of each of those synapses. The broad architecture of the simulation is further structured to emphasize a three step path: Inputs to the network impact most strongly on the spiny stellate cells of the module receiving the input; these cells in tum project to cortico-cortical pyramidal cells more strongly than they do to other cell types; and finally, the pathway from the cortico-cortical pyramids to the output pyramidal cells of the same module is also particularly strong. This general architecture (figure 2) has received empirical support in many regions of cortex (Jones, 1986). In distinction to this synaptic architecture, a fine-grain connectivity is defined in our simulated network as well. At a more microscopic level, connectivity in the network is random. Thus, within the confines of the architecture described above, the determination of which neuron of a particular class is connected to which other cell in a target class is done at random. Two distinct levels of connectivity have, therefore, been established (figure 3). Together they provide a middle ground between the completely arbitrary connectivity of many artificial neural networks and the problem specific connectivities of other artificial systems. This distinction between gross synaptic architecture and fine-grain connectivity also has intuitive appeal for theories of brain development and, as we shall see, has non-trivial effects on the computational capabilities of the network as a whole. With defintions for input integration within the local processors, that is within the neurons, and with the establishment of connectivity patterns, the network is complete and ready to perform as a computational unit. In order to judge the simulation's capabilities in some rough way, a qualitative analysis of its response to an input will suffice. Figure 4 719 shows the response of the network to an input composed of a small burst of action potentials arriving at a single module. The data is displayed as a raster in which time is mapped along the abscissa and all the cells of the network are arranged by module and cell class along the ordinate. Each marker on the graph represents a single action potential fIred by the appropriate neuron at the indicated time. Qualitatively, what is of importance is the fact that the network does not remain unresponsive, saturate with activity in all neurons, or oscillate in any way. Of course, that the network behave this way was predetermined by the combination of the properties of the neurons with a judicious selection of synaptic weights and path strengths. The properties of the neurons were fixed from physiological data, and once a synaptic architecture was found which produced the results in figure 4, that too was fixed. A more detailed analysis of the temporal firing pattern and of the distribution of activity over the different cell classes might reveal important network properties and the relative importance of various pathways to the overall function. Such an analysis of the sensitivity of the network to different path strengths and even to intracellular parameters will, however, have to be postponed. Suffice it to say at this point that the network, as structured, has some nonzero, finite, non-oscillatory response which, qualitatively, might not offend a physiologist judging cortical activity. Though the synaptic architecture was tailored manually and fixed so as to produce "reasonable" results, the fine-grain connectivity, i.e. the determination of exactly which cell in a class connects to which other cell, was random. An important property of artificial (and presumably biological) neural networks can be uncovered by exploiting the distinction between levels of connectivity described above. Before doing so, however, a detail of neural network design must be made explicit. Any network, either artificial or biological, must contend with the time it takes to communicate among the processing elements. In the brain, the time it takes for an action potential to travel from one neuron to another depends on the conduction velocity of the axon down which the spike is traveling and on the delay that occurs at the synapse connecting the cells. Roughly, the total transmission time from one cortical neuron to another lies between 1 and 5 milliseconds. In our simulation two 720 paradigms were used. In one case, the transmission times between all neurons were standardized at 1 msec.* Alternatively, the transmission times were fixed at random, though admittedly unphysiological, values between 0.1 and 2 msec. Now, if the time it takes for an action potential to travel from one neuron to another were fixed for all cells at 1 msec, different fine-grain connectivity patterns are found to produce entirely distinct network responses to the same input, in spite of the fact that the gross synaptic architecture remained constant. This was true no matter what particular synaptic architecture was used. If, on the other hand, one changes the transmission times so that they vary randomly between 0.1 and 2 msec, it becomes easy to find sets of synaptic strengths that were robust with respect to changes in the fine-grain connectivity. Thus, a wide search of path strengths failed to produce a network which was robust to changes in fine-grain connectivity in the case of identical transmission times, while a set of synaptic weights that produced robust responses was easy to find when the transmission times were randomized. Figure 5 summarizes this result. In the figure overall network activity is measured simply as the total number of action potentials generated by pyramidal cells during an experiment and robustness can be judged as the relative stability of this response. The abscissa plots distinct experiments using the same synaptic architecture with different fine-grain connectivity patterns. Thus, though the synaptic architecture remains constant, the different trials represent changes in which particular cell is connected to which other cell. The results show quite dramatically that the network in which the transmission times are randomly distributed is more robust with respect to changes in fine-grain connectivity than the network in which the transmission times are all 1 msec. It is important to note that in either case, both when the network was robust and when changes of fine-grain connectivity produced gross changes in network output, the synaptic architectures produced outputs like that in figure 4 with some fine-grain connectivities. If the response of the network to an input can be considered the result of * Because neurons receive varying amounts of input and because integration is performed by summating excitatory and inhibitory postsynaptic potentials in a nonlinear way, the time each neuron needs to summate its inputs and produce an action potential varies from neuron to neuron and from time to time. This then allows for asynchronous fuing in spite of the identical transmission times. 721 some computation, figure 5 reveals that the same computational capability is not robust with respect to changes in fine-grain connectivity when transmission times between neurons are all 1 msec, but is more robust when these times are randomized. Thus, a single computational capability, viz. a response like that in figure 4 to a single input, was found to exist in networks with different synaptic architectures and different transmission time paradigms; this computational capability, however, varied in terms of its robustness with respect to changes in fine-grain connectivity when present in either of the transmission time paradigms. A more complex computational capability emerged from the neural network simulation we have developed and described. If we label two neighboring modules C2 and C3, an input to C2 will suppress the response of C3 to a second input at C3 if the second input is delayed. A convenient way of representing this spatio-temporal integration property is given in figure 6. The ordinate plots the ratio of the normal response of one module (say C3) to the response of the module to the same input when an input to a neighboring module (say C2) preceeds the input to the original module (C3). Thus, a value of one on the ordinate means the earlier spatially distinct input had no effect on the response of the module in which this property is being measured. A value less than one represents suppression, while values greater than one represent enhancement. On the abscissa, the interstimulus interval is plotted. From figure 6, it can be seen that significant suppression of the pyramidal cell output, mostly of the output pyramidal cell output, occurs when the inputs are separated by 10 to 30 msec. This response can be characterized as a sort of dynamic lateral inhibition since an input is suppressing the ability of a neighboring region to respond when the input pairs have a particular time course. This property could playa variety of role in biological and artificial neural networks. One role for this spatio-temporal integration property, for example, might be in detecting the velocity of a moving stimulus. The emergent spatio-temporal property of the network just described was not explicitly built into the network. Moreover, no set of synaptic weights was able to give rise to this computational capability when transmission times were all set to 1 msec. Thus, in addition to providing robustness, the random transmission times also enabled a more complex property to emerge. The important factor in the appearances of both the robustness and the dynamic lateral inhibition was randomization; though it was implemented as randomly varying transmission times, random spontaneous activity would have played the same role. From the viewpoint, then, of the engineer designing artificial neural networks, the neural network presented here has instructional value in spite of the 722 fact that it was designed to synthesize biological data. Specifically, it motivates the consideration of randomness as a design constraint. From the prespective of the biologists attending this meeting, a simple fact will reveal the importance of synthetic simulations. The dynamic lateral inhibition presented in figure 6 is known to exist in rat somatosensory cortex (Simons, 1985). By deflecting the whiskers on a rat's face, Simons was able to stimulate individual barrels of the posteromedial somatosensory barrel field in combinations which revealed similar spatio-temporal interactions among the responses of the cortical neurons of the barrel field. The temporal suppression he reported even has a time course similar to that of the simulation. What the experiment did not reveal, however, was the class of cell in which suppression was seen; the simulation located most of the suppression in the output pyramidal cells. Hence, for a biologist, even a simple synthetic simulation like the one presented here can make defmitive predictions. What differentiates the predictions made by synthetic simulations from those of more general artificial neural systems, of course, is that the strong biological foundations of synthetic simulations provide an easily grasped and highly relevant framework for both predictions and experimental verification. One of the advertised purposes of this meeting was to "bring together neurobiologists, cognitive psychologists, engineers, and physicists with common interest in natural and artificial neural networks." Towards that end, synthetic computer simulations, i.e. simulations which follow known neurophysiological and neuroanatomical data as if they comprised a complex recipe, can provide an experimental medium which is useful for both biologists and engineers. The simulation of cerebral neocortex developed here has information regarding the role of randomness in the the robustness and presence of various computational capabilities as well as information regarding the value of distinct levels of connectivity to contribute to the design of artificial neural networks. At the same time, the synthetic nature of the network provides the biologist with an environment in which he can test notions of actual neural function as well as with a system which replicates known properties of biological systems and makes explicit predictions. Providing twoway interactions, synthetic simulations like this one will allow future generations of artificial neural networks to benefit from the empirical findings of biologists, while the slowly evolving theories of brain function benefit from the more generalizable results and methods of engineers. 723 References Crick, F. H. C. (1979) Thinking about the brain, Scientific American, 241:219 - 232. Houchin,1. (1975) Direction specificity in cortical responses to moving stimuli -- a simple model. Proceedings of the Physiological Society, 247:7 - 9. Jones, E. G. (1986) Connectivity of primate sensory-motor cortex, in Cerebral Cortex, vol. 5, E. G. Jones and A. Peters (eds), Plenum Press, New York. Koch, C., Poggio, T., and Torre, V. (1983) Nonlinear interactions in a dendritic tree: Localization, timing, and role in information processing. Proceedings of the National Academy of Science, USA, 80:2799 - 2802. Lewin, R. (1982) Neuroscientists look for theories, Science, 216:507. MacGregor, R.I. and Lewis, E.R. (1977) Neural Modeling, Plenum Press, New York. Mountcastle, V. B. (1978) An organizing principle for cerebral function: The unit module and the distributed system, in The Mindful Brain, G. M. Edelman and V. B. Mountcastle (eds.), MIT Press, Cambridge, MA. Shepherd, G.M. (in press) Basic circuit of cortical organization, in Perspectives in Memory Research, M.S. Gazzaniga (ed.). MIT Press, Cambridge, MA. Simons, D. J. (1985) Temporal and spatial integration in the rat SI vibrissa cortex, Journal of Neurophysiology, 54:615 - 635. Szenthagothai,1. (1978) Specificity versus (quasi-) randomness in cortical connectivity, in Architectonics of the Cerebral Cortex, M. A. B. Brazier and H. Petsche (eds.), Raven Press, New York. Szentagothai, J. (1979) Local neuron circuits in the neocortex, in The Neurosciences. Fourth Study Program, F. O. Schmitt and F. G. Worden (eds.), MIT Press, Cambridge, MA. Torre, V. and Poggio, T. (1978) A synaptic mechanism possibly underlying directional selectivity to motion, Proceeding of the Royal Society (London) B, 202:409 -416. Woolsey, T.A. and Van der Loos, H. (1970) Structural organization of layer IV in the somatosensory region (SI) of mouse cerebral cortex, Brain Research, 17:205-242. 724 Shunting Inhibition Simultaneous EPSP & IPSP IPSP Figure IA: Intracellular records of post-synaptic potentials resulting from single excitatory and inhibitory inputs to cells at different resting potentials. PSP Amplitude Dependence on Membrane Potential Resting Potential • ·40 mV EPSPs Resting Potential • -60 mV rResting Potential _ -80 mV L-_____ _ Resting I Potential C'=:-::----100 mV .... ------Resting ~ Potential _ -120 mV c= Resting Potential • 40 mV IPSPs Resting Potential 20 mV rResting Potential OmV Resting I Potential c----:----·20 mV L.. ______ _ Resting ~ Potential -40mV ~ Figure IB: Illustration of the current shunting nonlinearity present in the model neurons. Though the simultaneous arrival of postsynaptic potentials of equal and opposite amplitude would result in no deflection in the membrane potential of a simple linear neuron model, a variety of factors contribute to the nonlinear response of actual neurons and of the neurons modeled in the present simulation. """" '" '" ""'" ',," .................................. .. ;luu ,~, '; " , . , " ,~ ~ :-:"~:"::::::. C.a.lls :::: ""'XX"""" "," , . , , , , . , , , , . , , , , . , .. , ... , . .. .... .. Output Pyramids ~." '.'. ~' .. " .. " .. "' .. "" .. " .. ' .. .' .. " .. '.'.' .'."' .. ' .. ' ... ' .. '," ~,,; " ;, . . ,~ ~;. , , \. ~~ , , , , , " , ~ , .. , .. , .. , .. , .. , .. , .. , ........ , .. , .. , .. , .. , .. , .. , .. , .. ",,:.,'.,',,-.,-,,'.,'.,'.,-,,-,,-.,-,,',,-.,',,'.,'.,'.,'.,-.,'.,'.,-,,-,,'.,-.,-,,-,,-,,-.,',,',,',,'.,-,,',,'.,-.,-,,',, -,,-.,',,'.,'.,-,,',,',,-,,-.,-,,',,',,-.,'.,'., -.,',,',,', ...... .. ) .'.'.'.'.'.' .' .' Input Figure 2: A schematic representation of the simulated cortical network. Five modules are used, each containing sixty neurons. Neurons are divided into four classes. Numerals within the caricatured neurons represent the number of cells in that particular class that are simulated. Though all cell classes are connected to all other classes, the pathway from input to spiny stellate to cortico-cortical pyramids to output pyramids is particularly strong. ....::J ~ 01 726 Path Strength Number of synapses X ~,.,.....,.,. ... of 6. 6. 0 6,. 6. 0 6,. 6.6. 0 6 6. 0 6. 6. 0 6,. 6 6. 0 0 ° ° 6. ? 6. 6. 6 6 0°0 0 6.6. 6. 0 6~ ° ° t:.. ° ° Output Inhibitory Intracorlical Spiny Stellate Pyramidal Cells Cells Pyramidal Cells Cells Figure 3: Two levels of connectivity are defined in the network. Gross synaptic architecture is defined among classes of cells. Fine-grain connectivity specifies which cell connects to which other cell and is determined at random. Module S Module 4 Module 3 Module 2 Module 1 Sample Raster Input: 333 Hz input, 6 rns duration applied to Module 3 -.... . . . ~ ." . : . : . . . I . 10 ..... I 20 Time (ms) -Cortico-c ortical ramids --- py -............ Spin Inhibitory cells -Output pyr y stellate cells amids I 30 727 Figure 4: Sample response of the entire network to a small burst of action potentials delivered to module 3. 728 Robustness With Respect to Connectivity Pattern Synaptic Architecture Constant 400 III • I/) 300 c: 0 c. • C/) III a: Q) \ Delay times = 1 ms () iU 200 "C 'E • ~ >. a.. • iU "0 ~ 100 • • jDeray times random • • • I • • • • • • • • • • • • • Individual Trials with Different Fine-grain Connectivity Patterns Figure 5: Plot of an arbitrary activity measure (total spike activity in all pyramidal cells) versus various instatiations ofthe same connectional architecture. Along the abscissa are represented the different fine-grained patterns of connectivity within a fixed connectional architecture. In one case the conductance times between all cells was I msec and in the other case the times were selected at random from values between 0.1 msec and 2 msec. This experiment shows the greater overall stability produced by random conduction times. 2 Spatio-Temporal Integration Properties Q) fY. ... Outpyr '" c: ... C·Cpyr 0 a. .. Sst '" Q) -+GABA a: Q) > '-;a ~ a: Randomized Axonal Conduction Times o I ~ o 20 40 60 80 100 120 Interstimulus Interval Figure 6: Spatio-temporal integration within the network. Plot of the time course of response suppression in the various cell classes. The ordinate plots the ratio of average cell activity (in terms of spikes) to a direct input after the presentation of an input to a neighboring mod ule, and the average reponse to an input in the absence of prior input to an adjacent module. Values greater than one represent an enhancement of activity in response to the spatially distinct preceeding input, while values less than one represent a suppression of the normal reponse. The abscissa plots the interstimulus interval. Note that the response suppression is most striking in only one class of cells. -.J ~ t:O
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674 P A 'ITERN CLASS DEGENERACY IN AN UNRESTRICfED STORAGE DENSITY MEMORY Christopher L. Scofield, Douglas L. Reilly, Charles Elbaum, Leon N. Cooper Nestor, Inc., 1 Richmond Square, Providence, Rhode Island, 02906. ABSTRACT The study of distributed memory systems has produced a number of models which work well in limited domains. However, until recently, the application of such systems to realworld problems has been difficult because of storage limitations, and their inherent architectural (and for serial simulation, computational) complexity. Recent development of memories with unrestricted storage capacity and economical feedforward architectures has opened the way to the application of such systems to complex pattern recognition problems. However, such problems are sometimes underspecified by the features which describe the environment, and thus a significant portion of the pattern environment is often non-separable. We will review current work on high density memory systems and their network implementations. We will discuss a general learning algorithm for such high density memories and review its application to separable point sets. Finally, we will introduce an extension of this method for learning the probability distributions of non-separable point sets. INTRODUcnON Information storage in distributed content addressable memories has long been the topic of intense study. Early research focused on the development of correlation matrix memories 1, 2, 3, 4. Workers in the field found that memories of this sort allowed storage of a number of distinct memories no larger than the number of dimensions of the input space. Further storage beyond this number caused the system to give an incorrect output for a memorized input. @ American Institute of Physics 1988 675 Recent work on distributed memory systems has focused on single layer, recurrent networks. Hopfield 5, 6 introduced a method for the analysis of settling of activity in recurrent networks. This method defined the network as a dynamical system for which a global function called the 'energy' (actually a Liapunov function for the autonomous system describing the state of the network) could be defined. Hopfield showed that flow in state space is always toward the fixed points of the dynamical system if the matrix of recurrent connections satisfies certain conditions. With this property, Hopfield was able to define the fixed points as the sites of memories of network acti vity. Like its forerunners, the Hopfield network is limited in storage capacity. Empirical study of the system found that for randomly chosen memories, storage capacity was limited to m ~ O.lSN, where m is the number of memories that could be accurately recalled, and N is the dimensionality of the network (this has since been improved to m ~ N, 7, 8). The degradation of memory recall with increased storage density is directly related to the proliferation in the state space of unwanted local minima which serve as basins of flow. UNRESTRICIEn STORAGE DENSITY MEMORIES Bachman et al. 9 have studied another relaxation system similar in some respects to the Hopfield network. However, in contrast to Hopfield, they have focused on defining a dynamical system in which the locations of the minima are explicitly known. In particular, they have chosen a system with a Liapunov function given by E = -IlL ~ Qj I Il- Xj I - L, J (1) where E is the total 'energy' of the network, Il (0) is a vector describing the initial network activity caused by a test pattern, and Xj' the site of the jth memory, for m memories in RN. L is a parameter related to the network size. Then 1l(0) relaxes to Il(T) = Xj for some memory j according to 676 (2) This system is isomorphic to the classical electrostatic potential between a positive (unit) test charge, and negative charges Qj at the sites Xj (for a 3-dimensional input space, and L = 1). The Ndimensional Coulomb energy function then defines exactly m basins of attraction to the fixed points located at the charge sites Xj. It can been shown that convergence to the closest distinct memory is guaranteed, independent of the number of stored memories m, for proper choice of Nand L 9, to. Equation 1 shows that each cell receives feedback from the network in the form of a scalar ~ Q-I Jl- x-I- L J J J • (3) Importantly, this quantity is the same for all cells; it is as if a single virtual cell was computing the distance in activity space between the current state and stored states. The result of the computation is then broadcast to all of the cells in the network. A 2-layer feedforward network implementing such a system has been described elsewhere 10. The connectivity for this architecture is of order m·N, where m is the number of stored memories and N is the dimensionality of layer 1. This is significant since the addition of a new memory m' = m + 1 will change the connectivity by the addition of N + 1 connections, whereas in the Hopfield network, addition of a new memory requires the addition of 2N + 1 connections. An equilibrium feedforward network with similar properties has been under investigation for some time 11. This model does not employ a relaxation procedure, and thus was not originally framed in the language of Liapunov functions. However, it is possible to define a similar system if we identify the locations of the 'prototypes' of this model as· the locations in state space of potentials which satisfy the following conditions Ej = -Qj lRo for I j.t - Xj I < Aj (4) = 0 for I fl - Xj I > A]. 677 where Ro is a constant. This form of potential is often referred to as the 'square-well' potential. This potential may be viewed as a limit of the Ndimensional Coulomb potential, in which the l/R (L = l) well is replaced with a square well (for which L » l). Equation 4 describes an energy landscape which consists of plateaus of zero potential outside of wells with flat, zero slope basins. Since the landscape has only flat regions separated by discontinuous boundaries, the state of the network is always at equilibrium, and relaxation does not occur. For this reason, this system has been called an equilibrium model. This model, also referred to as the Restricted Coulomb Energy (RCE)14 model, shares the property of unrestricted storage density. LEARNING IN HIGH DENSITY MEMORIES A simple learning algorithm for the placement of the wells has been described in detail elsewhere 11, 12. Figurel: 3-layer feedforward network. Cell i computes the quantity IJl - xii and compares to internal threshold Ai. 678 Reilly et. al. have employed a three layer feedforward network (figure 1) which allows the generalization of a content addressable memory to a pattern classification memory. Because the locations of the minima are explicitly known in the equilibrium model, it is possible to dynamically program the energy function for an arbitrary energy landscape. This allows the construction of geographies of basins associated with the classes constituting the pattern environment. Rapid learning of complex, non-linear, disjoint, class regions is possible by this method 12, 13. LEARNING NON-SEPARABLE CLASS REGIONS Previous studies have focused on the acquisition of the geography and boundaries of non-linearly separable point sets. However, a method by which such high density models can acquire the probability distributions of non-separable sets has not been described. Non-separable sets are defined as point sets in the state space of a system which are labelled with multiple class affiliations. This can occur because the input space has not carried all of the features in the pattern environment, or because the pattern set itself is not separable. Points may be degenerate with respect to the explicit features of the space, however they may have different probability distributions within the environment. This structure in the environment is important information for the identification of patterns by such memories 10 the presence of feature space degeneracies. We now describe one possible mechanism for the acquisition of the probability distribution of non-separable points. It is assumed that all points in some region R of the state space of the network are the site of events Jl (0, Ci) which are examples of pattern classes C = {C1 , ... , CM }. A basin of attraction, xk( C i), defined by equation 4, is placed at each site fl(O, Ci) unless (5) that is, unless a memory at Xj (of the class Ci) already contains fl(O, Ci)· The initial values of Qo and Ro at xk(Ci) are a constant for all sites Xj. Thus as events of the classes C1, ... , C M occur at a particular site in R, multiple wells are placed at this location. 679 If a well x/ C i) correctly covers an event Jl (0, Ci), then the charge at that site (which defines the depth of the well) is incremented by a constant amount ~ Q o. In this manner, the region R is covered with wells of all classes {C1 , ... , CM }, with the depth of well XiCi) proportional to the frequency of occurence of Ci at Xj. The architecture of this network is exactly the same as that already described. As before, this network acquires a new cell for each well placed in the energy landscape. Thus we are able to describe the meaning of wells that overlap as the competition by multiple cells in layer 2 in firing for the pattern of activity in the input layer. APPLICATIONS This system has been applied to a problem in the area of risk assessment in mortgage lending. The input space consisted of feature detectors with continuous firing rates proportional to the values of 23 variables in the application for a mortgage. For this set of features, a significant portion of the space was nonseparable. Figures 2a and 2b illustrate the probability distributions of high and low risk applications for two of the features. It is clear that in this 2-dimensional subspace, the regions of high and low risk are non-separable but have different distributions. t-----------#llir----- Prob. = 1.0. 1000 Patterns Prob. = 0.5 0.0 Feature 1 1.0 Figure 2a: Probability distribution for High and Low risk patterns for feature 1. 680 1-----1----\--------- Prob. = 1.0. t 000 Patterns Prob. = 0.5 0.0 Feature 2 1.0 Figure 2b: Probability distribution for High and Low risk patterns for feature 2. Figure 3 depicts the probability distributions acquired by the system for this 2-dimensional subspace. In this image, circle radius is proportional to the degree of risk: Small circles are regions of low risk, and large circles are regions of high risk. 0 0 o 00 o 0 0 0 0 0 o 0:>0 o t?. 0 00 V 0 00 0 00 0 Feature 1 o o Figure 3: Probability distribition for Low and High risk. Small circles indicate low risk regIons and large circles indicate high risk regions. 681 Of particular interest is the clear clustering of high and low risk regions in the 2-d map. Note that the regions are in fact nonlinearly separable. DISCUSSION We have presented a simple method for the acquisition of probability distributions in non-separable point sets. This method generates an energy landscape of potential wells with depths that are proportional to the local probability density of the classes of patterns in the environment. These well depths set the probability of firing of class cells In a 3-layer feedforward network. Application of this method to a problem in risk assessment has shown that even completely non-separable subspaces may be modeled with surprising accuracy. This method improves pattern classification in such problems with little additional computational burden. This algorithm has been run in conjunction with the method described by Reilly et. al.II for separable regions. This combined system is able to generate non-linear decision surfaces between the separable zones, and approximate the probability distributions of the non-separable zones in a seemless manner. Further discussion of this system will appear in future reports. Current work is focused on the development of a more general method for modelling the scale of variations in the distributions. Sensitivity to this scale suggests that the transition from separable to non-separable regions is smooth and should not be handled with a 'hard' threshold. ACKNOWLEDGEMENTS We would like to thank Ed Collins and Sushmito Ghosh for their significant contributions to this work through the development of the mortgage risk assessment application. REFERENCES [1] Anderson, J .A.: A simple neural network generating an interactive memory. Math. Biosci. 14, 197-220 (1972). 682 [2] Cooper, L.N.: A possible organization of animal memory and learning. In: Proceedings of the Nobel Symposium on Collective Properties of Physical Systems, Lundquist, B., Lundquist, S. (eds.). (24), 252-264 London, New York: Academic Press 1973. [3] Kohonen, T.: Correlation matrix memories. IEEE Trans. Comput. 21, 353-359 (1972). [4] Kohonen, T.: Associative memory a system-theoretical approach. Berlin, Heidelberg, New York: Springer 1977. [5] Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79, 2554-2558 (April 1982). [6] Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81, 2088-3092 (May, 1984). [7] Hopfield, J.J., Feinstein, D.I., Palmer, R.G.: 'Unlearning' has a stabilizing effect in collective memories. Nature 304, 158-159 (July 1983). [8] Potter, T.W.: Ph.D. Dissertation in advanced technology, S.U.N.Y. Binghampton, (unpublished). [9] Bachmann, C.M., Cooper, L.N., Dembo, A., Zeitouni, 0.: A relaxation model for memory with high density storage. to be published in Proc. Nati. Acad. Sci. USA. [10] Dembo, A., Zeitouni, 0.: ARO Technical Report, Brown University, Center for Neural Science, Pr0vidence, R.I., (1987), also submitted to Phys. Rev. A. [11] Reilly, D.L., Cooper, L.N., Elbaum, C.: A neural model for category learning. BioI. Cybern. 45, 35 -41 (1982). [12] Reilly, D.L., Scofield, C., Elbaum, C., Cooper, L.N.: Learning system architectures composed of multiple learning modules. to appear in Proc. First In1'1. Conf. on Neural Networks (1987). [13] Rimey, R., Gouin, P., Scofield, C., Reilly, D.L.: Real-time 3-D object classification using a learning system. Intelligent Robots and Computer Vision, Proc. SPIE 726 (1986). [14] Reilly, D.L., Scofield, C. L., Elbaum, C., Cooper, L.N: Neural Networks with low connectivity and unrestricted memory storage density. To be published.
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850 Strategies for Teaching Layered Networks Classification Tasks Ben S. Wittner 1 and John S. Denker AT&T Bell Laboratories Holmdel, New Jersey 07733 Abstract There is a widespread misconception that the delta-rule is in some sense guaranteed to work on networks without hidden units. As previous authors have mentioned, there is no such guarantee for classification tasks. We will begin by presenting explicit counterexamples illustrating two different interesting ways in which the delta rule can fail. We go on to provide conditions which do guarantee that gradient descent will successfully train networks without hidden units to perform two-category classification tasks. We discuss the generalization of our ideas to networks with hidden units and to multicategory classification tasks. The Classification Task Consider networks of the form indicated in figure 1. We discuss various methods for training such a network, that is for adjusting its weight vector, w. If we call the input v, the output is g(w· v), where 9 is some function. The classification task we wish to train the network to perform is the following. Given two finite sets of vectors, Fl and F2, output a number greater than zero when a vector in Fl is input, and output a number less than zero when a vector in F2 is input. Without significant loss of generality, we assume that 9 is odd (Le. g( -s) == -g( s». In that case, the task can be reformulated as follows. Define 2 F :== Fl U {-v such that v E F2} (1) and output a number greater than zero when a vector in F is input. The former formulation is more natural in some sense, but the later formulation is somewhat more convenient for analysis and is the one we use. We call vectors in F, training vectors. A Class of Gradient Descent Algorithms We denote the solution set by W :== {w such that g(w· v) > 0 for all v E F}, lCurrently at NYNEX Science and Technology, 500 Westchester Ave., White Plains, NY 10604 2 We use both A := Band B =: A to denote "A is by definition B". @ American Institute of Physics 1988 (2) 851 output inputs Figure 1: a simple network and we are interested in rules for finding some weight vector in W. We restrict our attention to rules based upon gradient'descent down error functions E(w) of the form E(w) = L h(w . v). VEF The delta-rule is of this form with 1 h(w . v) = h6(W . v) := -(b - g(w . v))2 2 (3) (4) for some positive number b called the target (Rumelhart, McClelland, et al.). We call the delta rule error function E6. Failure of Delta-rule Using Obtainable Targets Let 9 be any function that is odd and differentiable with g'(s) > 0 for all s. In this section we assume that the target b is in the range of g. We construct a set F of training vectors such that even though M' is not empty, there is a local minimum of E6 not located in W. In order to facilitate visualization, we begin by assuming that 9 is linear. We will then indicate why the construction works for the nonlinear case as well. We guess that this is the type of counter-example alluded to by Duda and Hart (p. 151) and by Minsky and Papert (p. 15). The input vectors are two dimensional. The arrows in figure 2 represent the training vectors in F and the shaded region is W. There is one training vector, vI, in the second quadrant, and all the rest are in the first quadrant. The training vectors in the first quadrant are arranged in pairs symmetric about the ray R and ending on the line L. The line L is perpendicular to R, and intersects R at unit distance from the origin. Figure 2 only shows three of those symmetric pairs, but to make this construction work we might need many. The point p lies on R at a distance of g-l(b) from the origin. We first consider the contribution to E6 due to any single training vector, v. The contribution is (1/2)(b - g(w· v))2, (5) and is represented in figure 3 in the z-direction. Since 9 is linear and since b is in the , \ , R \ ,. \ , p ,'c" , , ..... \ " , , L \ X-axis 853 x-axis Figure 3: Error surface We now remove the assumption that 9 is linear. The key observation is that dh6/ds == h/(s) = (b - g(s»( -g'(s» (6) still only has a single zero at g-l(b) and so h(s) still has a single minimum at g-l(b). The contribution to E6 due to the training vectors in the first quadrant therefore still has a global minimum on the xy-plane at the point p. So, as in the linear case, if there are enough symmetric pairs of training vectors in the first quadrant, the value of Eo at p can be made arbitrarily lower than the value along some circle in the xy-plane centered around p, and E5 = Eo + El will have a local minimum arbitrarily near p. Q.E.D. Failure of Delta-rule Using Unobtainable Targets We now consider the case where the target b is greater than any number in the range of g. The kind of counter-example presented in the previous section no longer exists, but we will show that for some choices of g, including the traditional choices, the delta rule can still fail. Specifically, we construct a set F of training vectors such that even though W is not empty, for some choices of initial weights, the path traced out by going down the gradient of E5 never enters W. 854 y-axis :, , , , , , ",.,-P , q , __ .... ,:~ 4 J'=----~----~--~ L x-axis Figure 4: Counter-example for unobtainable targets We suppose that 9 has the following property. There exists a number r > 0 such that . hs'( -rs) hm h '() = o. _-00 5 S (7) An example of such a 9 is 2 9(S) = tanh(s) = 1 + e-2., - 1, (8) for which any r greater than 1 will do. The solid arrows in figure 4 represent the training vectors in F and the more darkly shaded region is W. The set F has two elements, and v 2=mm[n (9) The dotted ray, R lies on the diagonal {y = x}. Since (10) 855 the gradient descent algorithm follows the vector field -v E(w) = -h/(w· V1)V1 - h/(w. V2)V2 . (11) The reader can easily verify that for all won R, (12) So by equation (7), if we constrain w to move along R, . -h/(w. vI) hm , 2 = O. w ...... oo -ho (w . v ) (13) Combining equations (11) and (13) we see that there is a point q somewhere on R such that beyond q, - V E( w) points into the region to the right of R, as indicated by the dotted arrows in figure 4. Let L be the horizontal ray extending to the right from q. Since for all s, g'(s) > 0 and b> g(s), (14) we get that - h/(s) = (b - g(s»g'(s) > o. (15) So since both vI and v 2 have a positive y-component, -V E(w) also has a positive y-component for all w. So once the algorithm following -V E enters the region above L and to the right of R (indicated by light shading in figure 4), it never leaves. Q.E.D. Properties to Guarantee Gradient Descent Learning In this section we present three properties of an error function which guarantee that gradient descent will not fail to enter a non-empty W. We call an error function of the form presented in equation (3) well formed if h is differentiable and has the following three properties. 1. For all s, -h'( s) ~ 0 (i.e. h does not push in the wrong direction). 2. There exists some f > 0 such that -h'(s) ~ f for all s ~ 0 (i.e. h keeps pushing if there is a misclassification). 3. h is bounded below. Proposition 1 If the error junction is well formed, then gradient descent is guaranteed to enter W, provided W is not empty. 856 The proof proceeds by contradiction. Suppose for some starting weight vector the path traced out by gradient descent never enters W. Since W is not empty, there is some non-zero w* in W. Since F is finite, A := min{w*. v such that v E F} -:> O. (16) Let wet) be the path traced out by the gradient descent algorithm. So w'(t) = -VE(w(t» = I:: -h'(w(t) ·v)v for all t. (17) vEF Since we are assuming that at least one training vector is misclassified at all times, by properties 1 and 2 and equation (17), w* . w'(t) 2: fA for all t. (18) So Iw'(t)1 2: fA/lw*1 =: e > 0 for all t. (19) By equations (17) and (19), dE(w(t»/dt = V E· w'(t) = -w'(t) . w'(t) ~ -e < 0 for all t. (20) This means that E(w(t» --+ -00 as t --+ 00. (21) But property 3 and the fact that F is finite guarantee that E is bounded below. This contradicts equation (21) and finishes the proof. Consensus and Compromise So far we have been concerned with the case in which F is separable (i.e. W is not empty). What kind of behavior do we desire in the non-separable case? One might hope that the algorithm will choose weights which produce correct results for as many of the training vectors as possible. We suggest that this is what gradient descent using a well formed error function does. From investigations of many well formed error functions, we suspect the following well formed error function is representative. Let g( s) = s, and for some b > 0, let h(S)={ (b-s)2 ifs~~; o otherwIse. (22) In all four frames of figure 5 there are three training vectors. Training vectors 1 and 2 are held fixed while 3 is rotated to become increasingly inconsistent with the others. In frames (i) and (ii) F is separable. The training set in frame (iii) lies just on the border between separability and non-separability, and the one in frame (iv) is in the interior of 857 i) 3 ii ) 2 3 1 iii) 2 iv) 2 3 L.1 1 ... 3 Figure 5: The transition between seperability and non-seperability the non-separable regime. Regardless of the position of vector 3, the global minimum of the error function is the only minimum. In frames (i) and (ii), the error function is zero on the shaded region and the shaded region is contained in W. As we move training vector number 3 towards its position in frame (iii), the situation remains the same except the shaded region moves arbitrarily far from the origin. At frame (iii) there is a discontinuity; the region on which the error function is at its global minimum is now the one-dimensional ray indicated by the shading. Once training vector 3 has moved into the interior of the non-separable regime, the region on which the error function has its global minimum is a point closer to training vectors 1 and 2 than to 3 (as indicated by the "x" in frame (iv». If all the training vectors can be satisfied, the algorithm does so; otherwise, it tries to satisfy as many as possible, and there is a discontinuity between the two regimes. We summarize this by saying that it finds a consensus if possible, otherwise it devises a compromise. Hidden Layers For networks with hidden units, it is probably impossible to prove anything like proposition 1. The reason is that even though property 2 assures that the top layer of weights 858 gets a non-vanishing error signal for misclassified inputs, the lower layers might still get a vanishingly weak signal if the units above them are operating in the saturated regime. We believe it is nevertheless a good idea to use a well formed error function when training such networks. Based upon a probabilistic interpretation of the output of the network, Baum and Wilczek have suggested using an entropy error function (we thank J.J. Hopfield and D.W. Tank for bringing this to our attention). Their error function is well formed. Levin, Solla, and Fleisher report simulations in which switching to the entropy error function from the delta-rule introduced an order of magnitude speed-up of learning for a network with hidden units. Multiple Categories Often one wants to classify a given input vector into one of many categories. One popular way of implementing multiple categories in a feed-forward network is the following. Let the network have one output unit for each category. Denote by oj(w) the output of the j-th output unit when input v is presented to the network having weights w. The network is considered to have classified v as being in the k-th category if or(w) > oj(w) for all j ~ k. (23) The way such a network is usually trained is the generalized delta-rule (Rumelhart, McClelland, et al.). Specifically, denote by c(v) the desired classification of v and let b"! .= {b if j = c(v); 1 • -b otherwise, for some target b > O. One then uses the error function E(w):= EE (bj - oj (w») 2 • v . 3 (24) (25) This formulation has several bothersome aspects. For one, the error function is not will formed. Secondly, the error function is trying to adjust the outputs, but what we really care about is the differences between the outputs. A symptom of this is the fact that the change made to the weights of the connections to any output unit does not depend on any of the weights of the connections to any of the other output units. To remedy this and also the other defects of the delta rule we have been discussing, we suggest the following. For each v and j, define the relative coordinate (26) 859 What we really want is all the 13 to be positive, so use the error function E(w):= E E h (f3j(w)) (27) v #c(v) for some well formed h. In the simulations we have run, this does not always help, but sometimes it helps quite a bit. We have one further suggestion. Property 2 of a well formed error function (and the fact that derivatives are continuous) means that the algorithm will not be completely satisfied with positive 13; it will try to make them greater than zero by some non-zero margin. That is a good thing, because the training vectors are only representatives of the vectors one wants the network to correctly classify. Margins are critically important for obtaining robust performance on input vectors not in the training set. The problem is that the margin is expressed in meaningless units; it makes no sense to use the same numerical margin for an output unit which varies a lot as is used for an output unit which varies only a little. We suggest, therefore, that for each j and v, keep a running estimate of uj(w), the variance of f3J(w), and replace f3J(w) in equation (27) by f3J (w)/uj (w). (28) Of course, when beginning the gradient descent, it is difficult to have a meaningful estimate of uj(w) because w is changing so much, but as the algorithm begins to converge, your estimate can become increasingly meaningful. References 1. David Rumelhart, James McClelland, and the PDP Research Group, Parallel Distributed Processing, MIT Press, 1986 2. Richard Duda and Peter Hart, Pattern Classification and Scene Analysis, John Wiley & Sons, 1973. 3. Marvin Minsky and Seymour Papert, "On Perceptrons", Draft, 1987. 4. Eric Baum and Frank Wilczek, these proceedings. 5. Esther Levin, Sara A. Solla, and Michael Fleisher, private communications.
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830 Invariant Object Recognition Using a Distributed Associative Memory Harry Wechsler and George Lee Zimmerman Department or Electrical Engineering University or Minnesota Minneapolis, MN 55455 Abstract This paper describes an approach to 2-dimensional object recognition. Complex-log conformal mapping is combined with a distributed associative memory to create a system which recognizes objects regardless of changes in rotation or scale. Recalled information from the memorized database is used to classify an object, reconstruct the memorized version of the object, and estimate the magnitude of changes in scale or rotation. The system response is resistant to moderate amounts of noise and occlusion. Several experiments, using real, gray scale images, are presented to show the feasibility of our approach. Introduction The challenge of the visual recognition problem stems from the fact that the projection of an object onto an image can be confounded by several dimensions of variability such as uncertain perspective, changing orientation and scale, sensor noise, occlusion, and non-uniform illumination. A vision system must not only be able to sense the identity of an object despite this variability, but must also be able to characterize such variability -- because the variability inherently carries much of the valuable information about the world. Our goal is to derive the functional characteristics of image representations suitable for invariant recognition using a distributed associative memory. The main question is that of finding appropriate transformations such that interactions between the internal structure of the resulting representations and the distributed associative memory yield invariant recognition. As Simon [1] points out, all mathematical derivation can be viewed simply as a change of representation, making evident what was previously true but obscure. This view can be extended to all problem solving. Solving a problem then means transforming it so as to make the solution transparent. We approach the problem of object recognition with three requirements: classification, reconstruction, and characterization. Classification implies the ability to distinguish objects that were previously encountered. Reconstruction is the process by which memorized images can be drawn from memory given a distorted version exists at the input. Characterization involves extracting information about how the object has changed from the way in which it was memorized. Our goal in this paper is to discuss a system which is able to recognize memorized 2-dimensional objects regardless of geometric distortions like changes in scale and orientation, and can characterize those transformations. The system also allows for noise and occlusion and is tolerant of memory faults. The following sections, Invariant Representation and Distributed Associative Memory, respectively, describe the various components of the system in detail. The Experiments section presents the results from several experiments we have performed on real data. The paper concludes with a discussion of our results and their implications for future research. © American Institute of Physics 1988 831 1. Invariant Representation The goal of this section is to examine the various components used to produce the vectors which are associated in the distributed associative memory. The block diagram which describes the various functional units involved in obtaining an invariant image representation is shown in Figure 1. The image is complex-log conformally mapped so that rotation and scale changes become translation in the transform domain. Along with the conformal mapping, the image is also filtered by a space variant filter to reduce the effects of aliasing. The conformally mapped image is then processed through a Laplacian in order to solve some problems associated with the conformal mapping. The Fourier transform of both the conformally mapped image and the Laplacian processed image produce the four output vectors. The magnitude output vector I-II is invariant to linear transformations of the object in the input image. The phase output vector <1>2 contains information concerning the spatial properties of the object in the input image. 1.1 Complex-Log Mapping and Space Variant Filtering The first box of the block diagram given in Figure 1 consists of two components: Complex-log mapping and space variant filtering. Complex-log mapping transforms an image from rectangular coordinates to polar exponential coordinates. This transformation changes rotation and scale into translation. If the image is mapped onto a complex plane then each pixel (x,y) on the Cartesian plane can be described mathematically by z = x + jy. The complex-log mapped points ware described by w =In{z) =In(lzl} +jiJ z (1) Our system sampled 256x256 pixel images to construct 64x64 complex-log mapped images. Samples were taken along radial lines spaced 5.6 degrees apart. Along each radial line the step size between samples increased by powers of 1.08. These numbers are derived from the number of pixels in the original image and the number of samples in the complex-log mapped image. An excellent examination of the different conditions involved in selecting the appropriate number of samples for a complex-log mapped image is given in [2J. The non-linear sampling can be split into two distinct parts along each radial line. Toward the center of the image the samples are dense enough that no anti-aliasing filter is needed. Samples taken at the edge of the image are large and an anti-aliasing filter is necessary. The image filtered in this manner has a circular region around the center which corresponds to an area of highest resolution. The size of this region is a function of the number of angular samples and radial samples. The filtering is done, at the same time as the sampling, by convolving truncated Bessel functions with the image in the space domain. The width of the Bessel functions main lobe is inversely proportional to the eccentricity of the sample point. A problem associated with the complex-log mapping is sensitivity to center misalignment of the sampled image. Small shifts from the center causes dramatic distortions in the complex-log mapped image. Our system assumes that the object is centered in the image frame. Slight misalignments are considered noise. Large misalignments are considered as translations and could be accounted for by changing the gaze in such a way as to bring the object into the center of the frame. The decision about what to bring into the center of the frame is an active function and should be determined by the task. An example of a system which could be used to guide the translation process was developed by Anderson and Burt [3J. Their pyramid system analyzes the input image at different tem~ . Compl".lo, I ~-·-FO",i" -~ Mapping I I ' 2 I 1ransform and I I -1-1 2 I Space Variant I Filtering I Image Laplacian _~I Fourier Transform I-II Figure 1. Block Diagram of the System. Distributed Associative Memory ~ Inverse Processing and Reconstruction Rotation and Scale Estimation Classification 00 c..:> ~ 833 poral and spatial resolution levels. Their smart sensor was then able to shift its fixation such that interesting parts of the image (ie. something large and moving) was brought into the central part of the frame for recognition. 1.2 Fourier Transform The second box in the block diagram of Figure 1 is the Fourier transform. The Fourier transform of a 2-dimensional image f(x,y) is given by F(u,v) = j j f(x,y)e-i(ux+vy) dx dy (2) -00 -00 and can be described by two 2-dimensional functions corresponding to the magnitude IF(u,v)1 and phase <l>F(u,v). The magnitude component of the Fourier trans~rm which is invariant to translatIOn, carries much of the contrast information of the image. The phase component of the Fourier transform carries information about how things ar} placed in an image. Translation of f(x,y) corresponds to the addition of a linear phase cpmponent. The complex-log mapping transforms rotation and scale into translation and tije magnitude of the Fourier transform is invariant to those translations so that I-II ivill not change significantly with rotation and scale of the object in the image. 1.3 Laplacian The Laplacian that we use is a difference-of-Gaussians (DOG) approximation to the function as given by Marr [4). 2 2 'V2G =h [1 - r2/2oo2) e{ -r /200 } (3) '1rtT The result of convolving the Laplacian with an image can be viewed as a two step process. The image is blurred by a Gaussian kernel of a specified width oo. Then the isotropic second derivative of the blurred image is computed. The width of the Gaussian kernel is chosen such that the conformally mapped image is visible -- approximately 2 pixels in our experiments. The Laplacian sharpens the edges of the object in the image and sets any region that did not change much to zero. Below we describe the benefits from using the Laplacian. The Laplacian eliminates the stretching problem encountered by the complex-log mapping due to changes in object size. When an object is expanded the complex-log mapped image will translate. The pixels vacated by this translation will be filled with more pixels sampled from the center of the scaled object. These new pixels will not be significantly different than the displaced pixels so the result looks like a stretching in the complex-log mapped image. The Laplacian of the complex-log mapped image will set the new pixels to zero because they do not significantly change from their surrounding pixels. The Laplacian eliminates high frequency spreading due to the finite structure of the discrete Fourier transform and enhances the differences between memorized objects by accentuating edges and de-emphasizing areas of little change. 2. Distributed Associative Memory (DAM) The particular form of distributed associative memory that we deal with in this paper is a memory matrix which modifies the flow of information. Stimulus vectors are associated with response vectors and the result of this association is spread over the entire memory space. Distributing in this manner means that information about a small portion of the association can be found in a large area of the memory. New associations are placed 834 over the older ones and are allowed to interact. This means that the size of the memory matrix stays the same regardless of the number of associations that have been memorized. Because the associations are allowed to interact with each other an implicit representation of structural relationships and contextual information can develop, and as a consequence a very rich level of interactions can be captured. There are few restrictions on what vectors can be associated there can exist extensive indexing and cross-referencing in the memory. Distributed associative memory captures a distributed representation which is context dependent. This is quite different from the simplistic behavioral model [5]. The construction stage assumes that there are n pairs of m-dimensional vectors that are to be associated by the distributed associative memory. This can be written as "l.K:::+. = -r. ~or 1· 1 n IV~ I' = , ... , 1 1 (4) h -d h ·th . I d -d h .th d· were s. enotes tel stlmu us vector an r. enotes tel correspon mg response Vector. W~ want to construct a memory matrix M such that when the kth stimulus vector S; is projected onto the space defined by M the resulting projection will be the corresponding response vector r;. More specifically we want to solve the following equation: MS=R (5) h S [ 1 1 1 -] d R [ 1 1 1 -] A· I· ~ h· were = s1 1 s2 1 · ··1 S an = r 1 1 r 2 1 ···1 r. umque so utlOn lor t IS equation does not necessarily n exist for any arbitrary gr~up of associations that might be chosen. Usually, the number of associations n is smaller than m, the length of the vector to be associated, so the system of equations is underconstrained. The constraint used to solve for a unique matrix M is that of minimizing the square error, IIMS - RJ1 2, which results in the solution (6) where S+ is known as the Moore-Penrose generalized inverse of S [6J. The recall operation projects an unknown stimulus vector s onto the memory space M. The resulting projection yields the response vector r r =Ms (7) If the memorized stimulus vectors are independent and the unknown stimulus vector s is one of the memorized vectors S;, then the recalled vector will be the associated response vector r;. If the memorized stimulus vectors are dependent, then the vector recalled by one of the memorized stimulus vectors will contain the associated response vector and some crosstalk from the other stored response vectors. The recall can be viewed as the weighted sum of the response vectors. The recall begins by assigning weights according to how well the unknown stimulus vector matches with the memorized stimulus vector using a linear least squares classifier. The response vectors are multiplied by the weights and summed together to build the recalled response vector. The recalled response vector is usually dominated by the memorized response vector that is closest to the unknown stimulus vector. Assume that there are n associations in the memory and each of the associated stimulus and response vectors have m elements. This means that the memory matrix has m2 elements. Also assume that the noise that. is added to each element of a memorized 835 stimulus vector IS independent, Zero mean, with a variance of O'~ The recall from the 1 memory is then (8) where tt is the input noise vector and t1 is the output noise vector. The ratio of the average output noise variance to the averagg input noise variance is 2 2 1 [MMT] 0' /0'. = -Tr o 1 m (9) For the autoassociative case this simplifies to (10) This says that when a noisy version of a memorized input vector is applied to the memory the recall is improved by a factor corresponding to the ratio of the number of memorized vectors to the number of elements in the vectors. For the heteroassociative memory matrix a similar formula holds as long as n is less than m [7]. (11) Fault tolerance is a byproduct of the distributed nature and error correcting capabilities of the distributed associative memory. By distributing the information, no single memory cell carries a significant portion of the information critical to the overall performance of the memory. 3. Experiments In this section we discuss the result of computer simulations of our system. Images of objects are first preprocessed through the sUbsystem outlined in section 2. The output of such a subsystem is four vectors: I-I , <1>1' 1-12, and <1>2' We construct the memory by associating the stimulus vector I-II with £he response vector <1>2 for each object in the database. To perform a recall from the meJIlory the.. unknown image is preprocessed by the same_subsystem to produce the vectors I-II' <1>1' 1-12, and <1>2' The resulting stimulus vector I-I is projected onto the m~mory matrix to produce a respOJlse vector which is an ~stimatel of the memorized phase <1>2' The estimated phase vector cI> 2 and the magnitude I-II ate used to reconstruct the memorized object. The difference between the estimated phase <1>2 and the unknown phase <1>2 is used to estimate the amount of rotation and scale experienced by the object. The database of images consists of twelve objects: four keys, four mechanical parts, and four leaves. The objects were chosen for their essentially two-dimensional structure. Each object was photographed using a digitizing video camera against a black background. We emphasize that all of the images used in creating and testing the recognition system were taken at different times using various camera rotations and distances. The images are digitized to 256x256, eight bit quantized pixels, and each object covers an area of about 40x40 pixels. This small object size relative to the background is necessary due to the non-linear sampling of the complex-log mapping. The objects were centered within the frame by hand. This is the source of much of the noise and could have been done automatically using the object's center of mass or some other criteria determined by the task. The orientation of each memorized object was arbitrarily chosen such that their major axis 836 was vertical. The 2-dimensional images that are the output from the invariant representation subsystem are scanned horizontally to form the vectors for memorization. The database used for these experiments is shown in Figure 2. a) Original Figure 2. The Database of Objects Used in the Experiments b) Unknown c) Recall: rotated 135· Figure 3. :Recall Using a Rotated and scaled key d) Memory:6 SNR: -3.37 Db The first example of the operation of our system is shown in Figure 3. Figure 3a) is the image of one of the keys as it was memorized. Figure 3b) is the unknown object presented to our system. The unknown object in this caSe is the same key that has been rotated by 180 degrees and scaled. Figure 3c) is the recalled, reconstructed image. The 837 rounded edges of the recalled image are artifacts of the complex-log mapping. Notice that the reconstructed recall is the unrotated memorized key with some noise caused by errors in the recalled phase. Figure 3d) is a histogram which graphically displays the classification vector which corresponds to S+S. The histogram shows the interplay between the memorized images and the unknown image. The" 6" on the bargraph indicates which of the twelve classes the unknown object belongs. The histogram gives a value which is the best linear estimate of the image relative to the memorized objects. Another measure, the signal-to-noise ratio (SNR), is given at the bottom of the recalled image. SNR compares the variance of the ideal recall after processing with the variance of the difference between the ideal and actual recall. This is a measure of the amount of noise in the recall. The SNR does not carry rr.uch information about the q"Jality of the recall image because the noise measured by the SNP.. is jue to many factors such as misalignment of the center, changing reflections, and dependence between other memorized objects -- each affecting. quality in a variety of ways. Rotation and scale estimate~ are made using a vector_ D corresponding to the dlll'erence between the unknown vector <1>2 and the recalled vector <I> 2' In an ideal situation D will be a plane whose E;radient indicates the exact amount of r:.otation and scale the recalled object has experienced. In our system the recalled vector <I> 2 is corrupted with noise which means rotation...and scale have to be estim:ned. The estimate is made by letting the first order difference D at each point in the plane vote for a specified range of rotation or scale. a) Original b) Unknown c) Recall d) Memory:4 Figure 4 Recall Using Scaled and Rotated" S" with Occlusion Figure 4 is an example of occlusion. The unknown object in this case is an "s" curve which is larger and slightly tilted from the memorized "s" curve. A portion of the bottom curve was occluded. The resulting reconstruction is very noisy but has filled in the missing part of the bottom curve. The noisy recall is reflected in both the SNR and the interplay betw~en the memories shown by the hi~togram. a) Ideal recall b) 30% removed c) 50% removed d) 75% removed Figure 5. Recall for Memory Matrix Randomly Set to Zero Figure 5 is the result of randomly setting the elements of the memory matrix to 838 zero. Figure 5a) shows is the ideal recall. Figure 5b) is the recall after 30 percent of the memory matrix has been set to zero. Figure 5c) is the recall for 50 percent and Figure 5d) is the recall for 75 percent. Even when 90 percent of the memory matrix has been set to zero a faint outline of the pin could still be seen in the recall. This result is important in two ways. First, it shows that the distributed associative memory is robust in the presence of noise. Second, it shows that a completely connected network is not necessary and as a consequence a scheme for data compression of the memory matrix could be found. 4. Conclusion In this paper we demonstrate a computer vIsIon system which recognIzes 2dimensional objects invariant to rotation or scale. The system combines an invariant representation of the input images with a distributed associative memory such that objects can be classified, reconstructed, and characterized. The distributed associative memory is resistant to moderate amounts of noise and occlusion. Several experiments, demonstrating the ability of our computer vision system to operate on real, grey scale images, were presented. Neural network models, of which the di~tributed associative memory is one example, were originally developed to simulate biological memory. They are characterized by a large number of highly interconnected simple processors which operate in p2..rallel. An excellent review of the many neural network models is given in [8J. The distrib-uted associative memory we use is linear, and as a result there are certain desirable properties which will not be exhibited by our computer vision system. For example, feedback through our system will not improve recall from the memory. Recall could be improved if a non-linear element, such as a sigmoid function, is introduced into the feedback loop. Non-linear neural networks, such as those proposed by Hopfield [9] or Anderson et. al. [10J, can achieve this type of improvement because each memorized pattern js associated with sta~le points in an energy space. The price to be paid for the introduction of non-linearities into a memory system is that the system will be difficult to analyze and can be unstable. Implementing our computer vision system using non-linear distributed associative memory is a goal of our future research. We are presently extending our work toward 3-dimensional object recognition. Much of the present research in 3-dimensional object recognition is limited to polyhedral, nonoccluded objects' in a clean, highly controlled environment. Most systems are edge based and use a generate-and-test paradigm to estimate the position and orientation of recognized objects. We propose to use an approach based on characteristic views [llJ or aspects [12J which suggests that the infinite 2-dimensional projections of a 3-dimensional object can be grouped into a finite number of topological equivalence classes. An efficie:.t 3dimensional recognition system would require a parallel indexing method to search for object models in the presence of geometric distortions, noise, and occlusion. Our object recognition system using distributed associative memory can fulfill those requirements with respect to characteristic views. Referenees [lJ Simon, H. A., (1984), The Seienee of the Artifldal (2nd ed.), MIT Press. [2J Massone, L., G. Sandini, and V. Tagliasco (1985), "Form-invariant" topological mapping strategy for 2D shape recognition, CVGIP, 30, 169-188. [3J Anderson, C. H., P. J. Burt, and G. S. Van Der Wal (1985), Change detection and tracking using pyramid transform techniques, Proe. of the SPIE Conferenee on Intelligenee, Robots, and Computer Vision, Vol. 579, 72-78. 839 [4] Marr, D. (1982), Vision, W. H. Freeman, 1982. [5] Hebb, O. D. (1949), The Organization of Behavior, New York: Wiley. [6J Kohonen, T. (1984), Self-Organization and Associative-Memories, Springer-Verlag. [7] Stiles, G. S. and D. L. Denq (1985), On the effect of noise on the Moore-Penrose generalized inverse associative memory, IEEE Trans. on PAMI, 7, 3,358-360. [8J MCClelland, J. L., and D. E. Rumelhart, and the PDP Research Group (Eds.) (1986), Parallel Distributed, Processing, Vol. 1, 2, MIT Press. [9] Hopfield, J. J. (1982), Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, 79, April 1982. [10J Anderson, J. A., J. W. Silversteir., S. A. Ritz, and R. S. Jones (1977), Distinctive features, categorical perception, and probability learning: some applications of a neural model, Psychol. Rev., 84,413-451. [11] Chakravarty, I., and H. Freeman (1982), Characteristic views as a basis for 3-D object recognition, Proc. SPIE on Robot Vision, 336,37-45. [12] Koenderink, J. J., and A. J. Van Doorn (1979), Internal representation of solid shape with respect to vision, Bioi. Cybern., 32,4,211-216.
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290 CYCLES: A Simulation Tool for Studying Cyclic Neural Networks Michael T. Gately Texas Instruments Incorporated, Dallas, TX 75265 ABSTRACT A computer program has been designed and implemented to allow a researcher to analyze the oscillatory behavior of simulated neural networks with cyclic connectivity. The computer program, implemented on the Texas Instruments Explorer / Odyssey system, and the results of numerous experiments are discussed. The program, CYCLES, allows a user to construct, operate, and inspect neural networks containing cyclic connection paths with the aid of a powerful graphicsbased interface. Numerous cycles have been studied, including cycles with one or more activation points, non-interruptible cycles, cycles with variable path lengths, and interacting cycles. The final class, interacting cycles, is important due to its ability to implement time-dependent goal processing in neural networks. INTRODUCTION Neural networks are capable of many types of computation. However, the majority of researchers are currently limiting their studies to various forms of mapping systems; such as content addressable memories, expert system engines, and artificial retinas. Typically, these systems have one layer of fully connected neurons or several layers of neurons with limited (forward direction only) connectivity. I have defined a new neural network topology; a two-dimensional lattice of neurons connected in such a way that circular paths are possible. The neural networks defined can be viewed as a grid of neurons with one edge containing input neurons and the opposite edge containing output neurons [Figure 1]. Within the grid, any neuron can be connected to any other. Thus from one point of view, this is a multi-layered system with full connectivity. I view the weights of the connections as being the long term memory (LTM) of the system and the propagation of information through the grid as being it's short term memory (STM). The topology of connectivity between neurons can take on any number of patterns. Using the mammalian brain as a guide, I have limit~d the amount of connectivity to something much less then total. In addition to making analysis of such systems less complex, limiting the connectivity to some small percentage of the total number of neurons reduces the amount of memory used in computer simulations. In general, the connectivity can be purely random, or can form any of a number of patterns that are repeated across the grid of neurons. The program CYCLES allows the user to quickly describe the shape of the neural network grid, the source of input data, the destination of the output data, the pattern of connectivity. Once constructed, the network can be "run." during which time the STM may be viewed graphically. © American Institute of Physics 1988 input column output column ~ ~ -00000000000--00000000000--00000000000-~OOOOOOOOOOo-~OOOOOOOOOOo-~OOOOOOOOOOo-~OOOOOOOOOOo-~O 000 o 0 0 000-~O 00000-~O 00000-~O 0 000 0-~O o 0 000 0--000 o 0 0 0 000-t sample connectivity pattern -- replicated output column across all neurons Figure 1. COMPONENTS OF A CYCLES NEURAL NETWORK COMMAND WINDOW ..... vJI ...... '_,.lIr _". .... ;"r •• ,Ii 11.,61 ... " ~ .,~~,~~ · . ......... I'······· . Alt., GlobM VaN., .. .••...... ...... ,. , · .. , ...... 'nllI./n ' .... "., , · .. ' •..... I .,,_ fI/II~",*- ,~. · '., ... , .. · .......... .. ' ' •..... IIlIIZIII u.rou.F_tII'Y GRAPHICAL DISPLAY WINDOW HM. It_,~ ...... w_I' NYlI" ''''' " "'~ r .... "'_ 0' fI'." .... 11 ... 8' ,,,,,," 1 "" ... t_ ~ of .... t ' .. ,t .... ,,' "J •••• ~ , .. ,.._ IN,.,.... .. 0Jf '.Itt,,,,. ,,,_ .. rt.,..·,., y ... ,.. ,..,_t,.,,,,,_01 r ... ""'tU.raf' ....... \ II,\IIM ""_1 W .... ", .. ,,.. tVf'«"ltJ w,~,,_ USER INTERACTION WINDOW STI rus WINDOW Figure 2. NEURAL NETWORK WORKSTATION INTERFACE tv to I-' 292 IMPLEMENTATION CYCLES was implemented on a TI Explorer/Odyssey computer system with 8MB of RAM and 128MB of Virtual Memory. The program was written in Common LISP. The program was started in July of 1986, put aside for a while, and finished in March of 1987. Since that time, numerous small enhancements have been made - and the system has been used to test various theories of cyclic neural networks. The code was integrated into the Neural Network Workstation (NNW), an interface to various neural network algorithms. The NNW utilizes the window interface of the Explorer LISP machine to present a consistent command input and graphical output to a variety of neural network algorithms [Figure 2]. The backpropagation-like neurons are collected together into a large threedimensional array. The implementation actually allows the use of multiple twodimensional grids; to date, however, I have studied only single-grid systems. Each neuron in a CYCLES simulation consists of a list of information; the value of the neuron, the time that the neuron last fired, a temporary value used during the computation of the new value, and a list of the neurons connectivity. The connectivity list stores the location of a related neuron and the strength of the connection between the two neurons. Because the system is implemented in arrays and lists, large systems tend to be very slow. However, most of my analysis has taken place on very small systems « 80 neurons) and for this size the speed is acceptable. To help gauge the speed of CYCLES, a single grid system containing 100 neurons takes 0.8 seconds and 1235 cons cells (memory cells) to complete one update within the LISP machine. If the graphics interface is disabled, a test requiring 100 updates takes a total of 10.56 seconds. TYPES OF CYCLES As mentioned above, several types of cycles have been observed. Each of these can be used for different applications. Figure 3 shows some of these cycles. 1. SIMPLE cycles are those that have one or more points of activation traveling across a set number of neurons in a particular order. The path length can be any SIze. 2. NON-INTERRUPTABLE cycles are those that have sufficiently strong connectivity strengths that random flows of activation which interact with the cycle will not upset or vary the original cycle. 3. VARIABLE PATH LENGTH cycles can, based upon external information, change their path length. There must be one or more neurons that are always a part of the path. 4. INTERACTING cycles typically have one neuron in common. Each cycle must have at least one other neuron involved at the junction point in order to keep the cycles separate. This type of cycle has been shown to implement a complex form of a clock where the product of the two (or more) path lengths are the fundamental frequency. Figure 3. Types of Cycles [Simple and Interacting] • • • • • • • • • • • • • • • • • • • • • • • • • * • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Figure 4. Types of Connectivity [Nearest Neighbor and Gaussian] INPUT Intent Joint 3 Extended Joint 2 Centered Joint 1 Extended Chuck Opened Chuck Closed OUTPUT Completed Move Joint 3 Move Joint 2 Move Joint 1 Open Chuck Close Chuck Figure 5. Robot Arm used in Example 293 294 CONNECTIVITY Several types of connectivity have been investigated. These are shown in Figure 4. 1. In TOTAL connectivity, every neuron is connected to every other neuron. This particular pattern produces very complex interactions with no apparent stability. 2. With RANDOM connectivity, each neuron is connected to a random number of other neurons. These other neurons can be anywhere in the grid. 3. A very useful type of connectivity is to have a PATTERN. The patterns can be of any shape, typically having one neuron feed its nearest neighbors. 4. Finally, the GAUSSIAN pattern has been used with the most success. In this pattern, each neuron is connected to a set number of nodes - but the selection is random. Further, the distribution of nodes is in a Gaussian shape, centered around a point "forward" of itself. Thus the flow of information, in general, moves forward, but the connectivity allows cycles to be formed. \ ALGORITHM The algorithm currently being used in the system is a standard inner product equation with a sigmoidal threshold function. Each time a neuron's weight is to be calculated, the value of each contributing neuron on the connectivity list is multiplied by the strength of the connection and summed. This sum is passed through a sigmoidal thresholding function. The value of the neuron is changed to be the result of this threshold function. As you can see, the system updates neurons in an ordered fashion, thus certain interactions will not be observed. Since timing information is saved in the neurons, asynchron:' could be simulated. Initially, the weights of the connections are set randomly. A number of interesting cycles have been observed as a result of this randomness. However, several experiments have required specific weights. To accommodate this, an interface to the weight matrix is used. The user can create any set of connection strengths desired. I have experimented with several learning algorithms-that is, algorithms that change the connection weights. The first mechanism was a simple Hebbian rule that states that if two neurons both fire, and there is a connection between them, then strengthen the strength of that connection. A second algorithm I experimented with used a pain/pleasure indicator to strengthen or weaken weights. An algorithm that is currently under development actually presets the weights from a grammar of activity required of the network. Thus, the user can describe a process that must be controlled by a network using a simple grammar. This description is then "compiled" into a set of weights that contain cycles to indicate time-independent components of the activity. 295 USAGE Even without a biological background, it is easy to see that the processing power of the human brain is far more than present associative memories. Our repertoire of capabilities includes, among other things: memory of a time line, creativity, numerous types of biological clocks, and the ability to create and execute complex plans. The CYCLES algorithm has been shown to be capable of executing complex, time-variable plans. A plan can be defined as a sequence of actions that must be performed in some preset order. Under this definition, the execution of a plan would be very straightforward. However, when individual actions within the plan take an indeterminate length of time, it is necessary to construct an execution engine capable of dealing with unexpected time delays. Such a system must also be able to abort the processing of a plan based on new data. With careful programming of connection weights, I have been able to use CYCLES to execute time-variable plans. The particular example I have chosen is for a robot arm to change its tool. In this activity, once the controller receives the signal that the motion required, a series of actions take place that result in the tool being changed. As input to this system I have used a number of sensors that may be found in a robot; extension sensors in 2-D joints and pressure sensors in articulators. The outputs of this network are pulses that I have defined to activate motors on the robot arm. Figure 5 shows how this system could be implemented. Figure 6 indicates the steps required to perform the task. Simple time delays, such as found with binding motors and misplaced objects are accommodated with the built in time-independence. The small cycles that occur within the neural network can be thought of as short term memory. The cycle acts as a place holder - keeping track of the system's current place in a series of tasks. This type of pausing is necessary in many "real" activities such as simple process control or the analysis of time varying data. IMPLICATIONS The success of CYCLES to simple process control activities such as robot arm control implies that there is a whole new area of applications for neural networks beyond present associative memories. The exploitation of the flow of activation as a form of short term memory provides us with a technique for dealing with many of the "other" type of computations which humans perform. The future of the CYCLES algorithm will take two directions. First, the completion of a grammar and compiler for encoding process control tasks into a network. Second, other learning algorithms will be investigated which are capable of adding and removing connections and altering the strengths of connections based upon an abstract pain/pleasure indicator. 296 The robot gets a signal to begin the tool change process. A cycle is started that outputs a signal to the chuck motor. ~.-.-. -~ • • • • • • • • • • • • • • • • • • • • • • • • • • . ~ • • • • • • When the joint indicator indicates that the joint is centered, it changes the flow of activation to cause a cycle that activates the third joint. : : : ;}'\.:-. -..~ ... • • • • • • • • • • • • • • • • • • D When sensor indicates that the chuck is open. the first cycle is stopped and a second begins activ<lting the motor in the first joint. • • • • _ . • • • • • • .~ .... • :.J. • • • .p..~ • • • • • • • • • • Next, the chuck is closed around the new tool bit. • • • • • • ~.~ .. • ~~. • • • • • • • • · . . ......... . -"'. • • • . .. D D When the first joint is fully extended, the joint sensor sends a signal that stops that cycle, and begins one that outputs a signal to the second joint. • • • • • • · / .... -.(.:/. : :• • • • • • U • • • • • • The last signal ends the sequence of cycles and sends the completed signal. • • • • • .• • • • • • • • • • • • • • • • • • • • • • • • -.~. • • Figure 6. Example use of CYCLES to control a Robot Arm
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22 Abstract LEARNING ON A GENERAL NETWORK Amir F. Atiya Department of Electrical Engineering California Institute of Technology Ca 91125 This paper generalizes the backpropagation method to a general network containing feedback t;onnections. The network model considered consists of interconnected groups of neurons, where each group could be fully interconnected (it could have feedback connections, with possibly asymmetric weights), but no loops between the groups are allowed. A stochastic descent algorithm is applied, under a certain inequality constraint on each intra-group weight matrix which ensures for the network to possess a unique equilibrium state for every input. Introduction It has been shown in the last few years that large networks of interconnected "neuron" -like elemp.nts are quite suitable for performing a variety of computational and pattern recognition tasks. One of the well-known neural network models is the backpropagation model [1]-[4]. It is an elegant way for teaching a layered feedforward network by a set of given input/output examples. Neural network models having feedback connections, on the other hand, have also been devised (for example the Hopfield network [5]), and are shown to be quite successful in performing some computational tasks. It is important, though, to have a method for learning by examples for a feedback network, since this is a general way of design, and thus one can avoid using an ad hoc design method for each different computational task. The existence of feedback is expected to improve the computational abilities of a given network. This is because in feedback networks the state iterates until a stable state is reached. Thus processing is perforrr:.ed on several steps or recursions. This, in general allows more processing abilities than the "single step" feedforward case (note also the fact that a feedforward network is a special case of a feedback network). Therefore, in this work we consider the problem of developing a general learning algorithm for feedback networks. In developing a learning algorithm for feedback networks, one has to pay attention to the following (see Fig. 1 for an example of a configuration of a feedback network). The state of the network evolves in time until it goes to equilibrium, or possibly other types of behavior such as periodic or chaotic motion could occur. However, we are interested in having a steady and and fixed output for every input applied to the network. Therefore, we have the following two important requirements for the network. Beginning in any initial condition, the state should ultimately go to equilibrium. The other requirement is that we have to have a unique © American Institute of Physics 1988 23 equilibrium state. It is in fact that equilibrium state that determines the final output. The objective of the learning algorithm is to adjust the parameters (weights) of the network in small steps, so as to move the unique equilibrium state in a way that will result finally in an output as close as possible to the required one (for each given input). The existence of more than op.e equilibrium state for a given input causes the following problems. In some iterations one might be updating the weights so as to move one of the equilibrium states in a sought direction, while in other iterations (especially with different input examples) a different equilibrium state is moved. Another important point is that when implementing the network (after the completion oflearning), for a fixed input there can be more than one possible output. Independently, other work appeared recently on training a feedback network [6],[7],[8]. Learning algorithms were developed, but solving the problem of ensuring a unique equilibrium was not considered. This problem is addressed in this paper and an appropriate network and a learning algorithm are proposed. neuron 1 inputs outputs Fig. 1 A recurrent network The Feedback Network Consider a group of n neurons which could be fully inter-connected (see Fig. 1 for an example). The weight matrix W can be asymmetric (as opposed to the Hopfield network). The inputs are also weighted before entering into the network (let V be the weight matrix). Let x and y be the input and output vectors respectively. In our model y is governed by the following set of differential equations, proposed by Hopfield [5]: du Tdj = Wf(u) - u + Vx, y = f(u) (1) 24 where f(u) = (J(ud, ... , f(un)f, T denotes the transpose operator, f is a bounded and differentiable function, and.,. is a positive constant. For a given input, we would like the network after a short transient period to give a steady and fixed output, no matter what the initial network state was. This means that beginning any initial condition, the state is to be attracted towards a unique equilibrium. This leads to looking for a condition on the matrix W. Theorem: A network (not necessarily symmetric) satisfying L L w'fi < l/max(J')2, i i exhibits no other behavior except going to a unique equilibrium for a given input. Proof : Let udt) and U2(t) be two solutions of (1). Let where " II is the two-norm. Differentiating J with respect to time, one obtains Using (1) , the expression becomes dJ(t) 2 2 2 T [( ) ( )] -d- = --lluI(t) - u2(t))11 + -(uI(t) - U2(t)) W f uI(t) - f uz(t) . t 1" .,. Using Schwarz's Inequality, we obtain Again, by Schwarz's Inequality, i = 1, ... ,n where Wi denotes the ith row of W. Using the mean value theorem, we get Ilf(udt)) - f(U2(t))II ~ (maxl!'I)IIUl(t) - uz(t)ll. (3) Using (2),(3), and the expression for J(t), we get d~~t) ~ -aJ(t) (4) where (2) 25 By hypothesis of the Theorem, a is strictly positive. Multiplying both sides of (4) by exp( at), the inequality results, from which we obtain J(t) ~ J(O)e- at . From that and from the fact that J is non-negative, it follows that J(t) goes to zero as t -+ <Xl. Therefore, any two solutions corresponding to any two initial conditions ultimately approach each other. To show that this asymptotic solution is in fact an equilibrium, one simply takes U2(t) = Ul(t + T), where T is a constant, and applies the above argument (that J(t) -+ 0 as t -+ <Xl), and hence Ul(t + T) -+ udt) as t -+ <Xl for any T, and this completes the proof. For example, if the function I is of the following widely used sigmoid-shaped form, 1 I(u) = l+e- u ' then the sum of the square of the weights should be less than 16. Note that for any function I, scaling does not have an effect on the overall results. We have to work in our updating scheme subject to the constraint given in the Theorem. In many cases where a large network is necessary, this constraint might be too restrictive. Therefore we propose a general network, which is explained in the next Section. The General Network We propose the following network (for an example refer to Fig. 2). The neurons are partitioned into several groups. Within each group there are no restrictions on the connections and therefore the group could be fully interconnected (i.e. it could have feedback connections) . The groups are connected to each other, but in a way that there are no loops. The inputs to the whole network can be connected to the inputs of any of the groups (each input can have several connections to several groups). The outputs of the whole network are taken to be the outputs (or part of the outputs) of a certain group, say group I. The constraint given in the Theorem is applied on each intra-group weight matrix separately. Let (qa, s"), a = 1, ... , N be the input/output vector pairs of the function to be implemented. We would like to minimize the sum of the square error, given by a=l where M e" = I)y{ - si}2, i=l and yf is the output vector of group f upon giving input qa, and M is the dimension of vector s". The learning process is performed by feeding the input examples qU sequentially to the network, each time updating the weights in an attempt to minimize the error. 26 inputs Fig. 2 An example of a general network (each group represents a recurrent network) J---V outputs Now, consider a single group l. Let Wi be the intra-group weight matrix of group l, vrl be the matrix of weights between the outputs of group,. and the inputs of group l, and yl be the output vector of group I. Let the respective elements be w~i' V[~., and y~. Furthermore, let n, be the number of neurons of group l. Assume that the time constant l' is sufficiently small so as to allow the network to settle quickly to the equilibrium state, which is given by the solution of the equation yl = f(W'yl + L vrlyr). (5) r£A I where A, is the set of the indices of the groups whose outputs a.re connected to the inputs of group ,. We would like each iteration to update the weight matrices Wi and vrl so as to move the equilibrium in a direction to decrease the error. We need therefore to know the change in the error produced by a small change in the weight matrices. Let .:;';, , and aa~~, denote the matrices whose (i, j)th element are :~'.' and ::~ respectively. Let ~ be the column vector '1 '1 :r whose ith element is ~. We obtain the following relations: uy. 8ea = [A' _ (W')T] -1 8ea ( ')T 8W' 8yl Y , 8ea = [A' _ (W')T] -1 8ea ( r)T 8Vtl 8yl y , where A' is the diagonal matrix whose ith diagonal element is l/f'(Lk w!kY~ + LrLktJ[kyk) for a derivation refer to Appendix). The vector ~ associated with groUp l can be obtained in terms of the vectors ~, fEB" where B, is the set of the indices of the groups whose inputs are connected to the outputs of group ,. We get (refer to Appendix) 8ea = '" (V'i)T[Ai _ (Wi{r 1 8e".. (6) 8yl ~ 8y3 JlBI The matrix A' ~ (W')T for any group l can never be singular, so we will not face any problem in the updating process. To prove that, let z be a vector satisfying [A' - (W'f]z = o. We can write zdmaxlf' I ~ LW~.Zk' k i = I, ... , nl where Zi is the ,"th element of z. Using Schwarz's Inequality, we obtain i = I, ... ,nl Squaring both sides and adding the inequalities for i = I, ... , nl, we get L/; ~ max(J')2(Lz~) LL(w~i)2. (7) k i k Since the condition LL(W!k)2 < I/max(J')2), k 27 is enforced, it follows that (7) cannot be satisfied unless z is the zero vector. Thus, the matrix A' - (W')T cannot be singular. For each iteration we begin by updating the weights of group f (the group contammg the final outputs). For that group ~ equals simply 2(y{ - SI, ... , yf.t SM, 0, ... , O)T). Then we move backwards to the groups connected to that group and obtain their corresponding !!J: vectors using (6), update the weights, and proceed in the same manner until we complete updating all the groups. Updating the weights is performed using the following stochastic descent algorithm for each group, 8ea t:. V = -a3 8V + a4 eaR , where R is a noise matrix whose elements are characterized by independent zero-mean unityvariance Gaussian densities, and the a's are parameters. The purpose of adding noise is to allow escaping local minima if one gets stuck in any of them. Note that the control parameter is taken to be ea. Hence the variance of the added noise tends to decrease the more we approach the ideal zero-error solution. This makes sense because for a large error, i.e. for an unsatisfactory solution, it pays more to add noise to the weight matrices in order to escape local minima. On the other hand, if the error is small, then we are possibly near the global minimum or to an acceptable solution, and hence we do not want too much noise in order not to be thrown out of that basin. Note that once we reach the ideal zero-error solution the added noise as well as the gradient of ea become zero for all a and hence the increments of the weight matrices become zero. If after a certain iteration W happens to violate the constraint Liiwlj ~ constant < I/max(J')2, then its elements are scaled so as to project it back onto the surface of the hypershere. Implementation Example A pattern recognition example is considered. Fig. 3 shows a set of two-dimensional training patterns from three classes. It is required to design a neural network recognizer with 28 three output neurons. Each of the neurons should be on if a sample of the corresponding class is presented, and off otherwise, i.e. we would like to design a "winner-take-all" network. A singlelayer three neuron feedback network is implemented. We obtained 3.3% error. Performing the same experiment on a feedforward single-layer network with three neurons, we obtained 20% error. For satisfactory results, a feedforward network should be two-layer. With one neuron in the first layer and three in the second layer, we got 36.7% error. Finally, with two neurons in the first layer and three in the second layer, we got a match with the feedback case, with 3.3% error. z z z z z z z z z z z z z z z zil z 1 33 3 1 3 3 3 33 3 3 3 ~ 3 3 3 3 3 3 3 Fig. 3 A pattern recognition example Conclusion A way to extend the backpropagation method to feedback networks has been proposed. A condition on the weight matrix is obtained, to insure having only one fixed point, so as to prevent having more than one possible output for a fixed input. A general structure for networks is presented, in which the network consists of a number of feedback groups connected to each other in a feedforward manner. A stochastic descent rule is used to update the weights. The lJ!ethod is applied to a pattern recognition example. With a single-layer feedback network it obtained good results. On the other hand, the feedforward backpropagation method achieved good resuls only for the case of more than one layer, hence also with a larger number of neurons than the feedback case. 29 Acknow ledgement The author would like to gratefully acknowledge Dr. Y. Abu-Mostafa for the useful discussions. This work is supported by Air Force Office of Scientific Research under Grant AFOSR-86-0296. Appendix Differentiating (5), one obtains a I a I Yj '(')(,,", I Ym '6) -a I = f Zj L..,Wjm-a I +Yp jk , wkp m wkp k,p = 1, ... ,n, where if j = k otherwise, and We can write a~' = (A' _ Wi) -lbkz> awkp (A - 1) where b kp is the nt-dimensional vector whose ith component is given by By the chain rule, b~l> = {y~ • 0 ifi = k otherwise. aea _ ""' aea ay; -a I -L..,-a I-a I' wkp j Yj wkp which, upon substituting from (A - 1), can be put in the form y!,gk~' where gk is the kth column of (A' - Wt)-l. Finally, we obtain the required expression, which is ae" = [At _ (WI)T] -1 ae" ( ,)T aw' ayl y . Regarding a()~~I' it is obtained by differentiating (5) with respect to vr~,. We get similarly where Ckl' is the nt-dimensional vector whose ith component is given by if i = k otherwise. 30 A derivation very similar to the case of :~l results in the following required expression: Bea = [A' _ (w,)T] -1 Bea ( r)T. BVrl By' y 8 8 j 8 y J Now, finally consider ~. Let ~, jf.B, be the matrix whose (k,p)th element is ~. The elements of ~ can be obtained by differentiating the equation for the fixed point for group . uy J, as follows, Hence, :~~. = (Ai - Wi) -IV'i. (A - 2) Using the chain rule, one can write ·T Bea = ~(ByJ) Bea By' ~ Byl By;' JEEr We substitute from (A - 2) into the previous equation to complete the derivation by obtaining References 111 P. Werbos, "Beyond regression: New tools for prediction and analysis in behavioral sciences", Harvard University dissertation, 1974. [21 D. Parker, "Learning logic", MIT Tech Report TR-47, Center for Computational Research in Economics and Management Science, 1985. [31 Y. Le Cun, "A learning scheme for asymmetric threshold network", Proceedings of Cognitiva, Paris, June 1985. [41 D. Rumelhart, G.Hinton, and R. Williams, "Learning internal representations by error propagation", in D. Rumelhart, J. McLelland and the PDP research group (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition, Vol. 1, MIT Press, Cambridge, MA, 1986. 151 J. Hopfield, "Neurons with graded response have collective computational properties like those of two-state neurons", Proc. N atl. Acad. Sci. USA, May 1984. [61 L. Ahneida, " A learning rule for asynchronous perceptrons with feedback in a combinatorial environment", Proc. of the First Int. Annual Conf. on Neural Networks, San Diego, June 1987. [71 R. Rohwer, and B. Forrest, "Training time-dependence in neural networks", Proc. of the First Int. Annual Conf. on Neural Networks, San Diego, June 1987. [81 F. Pineda, "Generalization of back-propagation to recurrent neural networks", Phys. Rev. Lett., vol. 59, no. 19, 9 Nov. 1987.
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Neural Net and Traditional Classifiers1 William Y. Huang and Richard P. Lippmann MIT Lincoln Laboratory Lexington, MA 02173, USA 387 Abstract. Previous work on nets with continuous-valued inputs led to generative procedures to construct convex decision regions with two-layer perceptrons (one hidden layer) and arbitrary decision regions with three-layer perceptrons (two hidden layers). Here we demonstrate that two-layer perceptron classifiers trained with back propagation can form both convex and disjoint decision regions. Such classifiers are robust, train rapidly, and provide good performance with simple decision regions. When complex decision regions are required, however, convergence time can be excessively long and performance is often no better than that of k-nearest neighbor classifiers. Three neural net classifiers are presented that provide more rapid training under such situations. Two use fixed weights in the first one or two layers and are similar to classifiers that estimate probability density functions using histograms. A third "feature map classifier" uses both unsupervised and supervised training. It provides good performance with little supervised training in situations such as speech recognition where much unlabeled training data is available. The architecture of this classifier can be used to implement a neural net k-nearest neighbor classifier. 1. INTRODUCTION Neural net architectures can be used to construct many different types of classifiers [7]. In particular, multi-layer perceptron classifiers with continuous valued inputs trained with back propagation are robust, often train rapidly, and provide performance similar to that provided by Gaussian classifiers when decision regions are convex [12,7,5,8]. Generative procedures demonstrate that such classifiers can form convex decision regions with two-layer perceptrons (one hidden layer) and arbitrary decision regions with three-layer perceptrons (two hidden layers) [7,2,9]. More recent work has demonstrated that two-layer perceptrons can form non-convex and disjoint decision regions. Examples of hand crafted two-layer networks which generate such decision regions are presented in this paper along with Monte Carlo simulations where complex decision regions were generated using back propagation training. These and previous simulations [5,8] demonstrate that convergence time with back propagation can be excessive when complex decision regions are desired and performance is often no better than that obtained with k-nearest neighbor classifiers [4]. These results led us to explore other neural net classifiers that might provide faster convergence. Three classifiers called, "fixed weight," "hypercube," and "feature map" classifiers, were developed and evaluated. All classifiers were tested on illustrative problems with two continuous-valued inputs and two classes (A and B). A more restricted set of classifiers was tested with vowel formant data. 2. CAPABILITIES OF Two LAYER PERCEPTRONS Multi-layer perceptron classifiers with hard-limiting nonlinearities (node outputs of 0 or 1) and continuous-valued inputs can form complex decision regions. Simple constructive proofs demonstrate that a three-layer perceptron (two hidden layers) can 1 This work was sponsored by the Defense Advanced Research Projects Agency and the Department of the Air Force. The views expressed are those of the authors and do not reflect the policy or position of the U. S. Government. © American Institute of Physics 1988 388 DECISION REGION FOR CLASS A X2 b, b2 b4 , , , ~, ~-1 ~-1 I I 2 ----[J' -:: -:-: I --~, .:-): 1 f----- -":-<i:/ ___ _ I I I I I I o 2 3 FIG. 1. A two-layer perceptron that form! di!joint deci!ion region! for cia!! A (!haded area!). Connection weight! and node ojJ!eb are !hown in the left. Hyperplane! formed by all hidden node! are drawn a! da!hed line! with node labek Arrow! on theu line! point to the half plane where the hidden node output i! "high". form arbitrary decision regions and a two-layer perceptron (one hidden layer) can form single convex decision regions [7,2,9]. Recently, however, it has been demonstrated that two-layer perceptrons can form decision regions that are not simply convex [14]. Fig. 1, for example, shows how disjoint decision regions can be generated using a two-layer perceptron. The two disjoint shaded areas in this Fig. represent the decision region for class A (output node has a "high" output, y = 1). The remaining area represents the decision region for class B (output node has a "low" output, y = 0). Nodes in this Fig. contain hard-limiting nonlinearities. Connection weights and node offsets are indicated in the left diagram. Ten other complex decision regions formed using two-layer perceptrons are presented in Fig. 2. The above examples suggest that two-layer perceptrons can form decision regions with arbitrary shapes. We, however, know of no general proof of this capability. A 1965 book by Nilson discusses this issue and contains a proof that two-layer nets can divide a finite number of points into two arbitrary sets ([10] page 89). This proof involves separating M points using at most M - 1 parallel hyperplanes formed by firstlayer nodes where no hyperplane intersects two or more points. Proving that a given decision region can be formed in a two-layer net involves testing to determine whether the Boolean representations at the output of the first layer for all points within the decision region for class A are linearly separable from the Boolean representations for class B. One test for linear separability was presented in 1962 [13]. A problem with forming complex decision regions with two-layer perceptrons is that weights and offsets must be adjusted carefully because they interact extensively to form decision regions. Fig. 1 illustrates this sensitivity problem. Here it can be seen that weights to one hidden node form a hyperplane which influences decision regions in an entire half-plane. For example, small errors in first layer weights that results in a change in the slopes of hyperplanes bs and b6 might only slightly extend the Al region but completely eliminate the A2 region. This interdependence can be eliminated in three layer perceptrons. It is possible to train two-layer perceptrons to form complex decision regions using back propagation and sigmoidal nonlinearities despite weight interactions. Fig. 3, for example, shows disjoint decision regions formed using back propagation for the problem of Fig. 1. In this and all other simulations, inputs were presented alternately from classes A and B and selected from a uniform distribution covering the desired decision region. In addition, the back propagation rate of descent term, TJ, was set equal to the momentum gain term, a and TJ = a = .01. Small values for TJ and a were necessary to guarantee convergence for the difficult problems in Fig. 2. Other simulation details are 389 ~llll 5) IEl I I blEl mJ I I I I 9) 6) + 3) =(3 m1 rm I I I I I I 4 ) 10) 1= ~ftfI r I I I I FIG. 2. Ten complex deci6ion region6 formed by two-layer perceptron6. The number6 a66igned to each ca6e are the "ca6e" number6 u6ed in the re6t of thi6 paper. as in [5,8]. Also shown in Fig. 3 are hyperplanes formed by those first-layer nodes with the strongest connection weights to the output node. These hyperplanes and weights are similar to those in the networks created by hand except for sign inversions, the occurrence of multiple similar hyperplanes formed by two nodes, and the use of node offsets with values near zero. 3. COMPARATIVE RESULTS OF TWO-LAYERS VS. THREE-LAYERS Previous results [5,8], as well as the weight interactions mentioned above, suggest that three-layer perceptrons may be able to form complex decision regions faster with back propagation than two-layer perceptrons. This was explored using Monte Carlo simulations for the first nine cases of Fig. 2. All networks have 32 nodes in the first hidden layer. The number of nodes in the second hidden layer was twice the number of convex regions needed to form the decision region (2, 4, 6, 4, 6, 6, 8, 6 and 6 for Cases 1 through 9 respectively). Ten runs were typically averaged together to obtain a smooth curve of percentage error vs. time (number of training trials) and enough trials were run (to a limit of 250,000) until the curve appeared to flatten out with little improvement over time. The error curve was then low-pass filtered to determine the convergence time. Convergence time was defined as the time when the curve crossed a value 5 percentage points above the final percentage error. This definition provides a framework for comparing the convergence time of the different classifiers. It, however, is not the time after which error rates do not improve. Fig. 4 summarizes results in terms of convergence time and final percentage error. In those cases with disjoint decision regions, back propagation sometimes failed to form separate regions after 250,000 trials. For example, the two disjoint regions required in Case 2 were never fully separated with 390 , " I !... .7.2 : ' I ~2~ , I I ,.. ....-409 J I 12.7 ~ '9.3,4.5 7.6 J , __ ' , __ t..-[--=~I -]-(1--- ---- I ,--r ,----r-, I I I I " I " , I " I 210-11.9 I I I I ·2 ~_ .... I __ ---,I ___ ~I __ ...... I ____ I,-----, ·2 o 2 4 6 FIG. 3. Deci!ion region, formed u,ing bacle propagation for Ca!e! ! of Fig. !. Thiele !olid line! repre!ent deci,ion boundariu. Da,hed line! and arrow! have the lame meaning a! in Fig. 1. Only hyperplane! for hidden node, with large weight! to the output node are !hown. Over 300,000 training trial! were required to form !eparote N!gion!. a two-layer perceptron but were separated with a three-layer perceptron. This is noted by the use of filled symbols in Fig. 4. Fig. 4 shows that there is no significant performance difference between two and three layer perceptrons when forming complex decision regions using back propagation training. Both types of classifiers take an excessively long time (> 100,000 trials) to form complex decision regions. A minor difference is that in Cases 2 and 7 the two-layer network failed to separate disjoint regions after 250,000 trials whereas the three-layer network was able to do so. This, however, is not significant in terms of convergence time and error rate. Problems that are difficult for the two-layer networks are also difficult for the three-layer networks, and vice versa. 4. ALTERNATIVE CLASSIFIERS Results presented above and previous results [5,8] demonstrate that multi-layer perceptron classifiers can take very long to converge for complex decision regions. Three alternative classifiers were studied to determine whether other types of neural net classifiers could provide faster convergence. 4.1. FIXED WEIGHT CLASSIFIERS Fixed weight classifiers attempt to reduce training time by adapting only weights between upper layers of multi-layer perceptrons. Weights to the first layer are fixed before training and remain unchanged. These weights form fixed hyperplanes which can be used by upper layers to form decision regions. Performance will be good if the fixed hyperplanes are near the decision region boundaries that are required in a specific problem. Weights between upper layers are trained using back propagation as described above. Two methods were used to adjust weights to the first layer. Weights were adjusted to place hyperplanes randomly or in a grid in the region (-1 < Xl,X2 < 10). All decision regions in Fig. 2 fall within this region. Hyperplanes formed by first layer nodes for "fixed random" and "fixed grid" classifiers for Case 2 of Fig. 2 are shown as dashed lines in Fig. 5. Also shown in this Fig. are decision regions (shaded areas) formed 12 10 8 6 04 2 o 2-1ayers ERROR RATE o .~.~.~.~x.~:':".:: ...... . OL-__ L-__ L-__ L-__ L-__ L-__ L-__ L-__ L-~~~ 200000 CONVERGENCE TIME 391 FIG. 4. Percentage errOr (top) and convergence time (bottom) for Ca8e6 1 through 9 of Fig. 2 for two-and three-layer perceptron clauifier6 trained u6ing back propagation. Filled 6ymbol6 indicate that 6eparate di6joint region6 were not formed after 250,000 triak using back propagation to train only the upper network layers. These regions illustrate how fixed hyperplanes are combined to form decision regions. It can be seen that decision boundaries form along the available hyperplanes. A good solution is possible for the fixed grid classifier where desired decision region boundaries are near hyperplanes. The random grid classifier provides a poor solution because hyperplanes are not near desired decision boundaries. The performance of a fixed weight classifier depends both on the placement of hyperplanes and on the number of hyperplanes provided. 4.2. HYPERCUBE CLASSIFIER Many traditional classifiers estimate probability density functions of input variables for different classes using histogram techniques [41. Hypercube classifiers use this technique by fixing weights in the first two layers to break the input space into hypercubes (squares in the case of two inputs). Hypercube classifiers are similar to fixed weight classifiers, except weights to the first two layers are fixed, and only weights to output nodes are trained. Hypercube classifiers are also similar in structure to the CMAC model described by Albus [11. The output of a second layer node is "high" only if the input is in the hypercube corresponding to that node. This is illustrated in Fig. 6 for a network with two inputs. The top layer of a hypercube classifier can be trained using back propagation. A maximum likelihood approach, however, suggests a simpler training algorithm which consists of counting. The output of second layer node Hi is connected to the output node corresponding to that class with greatest frequency of occurrence of training inputs in hypercube Hi. That is, if a sample falls in hypercube Hi, then it is classified as class (J* where Nj,o. > Ni,O for all (J f:. (J •• (1) In this equation, Ni,O is the number of training tokens in hypercube Hi which belong to class (J. This will be called maximum likelihood (ML) training. It can be implemented by connection second-layer node Hi only to that output node corresponding to class (J. in Eq. (1). In all simulations hypercubes covered the area (-1 < Xl, X2 < 10). 392 RANDOM GRID o FIG. 5. Deci.ion region. formed with "fixed random" and "fixed grid" clal6ifier. for Ca.e ! from Fig. ! ruing back propagation training. Line! !hown are hyperplane! formed by the fird layer node!. Shaded area. repre.ent the deci.ion region for clau A. A B } INPUT TRAINED LAYER FIXED LAYERS "2 3 2 FOUR BINS CREATED BY FIXED LAYERS "1 FIG. 6. A hypercube clauifier (left) i! a three-layer perceptron with fixed weight! to the fird two layen, and trainable weight! to output node!. Weights are initialized !uch that output! of nodes HI through H. (left) are "high" only when the input i! in the corre!ponding hypercube (right). OUTPUT (Only One High) SElECT [ CLASS WITH MAJORITY IN TOP k SELECT TOP [ k EXEMPLARS CALCULATE CORRELATION TO STORED EXEMPLARS II, INPUT FIo. 1. Feature map clauifier. 4.3. FEATURE MAP CLASSIFIER SUPERVISED ASSOCIATIVE LEARNING UNSUPERVISED KOHONEN FEATURE MAP LEARNING 393 In many speech and image classification problems a large quantity of unlabeled training data can be obtained, but little labeled data is available. In such situations unsupervised training with unlabeled training data can substantially reduce the amount of supervised training required [3]. The feature map classifier shown in Fig. 7 uses combined supervised/unsupervised training, and is designed for such problems. It is similar to histogram classifiers used in discrete observation hidden Markov models [11] and the classifier used in [6]. The first layer of this classifier forms a feature map using a self organizing clustering algorithm described by Kohonen [6]. In all simulations reported in this paper 10,000 trials of unsupervised training were used. After unsupervised training, first-layer feature nodes sample the input space with node density proportional to the combined probability density of all classes. First layer feature map nodes perform a function similar to that of second layer hypercube nodes except each node has maximum output for input regions that are more general than hypercubes and only the output of the node with a maximum output is fed to the output nodes. Weights to output nodes are trained with supervision after the first layer has been trained. Back propagation, or maximum likelihood training can be used. Maximum likelihood training requires Ni,8 (Eq. 1) to be the number of times first layer node i has maximum output for inputs from class 8. In addition, during classification, the outputs of nodes with Ni,8 = 0 for all 8 (untrained nodes) are not considered when the first-layer node with the maximum output is selected. The network architecture of a feature map classifier can be used to implement a k-nearest neighbor classifier. In this case, the feedback connections in Fig. 7 (large circular summing nodes and triangular integrators) used to select those k nodes with the maximum outputs must be slightly modified. K is 1 for a feature map classifier and must be adjusted to the desired value of k for a k-nearest neighbor classifier. 5. COMPARISON BETWEEN CLASSIFIERS The results of Monte Carlo simulations using all classifiers for Case 2 are shown in Fig. 8. Error rates and convergence times were determined as in Section 3. All alter394 Percent Correct Conventional Fixed Weight Hypercube Feature Map 12 % 8 4 0 Tr ials Convergence Time 2500 1 77K 2000 I 2-1ay 1500 1000 • • I ~id 500 I 0 KNN GAUSS 32 3& 40 120 Number ot Hidden Nodes FIG. 8. Comparative performance of clauifier8for Ca8e 2. Training time of the feature map clauifier8 doe8 not include the 10,000 un8upervi8ed training trials. native classifiers had shorter convergence times than multi-layer perceptron classifiers trained with back propagation. The feature map classifier provided best performance. With 1,600 nodes, its error rate was similar to that of the k-nearest neighbor classifiers but it required fewer than 100 supervised training tokens. The larger fixed weight and hypercube classifiers performed well but required more supervised training than the feature map classifiers. These classifiers will work well when the combined probability density function of all classes varies smoothly and the domain where this function is non-zero is known. In this case weights and offsets can be set such that hyperplanes and hypercubes cover the domain and provide good performance. The feature map classifier automatically covers the domain. Fixed weight "random" classifiers performed substantially worse than fixed weight "grid" classifiers. Back propagation training (BP) was generally much slower than maximum likelihood training (ML). 6. VOWEL CLASSIFICATION Multi layer perceptron, feature map, and traditional classifiers were tested with vowel formant data from Peterson and Barney [11]. These data had been obtained by spectrographic analysis of vowels in /hVd/ context spoken by 67 men, women and children. First and second formant data of ten vowels was split into two sets, resulting in a total of 338 training tokens and 333 testing tokens. Fig. 9 shows the test data and the decision regions formed by a two-layer percept ron classifier trained with back propagation. The performance of classifiers is presented in Table I. All classifiers had similar error rates. The feature map classifier with only 100 nodes required less than 50 supervised training tokens (5 samples per vowel class) for convergence. The perceptron classifier trained with back propagation required more than 50,000 training tokens. The first stage of the feature map classifier and the multi-layer perceptron classifier were trained by randomly selecting entries from the 338 training tokens after labels had been removed and using tokens repetitively. 395 4000 D head .. hid D + hod • had . . .. hawed 2000 • heard o heed ( hud F2 (lIz) .. ) vho' d + .. " hood + 1000 .. + 500 1400 0 F1 (Hz) FIG. 9. DecilJion regionlJ formed by a two-layer network using BP after 200,000 training tokens from PeterlJon'lJ steadylJtate vowel data [PeterlJon, 1952}. AllJo shown are samplelJ of the telJting lJet. Legend IJhow example 0/ the pronunciation of the 10 vowels and the error within each vowel. I ALGORITHM I TRAINING TOKENS I % ERROR I TABLE I Performance of classifiers on IJteady IJtate vowel data. 396 7. CONCLUSIONS Neural net architectures form a flexible framework that can be used to construct many different types of classifiers. These include Gaussian, k-nearest neighbor, and multi-layer perceptron classifiers as well as classifiers such as the feature map classifier which use unsupervised training. Here we first demonstrated that two-layer perceptrons (one hidden layer) can form non-convex and disjoint decision regions. Back propagation training, however, can be extremely slow when forming complex decision regions with multi-layer perceptrons. Alternative classifiers were thus developed and tested. All provided faster training and many provided improved performance. Two were similar to traditional classifiers. One (hypercube classifier) can be used to implement a histogram classifier, and another (feature map classifier) can be used to implement a k-nearest neighbor classifier. The feature map classifier provided best overall performance. It used combined supervised/unsupervised training and attained the same error rate as a k-nearest neighbor classifier, but with fewer supervised training tokens. Furthermore, it required fewer nodes then a k-nearest neighbor classifier. REFERENCES [1] J. S. Albus, Brains, Behavior, and Robotics. McGraw-Hill, Petersborough, N.H., 1981. [2] D. J. Burr, "A neural network digit recognizer," in Proceedings of the International Conference on Systems, Man, and Cybernetics, IEEE, 1986. [3] D. B. Cooper and J. H. Freeman, "On the asymptotic improvement in the outcome of supervised learning provided by additional nonsupervised learning," IEEE Transactions on Computers, vol. C-19, pp. 1055-63, November 1970. [4] R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis. John-Wiley &. Sons, New York, 1973. [5] W. Y. Huang and R. P. Lippmann, "Comparisons between conventional and neural net classifiers," in 1st International Conference on Neural Network, IEEE, June 1987. [6] T. Kohonen, K. Makisara, and T. Saramaki, "Phonotopic maps insightful representation of phonological features for speech recognition," in Proceedings of the 7th International Conference on Pattern Recognition, IEEE, August 1984. [7] R. P. Lippmann, "An introduction to computing with neural nets," IEEE A SSP Magazine, vol. 4, pp. 4-22, April 1987. [8] R. P. Lippmann and B. Gold, "Neural classifiers useful for speech recognition," in 1st International Conference on Neural Network, IEEE, June 1987. [9] I. D. Longstaff and J. F. Cross, "A pattern recognition approach to understanding the multi-layer perceptron," Mem. 3936, Royal Signals and Radar Establishment, July 1986. [10] N. J. Nilsson, Learning Machines. McGraw Hill, N.Y., 1965. [11] T. Parsons, Voice and Speech Processing. McGraw-Hill, New York, 1986. [12] F. Rosenblatt, Perceptrons and the Theory of Brain Mechanisms. Spartan Books, 1962. [13] R. C. Singleton, "A test for linear separability as applied to self-organizing machines," in SelfOrganization Systems, 1962, (M. C. Yovits, G. T. Jacobi, and G. D. Goldstein, eds.), pp. 503524, Spartan Books, Washington, 1962. [14] A. Wieland and R. Leighton, "Geometric analysis of neural network capabilities," in 1st International Conference on Neural Networks, IEEE, June 1987.
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652 Scaling Properties of Coarse-Coded Symbol Memories Ronald Rosenfeld David S. Touretzky Computer Science Department Carnegie Mellon University Pittsburgh, Pennsylvania 15213 Abstract: Coarse-coded symbol memories have appeared in several neural network symbol processing models. In order to determine how these models would scale, one must first have some understanding of the mathematics of coarse-coded representations. We define the general structure of coarse-coded symbol memories and derive mathematical relationships among their essential parameters: memory 8ize, 8ymbol-8et size and capacity. The computed capacity of one of the schemes agrees well with actual measurements oC tbe coarse-coded working memory of DCPS, Touretzky and Hinton's distributed connectionist production system. 1 Introduction A di8tributed repre8entation is a memory scheme in which each entity (concept, symbol) is represented by a pattern of activity over many units [3]. If each unit participates in the representation of many entities, it is said to be coar8ely tuned, and the memory itself is called a coar8e-coded memory. Coarse-coded memories have been used for storing symbols in several neural network symbol processing models, such as Touretzky and Hinton's distributed connectionist production system DCPS [8,9], Touretzky's distributed implementation of linked list structures on a Boltzmann machine, BoltzCONS [10], and St. John and McClelland's PDP model of case role defaults [6]. In all of these models, memory capacity was measured empirically and parameters were adjusted by trial and error to obtain the desired behavior. We are now able to give a mathematical foundation to these experiments by analyzing the relationships among the fundamental memory parameters. There are several paradigms for coarse-coded memories. In a feature-based repre8entation, each unit stands for some semantic feature. Binary units can code features with binary values, whereas more complicated units or groups of units are required to code more complicated features, such as multi-valued properties or numerical values from a continuous scale. The units that form the representation of a concept define an intersection of features that constitutes that concept. Similarity between concepts composed of binary Ceatures can be measured by the Hamming distance between their representations. In a neural network implementation, relationships between concepts are implemented via connections among the units forming their representations. Certain types of generalization phenomena thereby emerge automatically. A different paradigm is used when representing points in a multidimensional continuous space [2,3]. Each unit encodes values in some subset of the space. Typically the @ American Institute of Physics 1988 653 subsets are hypercubes or hyperspheres, but they may be more coarsely tuned along some dimensions than others [1]. The point to be represented is in the subspace formed by the intersection of all active units. AB more units are turned on, the accuracy of the representation improves. The density and degree of overlap of the units' receptive fields determines the system's resolution [7]. Yet another paradigm for coarse-coded memories, and the one we will deal with exclusively, does not involve features. Each concept, or symbol, is represented by an arbitrary subset of the units, called its pattern. Unlike in feature-based representations, the units in the pattern bear no relationship to the meaning of the symbol represented. A symbol is stored in memory by turning on all the units in its pattern. A symbol is deemed present if all the units in its pattern are active.l The receptive field of each unit is defined as the set of all symbols in whose pattern it participates. We call such memories coarsecoded symbol memories (CCSMs). We use the term "symbol" instead of "concept" to emphasize that the internal structure of the entity to be represented is not involved in its representation. In CCSMs, a short Hamming distance between two symbols does not imply semantic similarity, and is in general an undesirable phenomenon. The efficiency with which CCSMs handle sparse memories is the major reason they have been used in many connectionist systems, and hence the major reason for studying them here. The unit-sharing strategy that gives rise to efficient encoding in CCSMs is also the source of their major weakness. Symbols share units with other symbols. AB more symbols are stored, more and more of the units are turned on. At some point, some symbol may be deemed present in memory because all of its units are turned on, even though it was not explicitly stored: a "ghost" is born. Ghosts are an unwanted phenomenon arising out of the overlap among the representations of the various symbols. The emergence of ghosts marks the limits of the system's capacity: the number of symbols it can store simultaneously and reliably. 2 Definitions and Fundamental Parameters A coarse coded symbol memory in its most general form consists of: • A set of N binary state units. • An alphabet of Q symbols to be represented. Symbols in this context are atomic entities: they have no constituent structure. • A memory scheme, which is a function that maps each symbol to a subset of the units - its pattern. The receptive field of a unit is defined as the set of all symbols to whose pattern it belongs (see Figure 1). The exact nature of the lThis criterion can be generalized by introducing a visibility threshold: a fraction of the pattern that should be on in order for a symbol to be considered present. Our analysis deals only with a visibility criterion of 100%, but can be generalized to accommodate nOise. 654 II II 81 I 82 I 88 I 8 4 I 85 I 86 I 87 I 88 II Ul • • • • U2 • • • • U8 • • • • U4 • • • U5 • • U6 • • • • • Figure 1: A memory scheme (N = 6, Q = 8) defined in terms of units Us and symbols 8;. The columns are the symbols' patterns. The rows are the units' receptive fieldB. memory scheme mapping determines the properties of the memory, and is the central target of our investigation. As symbols are stored, the memory fills up and ghosts eventually appear. It is not possible to detect a ghost simply by inspecting the contents of memory, since there is no general way of distinguishing a symbol that was stored from one that emerged out of overlaps with other symbols. (It is sometimes possible, however, to conclude that there are no ghosts.) Furthermore, a symbol that emerged as a ghost at one time may not be a ghost at a later time if it was subsequently stored into memory. Thus the definition of a ghost depends not only on the state of the memory but also on its history. Some memory schemes guarantee that no ghost will emerge as long as the number of symbols stored does not exceed some specified limit. In other schemes, the emergence of ghosts is an ever-present possibility, but its probability can be kept arbitrarily low by adjusting other parameters. We analyze systems of both types. First, two more bits of notation need to be introduced: Pghost: Probability of a ghost. The probability that at least one ghost will appear after some number of symbols have been stored. k: Capacity. The maximum number of symbols that can be stored simultaneously before the probability of a ghost exceeds a specified threshold. If the threshold is 0, we say that the capacity is guaranteed. A localist representation, where every symbol is represented by a single unit and every unit is dedicated to the representation of a single symbol, can now be viewed as a special case of coarse-coded memory, where k = N = Q and Pghost = o. Localist representations are well suited for memories that are not sparse. In these cases, coarsecoded memories are at a disadvantage. In designing coarse-coded symbol memories we are interested in cases where k « N « Q. The permissible probability for a ghost in these systems should be low enough so that its impact can be ignored. 655 3 Analysis of Four Memory Schemes 3.1 Bounded Overlap (guaranteed capacity) If we want to construct the memory scheme with the largest possible a (given Nand k) while guaranteeing Pghost = 0, the problem can be stated formally as: Given a set of size N, find the largest collection of subsets of it such that no union of k such subsets subsumes any other subset in the collection. This is a well known problem in Coding Theory, in slight disguise. Unfortunately, no complete analytical solution is known. We therefore simplify our task and consider only systems in which all symbols are represented by the same number of units (i.e. all patterns are of the same size). In mathematical terms, we restrict ourselves to constant weight codes. The problem then becomes: Given a set of size N, find the largest collection of subsets of size exactly L such that no union of k such subsets subsumes any other subset in the collection. There are no known complete analytical solutions for the size of the largest collection of patterns even when the patterns are of a fixed size. Nor is any efficient procedure for constructing such a collection known. We therefore simplify the problem further. We now restrict our consideration to patterns whose pairwise overlap is bounded by a given number. For a given pattern size L and desired capacity k, we require that no two patterns overlap in more than m units, where: _lL -1J m- -k (1) Memory schemes that obey this constraint are guaranteed a capacity of at least k symbols, since any k symbols taken together can overlap at most L - 1 units in the pattern of any other symbol - one unit short of making it a ghost. Based on this constraint, our mathematical problem now becomes: Given a set of size N, find the largest collection of subsets of size exactly L such that the intersection of any two such subsets is of size ~ m (where m is given by equation 1.) Coding theory has yet to produce a complete solution to this problem, but several methods of deriving upper bounds have been proposed (see for example [4]). The simple formula we use here is a variant of the Johnson Bound. Let abo denote the maximum number of symbols attainable in memory schemes that use bounded overlap. Then (m~l) (m~l) (2) 656 The Johnson bound is known to be an exact solution asymptotically (that is, when N, L, m -+ 00 and their ratios remain finite). Since we are free to choose the pattern size, we optimize our memory scheme by maximizing the above expression over all possible values of L. For the parameter subspace we are interested in here (N < 1000, k < 50) we use numerical approximation to obtain: < ( N )m+l max < LeII,N] L - m (3) (Recall that m is a function of Land k.) Thus the upper bound we derived depicts a simple exponential relationship between Q and N/k. Next, we try to construct memory schemes of this type. A Common Lisp program using a modified depth-first search constructed memory schemes for various parameter values, whose Q'S came within 80% to 90% of the upper bound. These results are far from conclusive, however, since only a small portion of the parameter space was tested. In evaluating the viability of this approach, its apparent optimality should be contrasted with two major weaknesses. First, this type of memory scheme is hard to construct computationally. It took our program several minutes of CPU time on a Symbolics 3600 to produce reasonable solutions for cases like N = 200, k = 5, m = 1, with an exponential increase in computing time for larger values of m. Second, if CCSMs are used as models of memory in naturally evolving systems (such as the brain), this approach places too great a burden on developmental mechanisms. The importance of the bounded overlap approach lies mainly in its role as an upper bound for all possible memory schemes, subject to the simplifications made earlier. All schemes with guaranteed capacities can be measured relative to equation 3. 3.2 Random Fixed Size Patterns (a stochastic approach) Randomly produced memory schemes are easy to implement and are attractive because of their naturalness. However, if the patterns of two symbols coincide, the guaranteed capacity will be zero (storing one of these symbols will render the other a ghost). We therefore abandon the goal of guaranteeing a certain capacity, and instead establish a tolerance level for ghosts, Pghost. For large enough memories, where stochastic behavior is more robust, we may expect reasonable capacity even with very small Pghost. In the first stochastic approach we analyze, patterns are randomly selected subsets of a fixed size L. Unlike in the previous approach, choosing k does not bound Q. We may define as many symbols as we wish, although at the cost of increased probability of a ghost (or, alternatively, decreased capacity). The probability of a ghost appearing after k symbols have been stored is given by Equation 4: (4) 657 TN,L(k, e) is the probability that exactly e units will be active after k symbols have been stored. It is defined recursively by Equation 5": TN,L(O,O) = 1 TN,L(k, e) = 0 for either k = 0 and e 1= 0, or k > 0 and e < L (5) TN,L(k, e) = E~=o T(k - 1, c - a) . (N-~-a)) . (~:~)/(~) We have constructed various coarse-coded memories with random fixed-size receptive fields and measured their capacities. The experimental results show good agreement with the above equation. The optimal pattern size for fixed values of N, k, and a can be determined by binary search on Equation 4, since Pghost(L) has exactly one maximum in the interval [1, N]. However, this may be expensive for large N. A computational shortcut can be achieved by estimating the optimal L and searching in a small interval around it. A good initial estimate is derived by replacing the summation in Equation 4 with a single term involving E[e]: the expected value of the number of active units after k symbols have been stored. The latter can be expressed as: The estimated L is the one that maximizes the following expression: An alternative formula, developed by Joseph Tebelskis, produces very good approximations to Eq. 4 and is much more efficient to compute. After storing k symbols in memory, the probability Pz that a single arbitrary symbol x has become a ghost is given by: L .(L) (N _ i)k (N)k Pz(N,L,k,a) = f.(-1)' i L / L (6) If we now assume that each symbol's Pz is independent of that of any other symbol, we obtain: (7) This assumption of independence is not strictly true, but the relative error was less than 0.1% for the parameter ranges we considered, when Pghost was no greater than 0.01. We have constructed the two-dimensional table TN,L(k, c) for a wide range of (N, L) values (70 ~ N ~ 1000, 7 ~ L ~ 43), and produced graphs of the relationships between N, k, a, and Pghost for optimum pattern sizes, as determined by Equation 4. The 658 results show an approximately exponential relationship between a and N /k [5]. Thus, for a fixed number of symbols, the capacity is proportional to the number of units. Let arlp denote the maximum number of symbols attainable in memory schemes that use random fixed-size patterns. Some typical relationships, derived from the data, are: arlP(Pghost = 0.01) ~ 0.0086. eO.46Sf arlp(Pghost = 0.001) ~ O.OOOS. eO.47Sf 3.3 Random Receptors (a stochastic approach) (8) A second stochastic approach is to have each unit assigned to each symbol with an independent fixed probability s. This method lends itself to easy mathematical analysis, resulting in a closed-form analytical solution. After storing k symbols, the probability that a given unit is active is 1 - (1 - s)k (independent of any other unit). For a given symbol to be a ghost, every unit must either be active or else not belong to that symbol's pattern. That will happen with a probability [1 - s . (1 - s)k] N, and thus the probability of a ghost is: Pghost(a, N, k,s) (9) Assuming Pghost « 1 and k « a (both hold in our case), the expression can be simplified to: Pghost(a,N,k,s) a· [1- s. (1- s)k]N from which a can be extracted: arr(N, k, 8, Pghost) (10) We can now optimize by finding the value of s that maximizes a, given any desired upper bound on the expected value of Pghost. This is done straightforwardly by solving Ba/Bs = o. Note that 8· N corresponds to L in the previous approach. The solution is s = l/(k + 1), which yields, after some algebraic manipulation: (11) A comparison of the results using the two stochastic approaches reveals an interesting similarity. For large k, with Pghost = 0.01 the term 0.468/k of Equation 8 can be seen as a numerical approximation to the log term in Equation 11, and the multiplicative factor of 0.0086 in Equation 8 approximates Pghost in Equation 11. This is hardly surprising, since the Law of Large Numbers implies that in the limit (N, k -+ 00, with 8 fixed) the two methods are equivalent. 659 Finally, it should be. noted that the stochastic approaches we analyzed generate a family of memory schemes, with non-identical ghost-probabilities. Pghost in our formulas is therefore better understood as an expected value, averaged over the entire family. 3.4 Partitioned Binary Coding (a reference point) The last memory scheme we analyze is not strictly distributed. Rather, it is somewhere in between a distributed and a localist representation, and is presented for comparison with the previous results. For a given number of units N and desired capacity k, the units are partitioned into k equal-size "slots," each consisting of N / k units (for simplicity we assume that k divides N). Each slot is capable of storing exactly one symbol. The most efficient representation for all possible symbols that may be stored into a slot is to assign them binary codes, using the N / k units of each slot as bits. This would allow 2NJic symbols to be represented. Using binary coding, however, will not give us the required capacity of 1 symbol, since binary patterns subsume one another. For example, storing the code '10110' into one of the slots will cause the codes '10010', '10100' and '00010' (as well as several other codes) to become ghosts. A possible solution is to use only half of the bits in each slot for a binary code, and set the other half to the binary complement of that code (we assume that N/k is even). This way, the codes are guaranteed not to subsume one another. Let Qpbc denote the number of symbols representable using a partitioned binary coding scheme. Then, '" _ 2N J2Ic - eO.847!:!..... pbc .. (12) Once again, Q is exponential in N /k. The form of the result closely resembles the estimated upper bound on the Bounded Overlap method given in Equation 3. There is also a strong resemblance to Equations 8 and 11, except that the fractional multiplier in front of the exponential, corresponding to Pghost, is missing. Pghost is 0 for the Partitioned Binary Coding method, but this is enforced by dividing the memory into disjoint sets of units rather than adjusting the patterns to reduce overlap among symbols. As mentioned previously, this memory scheme is not really distributed in the sense used in this paper, since there is no one pattern associated with a symbol. Instead, a symbol is represented by anyone of a set of k patterns, each N /k bits long, corresponding to its appearance in one of the k slots. To check whether a symbol is present, all k slots must be examined. To store a new symbol in memory, one must scan the k slots until an empty one is found. Equation 12 should therefore be used only as a point of reference. 4 Measurement of DCPS The three distributed schemes we have studied all use unstructured patterns, the only constraint being that patterns are at least roughly the same size. Imposing more complex structure on any of these schemes may is likely to reduce the capacity somewhat. In 660 Memory Scheme Result Bounded Overlap Qbo(N, k) < eO.367t Random Fixed-size Patterns Q,,!p(Pghost = 0.01) ~ 0.0086. e°.468r Q,,!p(Pghost = 0.001) ~ 0.0008 . e°.473f Random Receptors Q _ P . eN .1og (k+1)"'Tl/((k+l)"'Tl_k"') ,.,. ghost Partitioned Binary Coding Qpbc eO.347r Table 1 Summary of results for various memory schemes. order to quantify this effect, we measured the memory capacity of DCPS (BoltzCONS uses the same memory scheme) and compared the results with the theoretical models analyzed above. DCPS' memory scheme is a modified version of the Random Receptors method [5]. The symbol space is the set of all triples over a 25 letter alphabet. Units have fixed-size receptive fields organized as 6 x 6 x 6 subspaces. Patterns are manipulated to minimize the variance in pattern size across symbols. The parameters for DCPS are: N = 2000, Q = 253 = 15625, and the mean pattern size is (6/25)3 x 2000 = 27.65 with a standard deviation of 1.5. When Pghost = 0.01 the measured capacity was k = 48 symbols. By substituting for N in Equation 11 we find that the highest k value for which Q,.,. ~ 15625 is 51. There does not appear to be a significant cost for maintaining structure in the receptive fields. 5 Summary and Discussion Table 1 summarizes the results obtained for the four methods analyzed. Some differences must be emphasiz'ed: • Qbo and Qpbc deal with guaranteed capacity, whereas Q,.!p and Q,.,. are meaningful only for Pghost > O. • Qbo is only an upper bound. • Q,.!p is based on numerical estimates. • Qpbc is based on a scheme which is not strictly coarse-coded. The similar functional form of all the results, although not surprising, is aesthetically pleasing. Some of the functional dependencies among the various parameters_ can be derived informally using qualitative arguments. Only a rigorous analysis, however, can provide the definite answers that are needed for a better understanding of these systems and their scaling properties. 661 Acknowledgments We thank Geoffrey Hinton, Noga Alon and Victor Wei for helpful comments, and Joseph Tebelskis for sharing with us his formula for approximating Pghost in the case of fixed pattern sizes. This work was supported by National Science Foundation grants IST-8516330 and EET-8716324, and by the Office of Naval Research under contract number NOOO14-86K-0678. The first author was supported by a National Science Foundation graduate fellowship. References [1] Ballard, D H. (1986) Cortical connections and parallel processing: structure and function. Behavioral and Brain Sciences 9(1). [2] Feldman, J. A., and Ballard, D. H. (1982) Connectionist models and their properties. Cognitive Science 6, pp. 205-254. [3] Hinton, G. E., McClelland, J. L., and Rumelhart, D. E. (1986) Distributed representations. In D. E. Rumelhart and J. L. McClelland (eds.), Parallel Distributed Processing: Explorations in the Microstructure of Cognition, volume 1. Cambridge, MA: MIT Press. [4] Macwilliams, F.J., and Sloane, N.J.A. (1978). The Theory of Error-Correcting Codes, North-Holland. [5] Rosenfeld, R. and Touretzky, D. S. (1987) Four capacity models for coarse-coded symbol memories. Technical report CMU-CS-87-182, Carnegie Mellon University Computer Science Department, Pittsburgh, PA. [6] St. John, M. F. and McClelland, J. L. (1986) Reconstructive memory for sentences: a PDP approach. Proceedings of the Ohio University Inference Conference. [7] Sullins, J. (1985) Value cell encoding strategies. Technical report TR-165, Computer Science Department, University of Rochester, Rochester, NY. [8] Touretzky, D. S., and Hinton, G. E. (1985) Symbols among the neurons: details of a connectionist inference architecture. Proceedings of IJCAI-85, Los Angeles, CA, pp. 238-243. [9] Touretzky, D. S., and Hinton, G. E. (1986) A distributed connectionist production system. Technical report CMU-CS-86-172, Computer Science Department, Carnegie Mellon University, Pittsburgh, PA. [10] Touretzky, D. S. (1986) BoltzCONS: reconciling connectionism with the recursive nature of stacks and trees. Proceedings of the Eighth A nnual Conference of the Cognitive Science Society, Amherst, MA, pp. 522-530.
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824 SYNCHRONIZATION IN NEURAL NETS Jacques J. Vidal University of California Los Angeles, Los Angeles, Ca. 90024 John Haggerty· ABSTRACT The paper presents an artificial neural network concept (the Synchronizable Oscillator Networks) where the instants of individual firings in the form of point processes constitute the only form of information transmitted between joining neurons. This type of communication contrasts with that which is assumed in most other models which typically are continuous or discrete value-passing networks. Limiting the messages received by each processing unit to time markers that signal the firing of other units presents significant implemen tation advantages. In our model, neurons fire spontaneously and regularly in the absence of perturbation. When interaction is present, the scheduled firings are advanced or delayed by the firing of neighboring neurons. Networks of such neurons become global oscillators which exhibit multiple synchronizing attractors. From arbitrary initial states, energy minimization learning procedures can make the network converge to oscillatory modes that satisfy multi-dimensional constraints Such networks can directly represent routing and scheduling problems that conSist of ordering sequences of events. INTRODUCTION Most neural network models derive from variants of Rosenblatt's original perceptron and as such are value-passing networks. This is the case in particular with the networks proposed by Fukushima I, Hopfield2, Rumelhart3 and many others. In every case, the inputs to the processing elements are either binary or continuous amplitude signals which are weighted by synaptic gains and subsequently summed (integrated). The resulting activation is then passed through a sigmoid or threshold filter and again produce a continuous or quantized output which may become the input to other neurons. The behavior of these models can be related to that of living neurons even if they fall considerably short of accounting for their complexity. Indeed, it can be observed with many real neurons that action potentials (spikes) are fired and propagate down the axonal branches when the internal activation reaches some threshold and that higher John Haggerty is with Interactive Systems Los angeles 3030 W. 6th St. LA, Ca. 90020 @) American Institute of Physics 1988 825 input rates levels result in more rapid firing. Behind these traditional models, there is the assumption that the average frequency of action potentials is the carrier of information between neurons. Because of integration, the firings of individual neurons are considered effective only to the extent to which they contribute to the average intensities It is therefore assumed that the activity is simply "frequency coded". The exact timing of individual firing is ignored. This view however does not cover some other well known aspects of neural communication. Indeed, the precise timing of spike arrivals can make a crucial difference to the outcome of some neural interactions. One classic example is that of pre-synaptic inhibition, a widespread mechanism in the brain machinery. Several studies have also demonstrated the occurrence and functional importance of precise timing or phase relationship between cooperating neurons in local networks4. 5 . The model presented in this paper contrasts with the ones just mentioned in that in the networks each firing is considered as an individual output event. On the input side of each node, the firing of other nodes (the presynaptic neurons) either delay (inhibit) or advance (excite) the node firing. As seen earlier, this type of neuronal interaction which would be called phase-modulation in engineering systems, can also find its rationale in experimental neurophysiology. Neurophysiological plausibility however is not the major concern here. Rather, we propose to explore a potentially useful mechanism for parallel distributed computing. The merit of this approach for artificial neural networks is that digital pulses are used for internode communication instead of analog voltages. The model is particularly well suited to the time-ordering and sequencing found in a large class of routing and trajectory control problems. NEURONS AS SYNCHRONIZABLE OSCILLATORS: In our model, the proceSSing elements (the "neurons") are relaxation oscillators with built-in self-inhibition. A relaxation oscillator is a dynamic system that is capable of accumulating potential energy until some threshold or breakdown point is reached. At that point the energy is abruptly released, and a new cycle begins. The description above fits the dynamic behavior of neuronal membranes. A richly structured empirical model of this behavior is found in the well-established differential formulation of Hodgkin and Huxley6 and in a simplified version given by Fitzhugh7. These differential equations account for the foundations of neuronal activity and are also capable of representing subthreshold behavior and the refractoriness that follows each firing. When the membrane potential enters the critical region, an abrupt depolarization, i.e., a collapse of the potential difference across the membrane occurs followed by a somewhat slower recovery. This brief electrical 826 shorting of the membrane is called the action potential or "spike" and constitutes the output event for the neuron. If the causes for the initial depolarization are maintained, oscillation ( "limit-cycles") develops, generating multiple firings. Depending on input level and membrane parameters, the oscillation can be limited to a single spike, or may produce an oscillatory burst, or even continually sustained activity. The present model shares the same general properties but uses the much simpler description of relaxation oscillator illustrated on Figure 1. Exdt3tOIJ OJ Input Out InJrjh1~olJ Intemilf l!neJU Inpul Activation EnergyE(t) Input perturbation 1 u (t - ty ~~utl r t Figure 1 Relaxation Oscillator with perturbation input Firing occurs when the energy level E(t) reaches some critical level Ec. Assuming a constant rate of energy influx a, firing will occur with the natural period Ec· T=a:When pre-synaptic pulses impinge on the course of energy accumulation, the firing schedule is disturbed. Letting to represent the instant of the last firing of the cell and tj. U = 1.2 •... J), the intants of impinging arrivals from other cells: E(t - to) = aCt - to) + L Wj •• uo(t - til ; E $ Ec where uo(t) represents the unit impulse at t=O. The dramatic complexity of synchronization dynamics can be appreCiated by considering the simplest possible case, that of a master slave interaction between two regularly firing oscillator units A and B, with natural periods TA and TB. At the instants of firing, unit A unidirectionally sends a spike Signal to unit B which is received at some interval <I> measured from the last time B fired. 827 Upon reception the spike is transformed into a quantum of energy 6E which depends upon the post-firing arrival time 4>. The relationship 6E(4)) can be shaped to represent refractoriness and other post-spike properties. Here it is assumed to be a simple ramp function. If the interaction is inhibitory. the consequence of this arrival is that the next firing of unit B is delayed (with respect to what its schedule would have been in absence of perturbation) by some positive interval 5 (Figure 2). Because of the shape of 6E(4)) . the delaying action. nil immediately after firing. becomes longer for impinging pre-synaptic spikes that arrive later in the interval. If the interaction is excitatory. the delay is negative. Le. a shortening of the natural firing interval. Under very general assumptions regarding the function 6E( 4». B will tend to synchronize to A. Within a given range of coupling gains, the phase 4> will self-adjust until equilibrium is achieved. With a given 6E(4)) , this equilibrium corresponds to a distribution of maximum entropy, i.e., to the point where both cells receive the same amouint of activation. during their common cycle. I I ~h ~ $~~ ~ ~ .. ) Inhibition B ( .. ) Excitation Figure 2 Relationship between phase and delay when input effiCiently increases linearly in the after-spike interval The synchronization dynamiCS presents an attractor for each rational frequency pair. To each ratio is aSSOCiated a range of stability but only the ratios of lowest cardinality have wide zones of phaselocking (Figure 3). The wider stability wnes correspond to a one to one ratio between fA and fB (or between their inverses TA and TBl. Kohn and Segundo have demonstrated that such phase locking occurs in living invertebrate neurons and pointed out the paradoxical nature of phase-locked inhibition which, within each stability region, 828 takes the appearence of excitation since small increases in input firing rate will locally result in increased output rates 8, 5. The areas between these ranges of stability have the appearance of unstable transitions but in fact. as recently pOinted out by Bak9 • form an infinity of locking steps known as the Devil's Staircase. corresponding to the infinity of intermediate rational pairs (figure 3). Bak showed that the staircase is self-similar under scaling and that the transitions form a fractal Cantor set with a fractal dimension which is a universal constant of dynamic systems. 1/2 ~ I 1/2 ;;: ) Excitation Inhibiti~~v'/ I :7 lI?L It.?' Figure 3 Unilateral SynchroniZation: CONSTRAINT SATISFACTION IN OSCILLATOR NETWORKS The global synchronization of an interconnected network of mutually phase-locking oscillators is a constraint satisfaction problem. For each synchronization equilibrium, the nodes fire in interlocked patterns that organize inter-spike intervals into integer ratios. The often cited "Traveling Salesman Problem". the archetype for a class of important "hard" problems. is a special case when the ratio must be 1 / 1: all nodes must fire at the same frequency. Here the equilibrium condition is that every node will accumulate the the same amount of energy during the global cycle. Furthermore. the firings must be ordered along a minimal path. Using stochastic energy minimization and simulated annealing. the first simulations have demonstrated the feasibility of the approach with a limited number of nodes. The TSP is isomorphic to many other sequencing problems which involve distributed constraints. and fall into the oscillator array neural net paradigm in a particularly natural way. Work is being pursued to more rigorously establish the limits of applicability of the model.. I Annea/./ng ~ ~~~~~-L~~~~~~T171~~~~~~~~~-L~a~~~--~~--~----~-171--~L-~~--~~--~~c~--~~--~--~----~ Gf~--~~--~----L-----~ e t-A-----.&--------.£.---829 Figure 4. The Traveling Salesman Problem: In the global oscillation oj minimal energy each node is constrained to fire at the same rate in the order corresponding to the minimal path. ACKNOWLEDGEMENT Research supported in part by Aerojet Electro-Systems under the Aerojet-UCLA Cooperative Research Master Agreement No. D8412I1, and by NASA NAG 2-302. REFERENCES l. K. Fukushima. BioI. Cybern. 20. 121 (1975). 2. J.J. Hopfield. Proc. Nat. Acad. Sci. 79.2556 (1982). 3. D.E. Rumelhart. G.E. Hinton. and R.J. Williams. Parallel Distributed Processing: Explorations in the Microstructure oj Cognition, (MIT Press. Cambridge. MA .. 1986) p. 318. 4. J.P. Segundo. G.P. Moore. N.J. Stensaas. and T.H. Bullock. J. Exp. BioI. 40. 643. (1963). 5. J.P. Segundo and A.F. Kohn. BioI Cyber 40. 113 (1981). 6. A.L. Hodgkin and A.F. Huxley. J. PhysiOI. 117.500 (1952). 7. Fitzhugh. Biophysics J .. 1. 445 (1961). 8. A.F. Kohn. A. Freitas da Rocha. and J.P. Segundo. BioI. Cybem. 41. 5 (1981). 9. P. Bak. Phys. Today (Dec 1986) p. 38 . 10. J. Haggerty and J.J. Vidal. UCLA BCI Report. 1975.
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278 ABSTRACT THE HOPFIELD MODEL WITH MUL TI-LEVEL NEURONS Michael Fleisher Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel The Hopfield neural network. model for associative memory is generalized. The generalization replaces two state neurons by neurons taking a richer set of values. Two classes of neuron input output relations are developed guaranteeing convergence to stable states. The first is a class of "continuous" relations and the second is a class of allowed quantization rules for the neurons. The information capacity for networks from the second class is fOWld to be of order N 3 bits for a network with N neurons. A generalization of the sum of outer products learning rule is developed and investigated as well. © American Institute of Physics 1988 279 I. INTRODUCTION The ability to perfonn collective computation in a distributed system of flexible structure without global synchronization is an important engineering objective. Hopfield's neural network [1] is such a model of associative content addressable memory. An important property of the Hopfield neural network is its guaranteed convergence to stable states (interpreted as the stored memories). In this work we introduce a generalization of the Hopfield model by allowing the outputs of the neurons to take a richer set of values than Hopfield's original binary neurons. Sufficient conditions for preserving the convergence property are developed for the neuron input output relations. Two classes of relations are obtained. The first introduces neurons which simulate multi threshold functions, networks with such neurons will be called quantized neural networks (Q.N.N.). The second class introduces continuous neuron input output relations and networks with such neurons will be called continuous neural networks (C.N.N.). In Section II, we introduce Hopfield's neural network and show its convergence property. C.N.N. are introduced in Section m and a sufficient condition for the neuron input output continuous relations is developed for preserving convergence. In Section IV, Q.N.N. are introduced and their input output relations are analyzed in the same manner as in III. In Section IV we look further at Q.N.N. by using the definition of information capacity for neural networks of [2] to obtain a tight asymptotic estimate of the capacity for a Q.N.N. with N neurons. Section VI is a generalized sum of outer products learning for the Q.N.N. and section VII is the discussion. n. THE HOPFIELD NEURAL NETWORK A neural network consists of N pairwise connected neurons. The i 'th neuron can be in one of two states: Xi = -lor Xi = + 1. The connections are fixed real numbers denoted by W ij (the connection from neuron i to nelD'On j ). Defme the state vector X to be a binary vector whose i 'th component corresponds to the state of the i 'th neuron. Randomly and asynchronously, each neuron examines its input and decides its next output in the following manner. Let ti be the threshold voltage of the i 'th neuron. If the weighted sum of the present other N -1 neuron outputs (which compose the i 'th neuron input) is 280 greater or equal to ti' the next Xi (xt) is+l. ifnot.Xt is -1. This action is given in (1). N X·+ = sgn [ ~ W··X ·-t· ] I Li IJ J I (1) j=1 We give the following theorem Theorem 1 (of (1)) The network described with symmetric (Wij=Wji ) zero diagonal (Wi;=<» connection matrix W has the convergence property. Defme the quantity 1 N N N E(X) =- ~ ~ W··X·X· + ~ t·X· 2 Li Li IJ I J Li I I i j=1 i=1 (2) We show that E (X) caD only decrease as a result of the action of the network. Suppose that Xk changed to X t = Xl +Mk • the resulting change in E is given by N tJ.E = -llXk ( 1: WkjXj-tk) j=1 (3) (Eq. (3) is correct because of the restrictions on W). The term in brackets is exactly the argument of the sgn function in (1) and therefore the signs of IlXk and the term in brackets is the same (or IlXk =<» and we get!lE ~ O. Combining this with the fact that E (X) is bounded shows that eventually the network will remain in a local minimum of E (X). TlUs cornpJetcs the proof. The technique used in the proof of Theorem 1 is an important tool in analyzing neural networks. A network with a particular underlying E (X) function can be used to solve optimization problems with E (K) as the object of optimization. Thus we see another use of neural networks. 281 m. THE C.N.N. We ask ourselves the following question: How can we change the sgn function in (1) without affecling the convergence property? The new action rule for the i 'th neuron is N X·+=/·[ ~ W··X· ] , 1 kI IJ J j=l (4) Our attention is focused on possible choices for Ii ('). The following theorem gives a part of the answer. Theorem 2 The network described by (4) (with symmetric zero diagonal W) has the convergence property if Ii ( . ) are strictly increasing and bounded. Define (5) We show as before that E ex) can only decrease and since E is bounded (because of the boundedness of Ii's) the theorem is proved. Xj Usinggi(Xi ) = J li-l(u)dU we have o Using the intel111ediate value theorem we gel (6) 282 where C is a point between Xk and Xk +LlXk . Now, if Mk > 0 we have C S; Xk+LlXk = > Ik-I(C) S;fk-1(Xk+Mk ) and the term in brackets is greater or equal to zero => IlE SO. A similar argument holds for Mk < 0 (of course Mk =0 => llE =0). This comp~etes the proof. Some remarks: (a) Strictly increasing bounded neuron relations are not the whole class of relations conserving the convergence property. This is seen immediately from the fact that Hopfield's original model (1) is not in this class. (b) The E (X) in the C.N.N. coincides with Hopfield's continuous neural network [3]. The difference between the two networks lies in the updating scheme. In our C.N.N. the neurons update their outputs at the moments they examine their inputs while in [3] the updating is in the form of a set of differential equations featuring the time evolution of the network outputs. (c) The boundedness requirement of the neuron relations results from the boundedness of E (K). It is possible to impose further restrictions on W resulting in unbounded neuron relations but keeping E (X) bounded (from below). This was done in [4] where the neurons exhibit linear relations. IV. THE Q.N.N. We develop the class of quantization rules for the neurons, keeping the convergence property. Denote the set of possible neuron outputs by Yo < Y 1 < ... < Y n and the set of threshold values by t 1 < t 2 < ... < t n the action of the neurons is given by N xt = Y/ if t/ < L W;jXj ~ tl+l I=O, ... ,n j=1 The following theorem gives a class of quantization rules with the convergence property. (8) 283 Theorem 3 An.y quantization rule for the neurons which is an increasing step functioo that is Yo<Y < . .. y ,t < ... <t 1 n' 1 n (9) Yields a network with the convergence property (with a W symmetric and zero diagonal). We proceed to prove. Define (10) where G (X) is a piecewise linear convex U function defined by the relation (11) As before we show M ~ O. Suppose a change occurred in Xk such thatXk =Yi- 1.Xt=yi . We then have (12) A similar argument follows when Xk =Yi ,Xk+=Yi- 1 < Xk . Any bigger change in Xk (from Yi to Yj with I i - j I > 1) yields the same result since it can be viewed as a sequence of I i - j I changes from Y i to Yj each resulting in M ~O. The proof is completed by noting that LlX'e=O=>M =0 and E (X) is bounded. 284 CorollaIy Hopfield's original model is a special case of (9). V. INFORMATION CAPACITY OF THE Q.N.N. We use the definition of [2] for the information capacity of the Q.N.N. Definition 1 The information capacity of the Q.N.N. (bits) is the log (Base 2) of the number of distinguishable networks of N neurons. Two networks are distinguishable if observing the state transitions of the neurons yields different observations. For Hopfield's original model it was shown in [2] that the capacity C of a network of N neurons is bounded by C ~ log (2(N-l)2f = O(N3)b. It was also shown that C ~ Q(N3)b and thus is exactly of the order N 3b. It is obvious that in our case (which contains the original model) we must have C ~ Q(N3)b as well (since the lower bound cannot decrease in this richer case). It is shown in the Appendix that the number of multi threshold functions of N -1 variables with n+l oUlput levels is at most (n+lf2+N+1 since we have N neurons there will be ( (n+lf2+N+1f distinguishablenetworlcs and thus (14) 01 as before, C is exactly of O(N3)b. In fact, the rise in C is probably a faclOr of O(log2n) as can be seen from the upper bound. VI. "OUTER PRODUCT" LEARNING RULE For Hopfleld's origiDal network with two state neurons (taking the values ±1) a nalw-al and extensively investigated r l.t 1.£ ] learning rule is the so called sum of outer products construction. 1 K 1 1 W .. =- ~ X·X· 1) N ~ 1 ) 1=1 (15) where Xl, ... , X K are the desired stable states of the network. A well-known result for (15) is that the asymplOtic capacity K of the network is K= N-l +1 410gN 285 (16) In this section we introduce a natural generalization of (15) and prove a similar result for the asymptotic capacity. We first limit the possible quantization rules to: with Y < ... < Y o n with t.=~(y.+y. IJ J 2 J J(a) n+l is even (b) V i Yi -:# 0 (c) y. =-y . I n-l j=l, ... n i=O, ... ,n (17) N eAt we state that the desired stable vectors Xl, . . . X K are such that each component is picked independently at random from ( Yo ' . . . Y M } with equal probability. Thus. the K • N components of the X 's are zero mean i.i.D random variables. Our modified learning rule is w .. = -L ~ X!. [_1 ] IJ N ~ I Xl 1=1 j (18) Note that for Xi E (+1, -I} (18) is identical to (16). Define 286 ;~~ IYi -Yjl l¢oJ IY.12 A = max l iJ IYj I We state that PROPOsmON: The asymptotic capacity of the above network is given by N K=----16A 2 logN ,.., (6y)2 PROOF: Def"me { K vectors chosen randomly as deSCribed} P (K , N) = P r are stable states with the W of ( ) (19) (20) where Aij is the event that the i th component of j th vector is in error. We concentrate on the event All W.L.G. The input u 1 when X' is presented is given by (21) The first term is mapped by (17) into itself and corresponds to the desired Signal. The last term is a sum of (K -1 )(N -1) i.i.D zero mean random variables and corresponds to noise. 287 K-l 1 K-l The middle term -N X 1 is disposed of by assuming -N ~ O. (With a zero diagonal N -+00 choice of W (using (18) with i *' j) this term does not appear). P r (A 11) = P r { noise gets us out of range } Denoting the noise by I we have (K -1)(N-l)4A 2 (22) where the first inequality is from the defmition of .1Yand the second uses the lemma of [6] p. 58. We thus get ,.., (,1Y)2N2 P (K , N) ~ 1 - K • N . 2exp -~---'---~ 8(K -l)(N-l)A 2 (23) substituting (19) and taking N ~ 00 we get P (K , N) ~ 1 and this completes the proof. Vll. DISCUSSION Two classes of generalization of the Hopfield neural network model were presented. We give some remarks: (a) Any combination of neurons from the two classes will have the convergence property as well. (b) Our defmition of the information capacity for the eN.N. is useless since a full observation of the pos· sible state transitions of the netwock is impossible. 288 APPENDIX We prove the following theorem. Theorem An upper bound on the num~ of multi threshold functions with N inputs and M points in the domain (out of(n+l)N possible points) et/ is the solution of the recurrence relation eM - CM - 1 + n ·CM - 1 N N N-l (A.I) Let us look on the N dimensional weight space W. Each input point X divides the weight space N into n+l regions by n parallel hyperplanes L W;X;=tk k=l, ... ,n. We keep adding points in such ;=1 a way that the new n hypeq>1anes corresponding to each added point partition the W space into as many regions as possible. Assume M -1 points have made e t! -I regions and we add the M 'lh point. Each hyperplane (out of n) is divided into at most Cf/_l1 region, (being itself an N -1 dimensional space divided by (M -1)n hyperlines). We thus have after passing the n hyperplanes: eM - CM - I + n ·CA1- 1 N N N-I N-l[ M-1] is e tI = (n + 1).L i n i and the theorem is proved . • =0 The solution of the recurrence in the case M =(n + I f (all possible points) we have a bound on the number of multi threshold functions of N variables equal to and the result used is established. 289 LIST OF REFERENCES [1] Hopfield J. J.t "Neural networks and physical systems with emergent collective computational abilities", Proc. Nat. Acad. Sci. USA, Vol. 79 (1982), pp. 2554-2558. [2] Abu-Mostafa Y.S. and Jacques J. St, "lnfonnation capacity of the Hopfield model", IEEE Trans. on Info. Theory, Vol. IT-31 (1985. ppA61-464. [3] Hopfield J. J., "Neurons with graded response have collective computational properties like those of two state neurons", Proc. Nat. Acad. Sci. USA, Vol. 81 (1984). [4] Fleisher M., "Fast processing of autoregressive signals by a neural network", to be presented at IEEE Conference, Israel 1987. [5] Levin, E., Private communication. [6] Pettov, "Sums of independent random variables".
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174 A Neural Network Classifier Based on Coding Theory Tzt-Dar Chlueh and Rodney Goodman eanrornla Instltute of Technology. Pasadena. eanromla 91125 ABSTRACT The new neural network classifier we propose transforms the classification problem into the coding theory problem of decoding a noisy codeword. An input vector in the feature space is transformed into an internal representation which is a codeword in the code space, and then error correction decoded in this space to classify the input feature vector to its class. Two classes of codes which give high performance are the Hadamard matrix code and the maximal length sequence code. We show that the number of classes stored in an N-neuron system is linear in N and significantly more than that obtainable by using the Hopfield type memory as a classifier. I. INTRODUCTION Associative recall using neural networks has recently received a great deal of attention. Hopfield in his papers [1,2) deSCribes a mechanism which iterates through a feedback loop and stabilizes at the memory element that is nearest the input, provided that not many memory vectors are stored in the machine. He has also shown that the number of memories that can be stored in an N-neuron system is about O.15N for N between 30 and 100. McEliece et al. in their work (3) showed that for synchronous operation of the Hopfield memory about N /(2IogN) data vectors can be stored reliably when N is large. Abu-Mostafa (4) has predicted that the upper bound for the number of data vectors in an N-neuron Hopfield machine is N. We believe that one should be able to devise a machine with M, the number of data vectors, linear in N and larger than the O.15N achieved by the Hopfield method. N N Feature Space = B = {-1 , 1 } L L Code Space = B = {-1 • 1 } Figure 1 (a) Classification problems versus (b) Error control decoding problems In this paper we are specifically concerned with the problem of classification as in pattern recognition. We propose a new method of building a neural network classifier, based on the well established techniques of error control coding. ConSider a typical classification problem (Fig. l(a)), in which one is given a priori a set of classes, C( a), a = 1, .... M. Associated with each class is a feature vector which labels the class ( the exemplar of the class), I.e. it is the © American Institute of Physics 1988 175 most representative point in the class region. The input is classified into the class with the nearest exemplar to the input. Hence for each class there is a region in the N-dimensional binary feature space BN == (I. -I}N. in which every vector will be classified to the corresponding class. A similar problem is that of decoding a codeword in an error correcting code as shown in Fig. I(b). In this case codewords are constructed by design and are usually at least dmtn apart. The received corrupted codeword is the input to the decoder. which then finds the nearest codeword to the input. In principle then. if the distance between codewords is greater than 2t +1. it is possible to decode (or classify) a noisy codeword (feature vector) into the correct codeword (exemplar) provided that the Hamming distance between the noisy codeword and the correct codeword is no more than t. Note that there is no guarantee that the exemplars are uniformly distributed in BN. consequently the attraction radius (the maximum number of errors that can occur in any given feature vector such that the vector can st111 be correctly classified) will depend on the minimum distance between exemplars. Many solutions to the minimum Hamming distance classification have been proposed. the one commonly used is derived from the idea of matched filters in communication theory. Lippmann [5) proposed a two-stage neural network that solves this classification problem by first correlating the input with all exemplars and then picking the maximum by a "winner-take-all" circuit or a network composed of two-input comparators. In Figure 2. fI.f2 .... .fN are the N input bits. and SI.S2 .... SM are the matching score s(Similartty) of f with the M exemplars. The second block picks the maximum of sI.S2 ..... SM and produces the index of the exemplar with the largest score. The main disadvantage of such a classifier is the complexity of the maximum-picking circuit. for example a ''winner-take-all'' net needs connection weights of large dynamic range and graded-response neurons. whilst the comparator maximum net demands M-I comparators organized in log2M stages. M A X I M U M cloSS(f) (0:) f = d + e (0:) • g = c + e ,,-_;_~ .... ~ DECODER~SS(f) Feature Code Space Space Fig. 2 A matched filter type classifier Fig. 3 Structure of the proposed classifier Our main idea is thus to transform every vector in the feature space to a vector in some code space in such a way that every exemplar corresponds to a codeword in that code. The code should preferably (but not necessarily) have the property that codewords are uniformly distributed in the code space. that is, the Hamming distance between every pair of codewords is the same. With this transformation. we turn the problem of classification into the coding problem of decoding a noisy codeword. We then do error correction decoding on the vector in the code space to obtain the index of the noisy codeword and hence classify the original feature vector. as shown in Figure 3. This paper develops the construction of such a classification machine as follows. First we conSider the problem of transforming the input vectors from the feature space to the code space. We describe two hetero-associative memories for dOing this. the first method uses an outer product matrix technique Similar to 176 that of Hopfield's. and the second method generates its matrix by the pseudoinverse techruque[S.7J. Given that we have transformed the problem of associative recall. or classification. into the problem of decoding a noisy codeword. we next consider suitable codes for our machine. We require the codewords in this code to have the property of orthogonality or pseudo-orthogonality. that is. the ratio of the cross-correlation to the auto-correlation of the codewords is small. We show two classes of such good codes for this particular decoding problem l.e. the Hadamard matrix codes. and the maximal length sequence codes[8J. We next formulate the complete decoding algorithm. and describe the overall structure of the classifier in terms of a two layer neural network. The first layer performs the mapping operation on the input. and the second one decodes its output to produce the index of the class to which the input belongs. The second part of the paper is concerned with the performance of the classifier. We first analyze the performance of this new classifier by finding the relation between the maximum number of classes that can be stored and the classification error rate. We show (when using a transform based on the outer product method) that for negligible misclassification rate and large N. a not very tight lower bound on M. the number of stored classes. is 0.22N. We then present comprehensive simulation results that confirm and exceed our theoretical expectations. The Simulation results compare our method with the Hopfield model for both the outer product and pseudo-inverse method. and for both the analog and hard limited connection matrices. In all cases our classifier exceeds the performance of the Hopfield memory in terms of the number of classes that can be reliably recovered. D. TRANSFORM TECHNIQUES Our objective is to build a machine that can discriminate among input vectors and classify each one of them into the appropriate class. Suppose d(a) E BN is the exemplar ofthe corresponding class e(a.). a. = 1.2 ..... M . Given the input f . we want the machine to be able to identify the class whose exemplar is closest to f. that is. we want to calculate the follOWing function. class ( f) = a. if f I f - d( a) I < I f - dH3) I where I I denotes Hamming distance in BN. We approach the problem by seeking a transform ~ that maps each exemplar d(a) in BN to the corresponding codeword w(a) in BL. And an input feature vector f = dey) + e is thus mapped to a noisy codeword g = wlY) + e' where e is the error added to the exemplar, and e' is the corresponding error pattern in the code space. We then do error correction decoding on g to get the index of the corresponding codeword. Note that e' may not have the same Hamming weight as e, that is, the transformation ~ may either generate more errors or eliminate errors that are present in the original input feature vector. We require ~ to satisfy the following equation, 0.=0,1 ..... M-l and ~ will be implemented uSing a Single-layer feedfoIWard network. 177 Thus we first construct a matrix according to the sets of d(a)'s and w(a)'s, call it T, and define r:, as where sgn is the threshold operator that maps a vector in RL to BL and R is the field of real numbers. Let D be an N x M matrix whose <lth column is d( a) and W be an L x M matrix whose ~th column Is w(~). The two possible methods of constructing the matrix for r:, are as follows: Scheme A (outer product method) [3,6] : In this scheme the matrix T Is defined as the sum of outer products of all exemplar-codeword pairs, i.e. M-l T(A)y = L Wl(e:)· die:) or equivalently. T(A) = WDt Scheme B (pseudo-Inverse method) [6.7] : We want to find a matrix T(B) satisfying the follOWing equation, 'f{B) D = W In general D is not a square matrix, moreover D may be singular, so D-l may not exist. To circumvent this difficulty, we calculate the pseudo-inverse (denoted Dt) of the matrix D instead of its real Inverse, let Dt::= (DtD)-lDt. T(B) can be formulated as, 'f{B) = W Dt = W (ot D)-l nt m. CODES The codes we are looking for should preferably have the property that its codewords be distributed uniformly in BL, that is, the distance between each two codewords must be the same and as large as pOSSible. We thus seek classes of eqUidistant codes. Two such classes are the Hadamard matrix codes, and the maximal length sequence codes. First define the word pseudo-orthogonal. Defmition: Let w(a) = (wO(a),Wl (a), ..... , WL-l (a)) E BL be the ath codeword of code C, where a = 1,2, ... ,M. Code C is said to be pseudo-orthogonal iff L-l (w(a) , w(~)) = L Wl(a) Wl(~) 1=0 =(~ ::: where ( , ) denotes inner product of two vectors. where E «L Hadamard Matrices: An orthogonal code of length L whose L codewords are rows or columns of an L x L Hadamard matrix. In this case e = 0 and the distance between any two codewords is L/2. It is conjectured that there exist such codes for all L which are multiples of 4, thus providing a large class of codes[8]. 178 Mazlmal Length Sequence Codes: There exists a family of maximal length sequence (also called pseudo-random or PN sequence) codes(8). generated by shift registers. that satisfy pseudo-orthogonality with e = -1. Suppose 9 (x) is a primitive polynomial over OF (2) of degree D. and let L = 2D -1. and if 00 f(xl = l/g (xl = L ck· xk k=O then CO.Cl •.••... is a periodic sequence of period L ( since 9 (x) I x L - 1). If code C is made up of the L cyclic shifts of c = ( 1 - 2 CO. 1 - 2 cl •... 1 - 2 c L - Il then code C satisfies pseudo-orthogonality with E = -1. One then easily sees that the minimum distance of this code is (L - 1)/2 which gives a correcting power of approximately L/4 errors for large L. IV. OVERALL CLASSIFIER STRUCTURE We shall now describe the overall classifier structure. essentially it consists of the mapping ~ followed by the error correction decoder for the maximal length sequence code or Hadamard matrix code. The decoder operates by correlating the input vector with every codeword and then thresholding the result at (L + e)/2. The rationale of this algorithm is as follows. since the distance between every two codewords in this code is exactly (L - e)/2 bits. the decoder should be able to correct any error pattern with less than (L - e) / 4 errors if the threshold is set halfway between Land e I.e. (L + e )/2. Suppose the input vector to the decoder is g = w< a) + e and e has Hamming weight s (i.e. s nonzero components) then we have (g. w (0:)) = L - 2s (g • w (~) ~ 2s + E where ~ i= a From the above equation. if g is less than (L - e) /4 errors away from w( a) (I.e. s < (L - e)/4) then (g • W<a)) will be more than (L + e)/2 and (g • w(~)) will be less than (L + e)/2. for all ~ #= a. As a result. we arrive at the following decoding algorithm. deax1e (g) = sgn ( w t g - ( (L + E)/2)j ) where j = [ 1 1 ..... 1 )t • which is an M x 1 vector. In the case when E = -1 and less than (L+l)/4 errors in the input. the output will be a vector in SM == {l.-I}M with only one component positive (+1). the index of which is the index of the class that the input vector belongs. However if there are more than (L+ 1) / 4 errors. the output can be either the all negative( -1) vector (decoder failure) or another vector with one pOSitive component(decoder error). The function class can now be defined as the composition of ~ and decode. the overall structure of the new classifier is depicted in Figure 4. It can be viewed as a twO-layer neural network with L hidden units and M output neurons. The first layer is for mapping the input feature vector to a noisy codeword in the code space ( the "internal representation" ) while the second one decodes the first's output and produces the index of the class to which the input belongs. 179 T(A) or T (8) t 9 1 W f1 f2 • • • f N-1 fN 9L Figure 4 Overall architecture of the new neural network classifier v. PERFORMANCE ANALYSIS From the previous section, we know that our classifier will make an error only if the transformed vector in the code space, which is the input to the decoder, has no less than (L - e)/4 errors. We now proceed to find the error rate for this classifier in the case when the input is one of the exemplars (i.e. no error), say f = d(~) and an outer product connection matrix for~. Following the approach of McEl1ece et. al.[31, we have N-l M-l (~ d(~)h = sgl ( L L Wl(a) dj(a) dj(~) ) j=o a= 0 N-l M-l = sgn( N wl(f3) + L L Wl(a) dla ) dj(~) ) j=o a=O a~~ Assume without loss of generality that Wl(~) = -I, and if N-l M-l X == L L Wl(a) dla) dj(~) ~ N j=O a=o a~~ then Notice that we assumed all d(a)'s are random, namely each component of any d(a) is the outcome of a Bernoulli trial, accordingly, X is the sum of N(M-l) independent identically distributed random variables with mean 0 and variance 1. In the asymptotic case, when Nand M are both very large, X can be approximated by a normal distribution with mean 0, variance NM. Thus p Pr { (~d(~)h ~ Wl(~)} Q(vlN/M) where Q(x) _1_ fOO t2/2 = vi 2 TT x e dt 180 Next we calculate the misclassification rate of the new classifier as follows (assuming E« L), Pe = L ~ (L) pk(l_p)L-k k=IL/4J k where L J is the integer floor. Since in generallt is not possible to express the summation expliCitly, we use the Chernoff method to bound Pe from above. Multiplying each term in the summation by a number larger than unity ( et(k - L/4) with t > 0 ) and summing from k = 0 instead of k = L L/ 4 J ' L Pe < L (L ) P k (l-p) L -k e t(k -L/4) = e -L t/4 (1 _ P + p et ) L k=O k Differentiating the RHS of the above equation w.r.t. t and set it to 0, we find the optimal to as eto = (l-p)/3p. The condition that to > 0 implies that p < 1/4, and since we are dealing with the case where p is small, it is automatically satisfied. Substituting the optimal to, we obtain where c = 4/(33/ 4 ) = 1.7547654 From the expression for Pe ,we can estimate M, the number of classes that can be classified with negllgible misclassification rate, in the following way, suppose Pe = () where ()« land p « 1, then For small x we have g-l(Z) - ../2 Log ( i/z) and since () is a fixed value, as L approaches infinity, we have M> N =.l:L 810gc 4.5 From the above lower bound for M, one easily see that this new machlne is able to classify a constant times N classes, which is better than the number of memory items a Hopfield model can store Le. N/(210gN). Although the analysis is done assumlng N approaches lnfinlty, the simulation results in the next section show that when N is moderately large (e.g. 63) the above lower bound applles. VI. SIMULATION RESULTS AND A CHARACTER RECOGNITION EXAMPLE We have Simulated both the Hopfield model and our new machine(using maxlmallength sequence codes) for L = N = 31, 63 and for the following four cases respectively. (1) connection matrix generated by outer product method (ti) connection matrix generated by pseudo-inverse method (ill) connection matrix generated by outer product method, the components of the connection matrix are hard limited. (iv) connection matrix generated by pseudo-inverse method, the components of the connection matrix are hard limited. 181 For each case and each choice of N. the program fixes M and the number of errors in the input vector. then randomly generates 50 sets of M exemplars and computes the connection matrix for each machine. For each machine it randomly picks an exemplar and adds nOise to it by randomly complementing the specified number of bits to generate 20 trial input vectors. it then simulates the machine and checks whether or not the input is classified to the nearest class and reports the percentage of success for each machine. The simulation results are shown in Figure 5. in each graph the hOrizontal axis is M and the vertical axis is the attraction radius. The data we show are obtained by collecting only those cases when the success rate is more than 98%, that is for fixed M what is the largest attraction radius (number of bits in error of the input vector) that has a success rate of more than 98%. Here we use the attraction radiUS of -1 to denote that for this particular M. with the input being an exemplar. the success rate is less than 98% in that machine . _e_ Hopfield Model .0- New Classifier{Op) • New Classtfier{PI) N=31 Binary Connection Matrix -, "18101114 u ,. :tD " :1' I. II '0 § 23f ..... CIl 21 .... :s (,).~ '9~ ~'1.:! 17 , .... ... .. , ::;:0:: 15 • ..... 13 11 9 7 5 3 M (a) N=63 Binary Connection Matrix -1 ~~~~~~~++~~I 3 7 11 1519 23 27 31 35 39 43 47 51 55 59 83 M (c) N=31 Analog Connection Matrix - ,~~~ .. ++++++++ .. ~~~~~~ d 23 .2 en 21 tl.a 19 f! ~ 17 ~~ 15 <: 13 11 9 7 5 3 , " • a ,0 12 '" ,. 18 'o It lit II ,. '0 M (h) N=63 Analog Connection Matrix 3 7 11 1519 23 27 31 35 39 43 47 61 M (d) Figure 5 Simulation results of the Hopfield memory and the new classifier 182 _e_ Hopfield Model .0- New Classifier(OP.L=63) .... New Classifier(OP.L=31) ~ 23 1 .9 rIl 21 1 u.a 19 ."--II.I--".~o....... ~"C 17 ......... -=-.~ :::~15 ,~ < 13 -.~~ '~ \ ~-:~ -1 +---+-_e __ -4eO-,e ___ e_-4e __ .. e e_ .. e __ e 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 M Figure 6 Perfonnance of the new classifier using codes of different lengths In all cases our classifier exceeds the perfonnance of the Hopfield model in tenns of the number of classes that can be reliably recovered_ For example. consider the case of N = 63 and a hard limited connection matrix for both the new classifier and the Hopfield model. we find that for an attraction radius of zero. that is. no error in the input vector. the Hopfield model has a classification capacity of approximately 5. while our new model can store 47. Also. for an attraction radius of 8. that is. an average of N/8 errors in the input vector. the Hopfield model can reliably store 4 classes while our new model stores 27 classes. Another Simulation (Fig. 6) USing a shorter code (L = 31 instead of L = 63) reveals that by shortening the code. the performance of the classifier degrades only slightly. We therefore conjecture that it is pOSSible to use traditional error correcting codes (e.g. BCH code) as internal representations. however. by going to a higher rate code, one is trading minimum distance of the code (error tolerance) for complexity (number of hidden units). which implies pOSSibly poorer performance of the classifier. We also notice that the superiority of the pseudoinverse method over the outer product method appears only when the connection matrices are hard limited. The reason for this is that the pseudOinverse method is best for decorrelating the dependency among exemplars, yet the exemplars in this simulation are generated randomly and are presumably independent. consequently one can not see the advantage of pseudoinverse method. For correlated exemplars, we expect the pseudoinverse method to be clearly better (see next example). Next we present an example of applying this classifier to recognizing characters. Each character is represented by a 9 x 7 pixel array, the input is generated by flipping every pixel with 0.1 and 0.2 probability. The input is then passed to five machines: Hopfield memory. the new classifier with either the pseudotnverse method or outer product method, and L = 7 or L = 31. Figure 7 and 8 show the results of all 5 machines for 0.1 and 0.2 pixel flipping probability respectively, a blank output means that the classifier refuses to make a decision. First note that the L = 7 case is not necessarily worse than the L = 31 case. this confirms the earlier conjecture that fewer hidden units (shorter code) only degrades perfonnance slightly. Also one eaSily sees that the pseudoinverse method is better than the outer product method because of the correlation between exemplars. Both methods outperform the Hopfield memory since the latter mixes exemplars that are to be remembered and produces a blend of exemplars rather than the exemplars themselves, accordingly it cannot classify the input without mistakes. (a) " · . · . ~ . · . :':'J L:.) . ~ . · . · . ... · . · . · . .... _. · . · . . (b) " · . · " ... · . · . . . ~. .. · · .-· . · . · . · , .. t; .. · . · · · ., · . (c) ,'. . . .. , , .. . ; : .. ' L (d) " · . ... . .. · . · . ~; :-' . . · · . .. ... , · -· .. F ,.', L-; (e) . " · --· . · . .. · :.. L. _. · ; ; . t'::l (f) " · . --· . · . · . .. · · ; . :... · . · · .. · -U .•. · . -· t: : : .. · . l.:1i ~~ .. · . · . . .. .. · . · · · --· . · . · . · . -. _. · [~ .j' . . · Lo:: Figure 7 The character recognition example with 10% pixel reverse probability (a) input (b) correct output (c) Hopfield Model (d)-(g) new classifier (d) OP, L = 7 (e)OP, L = 31 (1) PI, L = 7 (g) PI, L = 31 (a) .. " . -.. --· . · . J'"l ~; _. · _. --... " . " '.:..' · .. · . · . · , · . ." · . ~. .. · . " II " .. _ .. · (b) " -. ..-· . ~ ~ · . · . · . , . · . .. --· . · . .-· . · . · . · . _ . · · . . · · _. · · · · . · · · · , (c) .'. .-· . · . · . E -· · · · ~.' · · · .. . , · · , .. . · · · · : .... (d) " · . · -· . .•. · . · . , . .. · · . · . · . · . -. .. · -U F .. i :.::: (e) . " , . --· . · . · . · . :' L' --· · :-· -· . · . · . , . · . · (f) " · . _.-· . · . -b -· . L . . · -· . L.J ..-· L. .. L_ ! · . · · · .. ' .. -: {g} · . -. · . · . · . · . ' .. · .. · . . . .. · · · · .. -· : : L~ -· , · _ . -· · . . , · · . Figure 8 The character recognition example with 20016 pixel reverse probability (a) input (b) correct output (c) Hopfield Model (d)-(g) new classifier (d) OP, L = 7 (e)OP. L = 31 (1) PI. L = 7 (g) PI. L = 31 Vll. CONCLUSION 183 In this paper we have presented a new neural network classifier design based on coding theory techniques. The classifier uses codewords from an error correcting code as its internal representations. Two classes of codes which give high performance are the Hadamard matrix codes and the maximal length sequence codes. In penormance terms we have shown that the new machine is significantly better than using the Hopfield model as a classifier. We should also note that when comparing the new classifier with the Hopfield model. the increased performance of the new classifier does not entail extra complexity. since it needs only L + M hard limiter neurons and L(N + M) connection weights versus N neurons and N2 weights in a Hopfield memory. In conclusion we believe that our model forms the basis of a fast. practical method of classification with an effiCiency greater than other previOUS neural network techniques. REFERENCES [1) J. J. Hopfield. Proc. Nat. Acad. Set USA, Vol. 79. pp. 2554-2558 (1982). [2) J. J. Hopfield. Proc. Nat. Acad. Set USA, Vol. 81, pp. 3088-3092 (1984). [3) R J. McEliece, et. aI, IEEE Tran. on Infonnation. Theory. Vol. IT-33. pp. 461-482 (1987). [4) Y. S. Abu-Mostafa and J. St. Jacques. IEEE Tran. on Information Theory • Vol. IT-3I, pp. 461-464 (1985). [5) R Lippmann, IEEEASSP Magazine, Vol. 4, No.2. pp. 4-22 (April 1987). [6) T. Kohonen. Associative Memory A System-Theoretical Approach (Springer-Verlag. Berlin Heidelberg. 1977). [7) S. S. Venkatesh,Linear Map with Point Rules ,Ph. D Thesis, Caltech, 1987. [8) E. R Berlekamp. Algebraic Coding Theory. Aegean Park Press. 1984.
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515 MICROELECTRONIC IMPLEMENTATIONS OF CONNECTIONIST NEURAL NETWORKS Stuart Mackie, Hans P. Graf, Daniel B. Schwartz, and John S. Denker AT&T Bell Labs, Holmdel, NJ 07733 Abstract In this paper we discuss why special purpose chips are needed for useful implementations of connectionist neural networks in such applications as pattern recognition and classification. Three chip designs are described: a hybrid digital/analog programmable connection matrix, an analog connection matrix with adjustable connection strengths, and a digital pipe lined best-match chip. The common feature of the designs is the distribution of arithmetic processing power amongst the data storage to minimize data movement. ... 0 Q)Q) ,c"C E~ ::::S,.... ZO 1 1 RAMs ••••• /..... Distributed / '. '. co mputati on chips / ..... . ''iiit:::::::;:::::, ••• .. .. • • • • .. Conventional CPUs 10 3 10 6 10 9 Node Complexity (No. of Transistors) Figure 1. A schematic graph of addressable node complexity and size for conventional computer chips. Memories can contain millions of very simple nodes each with a very few transistors but with no processing power. CPU chips are essentially one very complex node. Neural network chips are in the distributed computation region where chips contain many simple fixed instruction processors local to data storage. (After Reece and Treleaven 1 ) © American Institute of Physics 1988 516 Introduction It is clear that conventional computers lag far behind organic computers when it comes to dealing with very large data rates in problems such as computer vision and speech recognition. Why is this? The reason is that the brain performs a huge number of operations in parallel whereas in a conventional computer there is a very fast processor that can perform a variety of instructions very quickly, but operates on only two pieces of data at a time. The rest of the many megabytes of RAM is idle during any instruction cycle. The duty cycle of the processor is close to 100%, but that of the stored data is very close to zero. If we wish to make better use of the data, we have to distribute processing power amongst the stored data, in a similar fashion to the brain. Figure 1 illustrates where distributed computation chips lie in comparison to conventional computer chips as regard number and complexity of addressable nodes per chip. In order for a distributed strategy to work, each processing element must be small in order to accommodate many on a chip, and communication must be local and hardwired. Whereas the processing element in a conventional computer may be able to execute many hundred different operations, in our scheme the processor is hard-wired to perform just one. This operation should be tailored to some particular application. In neural network and pattern recognition algorithms, the dot products of an input vector with a series of stored vectors (referred to as features or memories) is often required. The general calculation is: Sum of Products v . F(i) = L v. f.. . J IJ J where V is the input vector and F(i) is one of the stored feature vectors. Two variations of this are of particular interest. In feature extraction, we wish to find all the features for which the dot product with the input vector is greater than some threshold T, in which case we say that such features are present in the input vector. Feature Extraction v . F(i) = L v. f.. . J IJ J In pattern classification we wish to find the stored vector that has the largest dot product with the input vector, and we say that the the input is a member of the class represented by that feature, or simply that that stored vector is closest to input vector. Classification max(V. F(i) = LV. f.. . J IJ J The chips described here are each designed to perform one or more of the above functions with an input vector and a number of feature vectors in parallel. The overall strategy may be summed up as follows: we recognize that in typical pattern recognition applications, the feature vectors need to be changed infrequently compared to the input 517 vectors, and the calculation that is perfonned is fixed and low-precision, we therefore distribute simple fixed-instruction processors throughout the data storage area, thus minimizing the data movement and optimizing the use of silicon. Our ideal is to have every transistor on the chip doing something useful during every instruction cycle. Analog Sum-or-Products U sing an idea slightly reminiscent of synapses and neurons from the brain, in two of the chips we store elements of features as connections from input wires on which the elements of the input vectors appear as voltages to summing wires where a sum-ofproducts is perfonned. The voltage resulting from the current summing is applied to the input of an amplifier whose output is then read to determine the result of the calculation. A schematic arrangement is shown in Figure 2 with the vertical inputs connected to the horizontal summing wires through resistors chosen such that the conductance is proportional to the magnitude of the feature element. When both positive and negative values are required, inverted input lines are also necessary. Resistor matrices have been fabricated using amorphous silicon connections and metal linewidths. These were programmed during fabrication by electron beam lithography to store names using the distributed feedback method described by Hopfield2,3. This work is described more fully elsewhere.4,5 Hard-wired resistor matrices are very compact, but also very inflexible. In many applications it is desirable to be able to reprogram the matrix without having to fabricate a new chip. For this reason, a series of programmable chips has been designed. Input lines Feature 4 -t-----tI----4t---t---f--.---1 Feature 3 -+--4II--I--'--+---+--4~ o c: "'C c: Feature 2 ~--+-""'-+--4---1~--I (I) Feature 1 Figure 2. A schematic arrangement for calculating parallel sum-of-products with a resistor matrix. Features are stored as connections along summing wires and the input elements are applied as voltages on the input wires. The voltage generated by the current summing is thresholded by the amplifer whose output is read out at the end of the calculation. Feedback connections may be 518 made to give mutual inhibition and allow only one feature amplifier to tum on, or allow the matrix to be used as a distributed feedback memory. Programmable Connection Matrix Figure 3 is a schematic diagram of a programmable connection using the contents of two RAM cells to control current sinking or sourcing into the summing wire. The switches are pass transistors and the 'resistors' are transistors with gates connected to their drains. Current is sourced or sunk if the appropriate RAM cell contains a '1' and the input Vi is high thus closing both switches in the path. Feature elements can therefore take on values (a,O,-b) where the values of a and b are determined by the conductivities of the n- and p-transistors obtained during processing. A matrix with 2916 such connections allowing full interconnection of the inputs and outputs of 54 amplifiers was designed and fabricated in 2.5Jlm CMOS (Figure 4). Each connection is about 100x100Jlm, the chip is 7x7mm and contains about 75,000 transistors. When loaded with 49 49-bit features (7x7 kernel), and presented with a 49-bit input vector, the chip performs 49 dot products in parallel in under 1Jls. This is equivalent to 2.4 billion bit operations/sec. The flexibility of the design allows the chip to be operated in several modes. The chip was programmed as a distributed feedback memory (associative memory), but this did not work well because the current sinking capability of the n-type transistors was 6 times that of the p-types. An associative memory was implemented by using a 'grandmother cell' representation, where the memories were stored along the input lines of amplifiers, as for feature extraction, but mutually inhibitory connections were also made that allowed only one output to tum on. With 10 stored vectors each 40 bits long, the best match was found in 50-600ns, depending on the data. The circuit can also be programmed to recognize sequences of vectors and to do error correction when vectors were omitted or wrong vectors were inserted into the sequences. The details of operation of the chip are described more fully elsewhere6. This chip has been interfaced to a UNIX minicomputer and is in everyday use as an accelerator for feature extraction in optical character recognition of handwritten numerals. The chip speeds up this time consuming calculation by a factor of more than 1000. The use of the chip enables experiments to be done which would be too time consuming to simulate. Experience with this device has led to the design of four new chips, which are currently being tested. These have no feedback capability and are intended exclusively for feature extraction. The designs each incorporate new features which are being tested separately, but all are based on a connection matrix which stores 46 vectors each 96 bits long. The chip will perform a full parallel calculation in lOOns. 519 VDD ~ , Vj ~ Excitatory Output(!) <:] Inhibitory V· Ivss J Figure 3. Schematic diagram of a programmable connection. A current sourcing or sinking connection is made if a RAM cell contains a '1' and the input Vi is high. The currents are summed on the input wire of the amplifier. ®1 Pads § Row Decoders r:--3 Connections ITII1 Amplifie rs Figure 4. Programmable connection matrix chip. The chip contains 75,000 transistors in 7x7mm, and was fabricated using 2.5Jlm design rules. 520 Adaptive Connection Matrix Many problems require analog depth in the connection strengths, and this is especially important if the chip is to be used for learning, where small adjustments are required during training. Typical approaches which use transistors sized in powers of two to give conductance variability take up an area equivalent to the same number of minimum sized transistors as the dynamic range, which is expensive in area and enables only a few connections to be put on a chip. We have designed a fully analog connection based on a DRAM structure that can be fabricated using conventional CMOS technology. A schematic of a connection and a connection matrix is shown in Figure 5. The connection strength is represented by the difference in voltages stored on two MOS capacitors. The capacitors are 33Jlm on edge and lose about 1 % of their charge in five minutes at room temperature. The leakage rate can be reduced by three orders of magnitude by cooling the the capacitors to -50°C and by five orders of magnitude by cooling to -100°C. The output is a current proportional to the product of the input voltage and the connection strength. The output currents are summed on a wire and are sent off chip to external amplifiers. The connection strengths can be adjusted using transferring charge between the capacitors through a chain of transistors. The connections strengths may be of either polarity and it is expected that the connections will have about 7 bits of analog depth. A chip has been designed in 1.25Jlm CMOS containing 1104 connections in an array with 46 inputs and 24 outputs. Weight update and decay by shifting charge .1 ..L 1.. 02 '-1'--, Input w (l (01-02) Output=w*lnput :or: Input ... 1" .. .4111. 'r" ...... , . .. '" ... ... ... ...... .. ...... ... ...... .. ...... ... .... -... ! Output through external amplifiers Figure 5. Analog connection. The connection strength is represented by the difference in voltages stored on two capacitors. The output is a current proprtional to the product of the input voltage and the connection strength. Each connection is 70x240Jlm. The design has been sent to foundry, and testing is expected to start in April 1988. The chip has been designed to perform a network calculation in <30ns, i.e., the chip will perform at a rate of 33 billion multiplies/sec. It can be used simply as a fast analog convolver for feature extraction, or as a learning 521 engine in a gradient descent algorithm using external logic for connection strength adjustment. Because the inputs and outputs are true analog, larger networks may be formed by tiling chips, and layered networks may be made by cascading through amplifiers acting as hidden units. Digital Classifier Chip The third design is a digital implementation of a classifier whose architecture is not a connectionist matrix. It is nearing completion of the design stage, and will be fabricated using 1.25Jlm CMOS. It calculates the largest five V·P(i) using an alldigital pipeline of identical processors, each attached to one stored word. Each processor is also internally pipelined to the extent that no stage contains more than two gate delays. This is important, since the throughput of the processor is limited by the speed of the slowest stage. Each processor calculates the Hamming distance (number of difference bits) between an input word and its stored word, and then compares that distance with each of the smallest 5 values previously found for that input word. An updated list of 5 best matches is then passed to the next processor in the pipeline. At the end of the pipeline the best 5 matches overall are output. Data Best match list (1 ) Features stored in pipeline pipeline r ing shift register ~ ~ Tag register ~ ~ + :{ it :i:: it {{ :::::: Ii :::::;::::::. f:: Jr }/ ::{ it :r :::::: :mIfl ::t:t if::::: t::}; {}} [, II:::::I :::::::; ::I:::::::I::;::::::::{; ::::::I:;: :::: I[HI tm. : ~L!. ~\ .. _-... -t!~[1 / Pf -, (2) Input and feature are compared bit-serially (3) Accumulator (4) Comparator inserts dumps distance new match and tag into into comparison list when better than register at end old match of input word Pig. 6 Schematic of one of the 50 processors in the digital classifier chip. The Hamming distance of the input vector to the feature vector is calculated, and if better than one of the five best matches found so far, is inserted into the match list together with the tag and passed onto the next processor. At the end of the pipeline the best five matches overall are output 522 The data paths on chip are one bit wide and all calculations are bit serial. This means that the processing elements and the data paths are compact and maximizes the number of stored words per chip. The layout of a single processor is shown in Fig. 6. The features are stored as 128-bit words in 8 16-bit ring shift registers and associated with each feature is a 14-bit tag or name string that is stored in a static register. The input vector passes through the chip and is compared bit-by-bit to each stored vector, whose shift registers are cycled in tum. The total number of bits difference is summed in an accumulator. After a vector has passed through a processor, the total Hamming distance is loaded into the comparison register together with the tag. At this time, the match list for the input vector arrives at the comparator. It is an ordered list of the 5 lowest Hamming distances found in the pipeline so far, together with associated tag strings. The distance just calculated is compared bit-serially with each of the values in the list in turn. If the current distance is smaller than one of the ones in the list, the output streams of the comparator are switched, having the effect of inserting the current match and tag into the list and deleting the previous fifth best match. After the last processor in the pipeline, the list stream contains the best five distances overall, together with the tags of the stored vectors that generated them. The data stream and the list stream are loaded into 16-bit wide registers ready for output. The design enables chips to be connected together to extend the pipeline if more than 50 stored vectors are required. The throughput is constant, irrespective of the number of chips connected together; only the latency increases as the number of chips increases. The chip has been designed to operate with an on-chip clock frequency of at least l00MHz. This high speed is possible because stage sizes are very small and data paths have been kept short. The computational efficiency is not as high as in the analog chips because each processor only deals with one bit of stored data at a time. However, the overall throughput is high because of the high clock speed. Assuming a clock frequency of l00MHz, the chip will produce a list of 5 best distances with tag strings every 1.3Jls, with a latency of about 2.5Jls. Even if a thousand chips containing 50,000 stored vectors were pipelined together, the latency would be 2.5ms, low enough for most real time applications. The chip is expected to perform 5 billion bit operation/sec. While it is important to have high clock frequencies on the chip, it is also important to have them much lower off the chip, since frequencies above 50MHz are hard to deal on circuit boards. The 16-bit wide communication paths onto and off the chip ensure that this is not a problem here. Conclusion The two approaches discussed here, analog and digital, represent opposites in computational approach. In one, a single global computation is performed for each match, in the other many local calculations are done. Both the approaches have their advantages and it remains to be seen which type of circuit will be more efficient in applications, and how closely an electronic implementation of a neural network should resemble the highly interconnected nature of a biolOgical network. These designs represent some of the first distributed computation chips. They are characterized by having simple processors distributed amongst data storage. The operation performed by the processor is tailored to the application. It is interesting to note some of the reasons why these designs can now be made: minimum linewidths on 523 circuits are now small enough that enough processors can be put on one chip to make these designs of a useful size, sophisticated design tools are now available that enable a single person to design and simulate a complete circuit in a matter of months, and fabrication costs are low enough that highly speculative circuits can be made without requiring future volume production to offset prototype costs. We expect a flurry of similar designs in the coming years, with circuits becoming more and more optimized for particular applications. However, it should be noted that the impressive speed gain achieved by putting an algorithm into custom silicon can only be done once. Further gains in speed will be closely tied to mainstream technological advances in such areas as transistor size reduction and wafer-scale integration. It remains to be seen what influence these kinds of custom circuits will have in useful technology since at present their functions cannot even be simulated in reasonable time. What can be achieved with these circuits is very limited when compared with a three dimensional, highly complex biological system, but is a vast improvement over conventional computer architectures. The authors gratefully acknowledge the contributions made by L.D. Jackel, and R.E. Howard References 1 M. Reece and P.C. Treleaven, "Parallel Architectures for Neural Computers", Neural Computers, R. Eckmiller and C. v.d. Malsburg, eds (Springer-Verlag, Heidelberg, 1988) 2 J.I. Hopfield, Proc. Nat. Acad. Sci. 79.2554 (1982). 3 J.S. Denker, Physica 22D, 216 (1986). 4 R.E. Howard, D.B. Schwartz, J.S. Denker, R.W. Epworth, H.P. Graf, W .E. Hubbard, L.D. Jackel, B.L. Straughn, and D.M. Tennant, IEEE Trans. Electron Devices ED-34, 1553, (1987) 5 H.P. Oraf and P. deVegvar, "A CMOS Implementation of a Neural Network Model", in "Advanced Research in VLSI", Proceedings of the 1987 Stanford Conference, P. Losleben (ed.), (MIT Press 1987). 6 H.P. Oraf and P. deVegvar, "A CMOS Associative Memory Chip Based on Neural Networks", Tech. Digest, 1987 IEEE International Solid-State Circuits Conference.
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730 Analysis of distributed representation of constituent structure in connectionist systems Paul Smolensky Department of Computer Science, University of Colorado, Boulder, CO 80309-0430 Abstract A general method, the tensor product representation, is described for the distributed representation of value/variable bindings. The method allows the fully distributed representation of symbolic structures: the roles in the structures, as well as the fillers for those roles, can be arbitrarily non-local. Fully and partially localized special cases reduce to existing cases of connectionist representations of structured data; the tensor product representation generalizes these and the few existing examples of fuUy distributed representations of structures. The representation saturates gracefully as larger structures are represented; it penn its recursive construction of complex representations from simpler ones; it respects the independence of the capacities to generate and maintain multiple bindings in parallel; it extends naturally to continuous structures and continuous representational patterns; it pennits values to also serve as variables; it enables analysis of the interference of symbolic structures stored in associative memories; and it leads to characterization of optimal distributed representations of roles and a recirculation algorithm for learning them. Introduction Any model of complex infonnation processing in networks of simple processors must solve the problem of representing complex structures over network elements. Connectionist models of realistic natural language processing, for example, must employ computationally adequate representations of complex sentences. Many connectionists feel that to develop connectionist systems with the computational power required by complex tasks, distributed representations must be used: an individual processing unit must participate in the representation of multiple items, and each item must be represented as a pattern of activity of multiple processors. Connectionist models have used more or less distributed representations of more or less complex structures, but little if any general analysis of the problem of distributed representation of complex infonnation has been carried out This paper reports results of an analysis of a general method called the tensor product representation. The language-based fonnalisms traditional in AI pennit the construction of arbitrarily complex structures by piecing together constituents. The tensor product representation is a connectionist method of combining representations of constituents into representations of complex structures. If the constituents that are combined have distributed representations, completely distributed representations of complex structures can result each part of the network is responsible for representing multiple constituents in the structure, and each constituent is represented over multiple units. The tensor product representation is a general technique, of which the few existing examples of fully distributed representations of structures are particular cases. The tensor product representation rests on identifying natural counterparts within connectionist computation of certain fundamental elements of symbolic computation. In the present analysis, the problem of distributed representation of symbolic structures is characterized as the problem of taking complex structures with certain relations to their constituent symbols and mapping them into activity vectors--patterns of activation-with corresponding relations to the activity vectors representing their constituents. Central to the analysis is identifying a connectionist counterpart of variable binding: a method for binding together a distributed representation of a variable and a distributed representation of a value into a distributed representation of a variable/value binding-a representation which can co-exist on exactly the same network units with representations of other variable/value bindings, with @ American Institute of Physics 1988 731 limited confusion of which variables are bound to which values. In summary, the analysis of the tensor product representation (1) provides a general technique for constructing fully distributed representations of arbitrarily complex structures; (2) clarifies existing representations found in particular models by showing what particular design decisions they embody; (3) allows the proof of a number of general computational properties of the representation; (4) identifies natural counterparts within connectionist computation of elements of symbolic computation, in particular, variable binding. The recent emergence to prominence of the connectionist approach to AI raises the question of the relation between the non symbolic computation occurring in connectionist systems and the symbolic computation traditional in AI. The research reported here is part of an attempt to marry the two types of computation, to develop for AI a form of computation that adds crucial aspects of the power of symbolic computation to the power of connectionist computation: massively parallel soft constraint satisfaction. One way to marry these approaches is to implement serial symbol manipulation in a connectionist system 1.2. The research described here takes a different tack. In a massively parallel system the processing of symbolic structures-for example, representations of parsed sentences-need not be limited to a series of elementary manipulations: indeed one would expect the processing to involve massively parallel soft constraint satisfaction. But in order for such processing to occur, a satisfactory answer must be found for the question: How can symbolic structures. or structured data in general. be naturally represented in connectionist systems? The difficulty here turns on one of the most fundamental problems for relating symbolic and connectionist computation: How can variable binding be naturally performed in connectionist systems? This paper provides an overview of a lengthy analysis reported elsewhere3 of a general connectionist method for variable binding and an associated method for representing structured data. The purpose of this paper is to introduce the basic idea of the method and survey some of the results; the reader is referred to the full report for precise defmitions and theorems, more extended examples. and proofs. The problem Suppose we want to represent a simple structured object, say a sequence of elements, in a connectionist system. The simplest method, which has been used in many models, is to dedicate a network processing unit to each possible element in each possible position4-9. This is a purely local representation. One way of looking at the purely local representation is that the binding of constituents to the variables representing their positions is achieved by dedicating a separate unit to every possible binding, and then by activating the appropriate individual units. Purely local representations of this sort have some advantages 10, but they have a number of serious problems. Three immediately relevant problems are these: (1) The number of units needed is #elements * #positions; most of these processors are inactive and doing no work at any given time. (2) The number of positions in the structures that can be represented has a fixed, rigid upper limit. (3) If there is a notion of similar elements, the representation does not take advantage of this: similar sequences do not have similar representations. The technique of distributed representation is a well-known way of coping with the first and third problemsll- 14. If elements are represented as patterns of activity over a population of processing units, and if each unit can participate in the representation of many elements, then the number of elements that can be represented is much greater than the number of units, and similar elements can be represented by similar patterns, greatly enhancing the power of the network to learn and take advantage of generalizations. 732 Distributed representations of elements in structures (like sequences) have been successfully used in many modelsl.4.S.1S-18. For each position in the structure, a pool of units is dedicated. The element occurring in that position is represented by a pattern of activity over the units in the pool. Note that this technique goes only part of the way towards a truly distributed representation of the entire structure. While the values of the variables defming the roles in the structure are represented by distributed patterns instead of dedicated units, the variables themselves are represented by localized, dedicated pools. For this reason I will call this type of representation semi-local. Because the representation of variables is still local, semi-local representations retain the second of the problems of purely local representations listed above. While the generic behavior of connectionist systems is to gradually overload as they attempt to hold more and more information, with dedicated pools representing role variables in structures, there is no loading at all until the pools are exhausted-and then there is complete saturation. The pools are essentially registers, and the representation of the structure as a whole has more of the characteristics of von Neumann storage than connectionist representation. A fully distributed connectionist representation of structured data would saturate gracefully. Because the representation of variables in semi-local representations is local, semi-local representations also retain part of the third problem of purely local representations. Similar elements have similar representations only if they occupy exactly the same role in the structure. A notion of similarity of roles cannot be incorporated in the semi-local representation. Tensor product binding There is a way of viewing both the local and semi-local representations of structures that makes a generalization to fully distributed representations immediately apparent. Consider the following structure: strings of length no more than four letters. Fig. 1 shows a purely local representation and Fig. 2 shows a semi-local representation (both of which appeared in the letter-perception model of McClelland and Rumelharr4,s). In each case, the variable binding has been viewed in the same way. On the left edge is a set of imagined units which can represent an element in the structure-a ftller of a role; these are the filler units. On the bottom edge is a set of imagined units which can represent a role: these are the role units. The remaining units are the ones really used to represent the structure: the binding units. They are arranged so that there is one for each pair offiller and role units. In the purely local case, both the filler and the role are represented by a "pattern of activity" localized to a single unit In the semi-local case, the ftiler is represented by a distributed pattern of activity but the role is still represented by a localized pattern. In either case, the binding of the filler to the role is achieved by a simple product operation: the activity of each binding unit is the product of the activities of the associated ftller and role unit In the vocabulary of matrix algebra, the activity representing the value/variable binding forms a matrix which is the outer product of the activity vector representing the value and the activity vector representing the variable. In the terminology of vector spaces, the value/variable binding vector is the tensor product of the value vector and the variable vector. This is what I refer to as the tensor product representation for variable bindings. Since the activity vectors for roles in Figs. 1 and 2 consist of all zeroes except for a single activity of 1, the tensor product operation is utterly trivial. The local and semi-local cases are trivial special cases of a general binding procedure capable of producing completely distributed representations. Fig. 3 shows a distributed case designed for visual transparency. Imagine we are representing speech data, and have a sequence of values for the energy in a particular formant at successive times. In Fig. 3, distributed patterns are used to represent both the energy value and the variable to which it is bound: the position in time. The particular binding shown is of an energy value 2 (on a scale of 1-4) to the time 4. The peaks in the patterns indicate the value and variable being represented. 733 0 (0 (8 8 8 [i] • 0 • 0 0 G) (8 ,e 8 e El 0 0 0 0 0 Filler !3 0 0 0 0 0 .? 0 0 0 0 0 .!:: (L.".' ) 0 6) @ @ 8 0 • 0 • 0 0 0 @) @> @ § El • 0 • 0 0 o • o o ROI. (Po,'tlon) Rol. (Po,lIIon) Fig. 1. Purely local representation of strings. Fig. 2. Semi-local representation of strings. If the patterns representing the value and variable being bound together are not as simple as those used in Fig. 3, the tensor product pattern representing the binding will not of course be particularly visually infonnative. Such would be the case if the patterns for the fIllers and roles in a structure were defmed with respect to a set of filler and role features: such distributed bindings have been used effectively by McClelland and Kawamoto18 and by Derthick19,20. The extreme mathematical simplicity of the tensor product operation makes feasible an analysis of the general, fully distributed case. Each binding unit in the tensor product representation corresponds to a pair of imaginary role and filler units. A binding unit can be readily interpreted semantically if its corresponding fIller and role units can. The activity of the binding unit indicates that in the structure being represented an element is present which possesses the feature indicated by the corresponding filler unit and which occupies a role in the structure which possesses the feature indicated by the corresponding role unit. The binding unit thus detects a conjunction of a pair of fIller and role features. (Higher-order conjunctions will arise later.) A structure consists of multiple fIller/role bindings. So far we have only discussed the representation of a single binding. In the purely local and semi-local cases, there are separate pools for different roles, and it is obvious how to combine bindings: simultaneously represent them in the separate pools. In the case of a fully distributed tensor product binding (eg., Fig. 3), each single binding is a pattern of activity that extends across the entire set of binding units. The simplest possibility for combining these patterns is simply to add them up; that is, to superimpose all the bindings on top of each other. In the special cases of purely local and semi-local representations, this procedure reduces trivially to simultaneously representing the individual fillers in the separate pools. 734 QB 0 0 0 @ 0 ~V~>· : :>-~ .: . , ,:,,~.~~:·;t' ·· 0 0 ~ . "~.}. Filler "i:' (Energy) 0 () .. 0 0 ~ :. jot ';, ~-} o ~ Role (Time) Fig. 3. A visually transparent fully distributed tensor product representation. The process of superimposing the separate bindings produces a representation of structures with the usual connectionist properties. If the patterns representing the roles are not too similar, the separate bindings can all be kept straight. It is not necessary for the role patterns to be nonoverlapping, as they are in the purely local and semi-local cases; it is sufficient that the patterns be linearly independent. Then there is a simple operation that will correctly extract the filler for any role from the representation of the structure as a whole. If the patterns are not just linearly independent, but are also orthogonal, this operation becomes quite direct; we will get to it shortly. For now, the point is that simply superimposing the separate bindings is sufficient. If the role patterns are not too similar, the separate bindings do not interfere. The representation gracefully saturates if more and more roles are filled, since the role patterns being used lose their distincmess once their number approaches that of the role units. Thus problem (2) listed above, shared by purely local and semi-local representations, is at last removed in fully distributed tensor product representations: they do not accomodate structures only up to a certain rigid limit, beyond which they are completely saturated; rather, they saturate gracefully. The third problem is also fully addressed, as similar roles can be represented by similar patterns in the tensor product representation and then generalizations both across similar fillers and across similar roles can be learned and exploited. The defmition of the tensor product representation of structured data can be summed up as follows: (a) Let a set S of structured objects be given a role decomposition: a set of fillers, F, a set of roles R , and for each object s a corresponding set of bindings P(s) = {f Ir : f fills role r in s }. (b) Let a connectionist representation of the fillers F be given; f is represented by the activity vector r. (c) Let a connectionist representation of the roles R be given; r is represented by the activity 735 vector r. (d) Then the corresponding tensor product representation of s is Y feB> r (where eB> !Irtl(s) denotes the tensor product operation). In the next section I will discuss a model using a fully distributed tensor product representation, which will require a brief consideration of role decompositions. I will then go on to summarize general properties of the tensor product representation. Role decompositions The most obvious role decompositions are positional decompositions that involve fixed position slots within a structure of pre-detennined fonn. In the case of a string, such a role would be the it" position in the string; this was the decomposition used in the examples of Figs. 1 through 3. Another example comes from McClelland and Kawamoto's modeI18 for learning to assign case roles. They considered sentences of the form The N 1 V the N 2 with the N 3; the four roles were the slots for the three nouns and the verb. A less obvious but sometimes quite powerful role decomposition involves not fixed positions of elements but rather their local context. As an example, in the case of strings of letters, such roles might be r"y = is preceded by x andfollowed by y, for various letters x and y. Such a local context decomposition was used to considerable advantage by Rumelhart and McClelland in their model of learning the morphophonology of the English past tense2l • Their structures were strings of phonetic segments, and the context decomposition was well-suited for the task because the generalizations the model needed to learn involved the transformations undergone by phonemes occurring in different local contexts. Rumelhart and McClelland's representation of phonetic strings is an example of a fully distributed tensor product representation. The fillers were phonetic segments, which were represented by a pattern of phonetic features, and the roles were nearest-neighbor phonetic contexts, which were also represented as distributed patterns. The distributed representation of the roles was in fact itself a tensor product representation: the roles themselves have a constituent structure which can be further broken down through another role decomposition. The roles are indexed by a left and right neighbor; in essence, a string of two phonetic segments. This string too can be decomposed by a context decomposition; the filler can be taken to be the left neighbor, and the role can be indexed by the right neighbor. Thus the vowel [i] in the word week is bound to the role rw kt and this role is in turn a binding of the filler [w] in the sub-role r' 1. The pattern for [i] is a vectOr i of phonetic features; the pattern for [w] is another such vector o(features w, and the pattern for the sub-role r'_l is a third vector k consisting of the phonetic features of [k]. The binding for the [i] in week is thus i0 (weB> k). Each unit in the representation represents a third-order conjunction of a phonetic feature for a central segment together with two phonetic features for its left and right neighbors. [To get precisely the representation used by Rumelhart and McClelland, we have to take this tensor product representation of the roles (eg. rW_1) and throw out a number of the binding units generated in this further decomposition; only certain combinations of features of the left and right neighbors were used. The distributed representation of letter triples used by Touretzky and Hintonl can be viewed as a similar third-order tensor product derived from nested context decompositions, with some binding units thrown away-in fact, all binding units off the main diagonal were discarded.] This example illustrates how role decompositions can be iterated, leading to iterated tensor product representations. Whenever the fillers or roles of one decomposition are structured objects, they can themselves be further reduced by another role decomposition. It is often useful to consider recursive role decompositions in which the fiDers are the same type of object as the original structure. It is clear from the above definition that such a decomposition cannot be used to generate a tensor product representation. Nonetheless, recursive role decompositions can be used to relate the tensor product representation of complex structures to the tensor product representations of simpler structures. For example, consider Lisp binary tree structures built from a set A of atoms. A non-recursive decomposition uses A as the set of fIllers, with each role being the 736 occupation of a certain position in the tree by an atom. From this decomposition a tensor product representation can be constructed. Then it can be seen that the operations car, cdr, and cons correspond to certain linear operators car, cdr, and cons in the vector space of activation vectors. Just as complex S-expressions can be constructed from atoms using cons, so their connectionist representations can be constructed from the simple representation of atoms by the application of cons. (This serial "construction" of the complex representation from the simpler ones is done by the analyst, not necessarily by the network; cons is a static, descriptive, mathematical operator-not necessarily a transformation to be carried out by a network.) Binding and unbinding in connectionist networks So far, the operation of binding a value to a variable has been described mathematically and pictured in Figs. 1-3 in terms of "imagined" filler units and role units. Of course, the binding operation can actually be performed in a network if the filler and role units are really there. Fig. 4 shows one way this can be done. The triangular junctions are Hinton's multiplicative connections22: the incoming activities from the role and filler units are multiplied at the junction and passed on to the binding unit. Fi lIer Units Binding Units Role Units Fig. 4. A network for tensor product binding and unbinding. "Unbinding" can also be performed by the network of Fig. 4. Suppose the tensor product representation of a structure is present in the binding units, and we want to extract the filler for a particular role. As mentioned above, this can be done accurately if the role patterns are linearly independent (and if each role is bound to only one filler). It can be shown that in this case, for each role there is a pattern of activity which, if set up on the role units, will lead to a pattern on the filler units that represents the corresponding filler. (If the role vectors are orthogonal, this pattern is the same as the role pattern.) As in Hinton's model20, it is assumed here that the triangular junctions work in all directions, so that now they take the product of activity coming in from the binding and role units and pass it on to the filler units, which sum all incoming activity. 737 The network of Fig. 4 can bind one value/variable pair at a time. In order to build up the representation of an entire structure, the binding units would have to accumulate activity over an extended period of time during which all the individual bindings would be performed serially. Multiple bindings could occur in parallel if part of the apparatus of Fig. 4 were duplicated: this requires several copies of the sets of filler and role units, paired up with triangular junctions, all feeding into a single set of binding units. Notice that in this scheme there are two independent capacities for parallel binding: the capacity to generate bindings in parallel, and the capacity to maintain bindings simultaneously. The former is determined by the degree of duplication of the ftller/role unit sets (in Fig. 4, for example, the parallel generation capacity is 1). The parallel maintenance capacity is determined by the number of possible linearly independent role patterns, i.e. the number of role units in each set It is logical that these two capacities should be independent, and in the case of the human visual and linguistic systems it seems that our maintenance capacity far exceeds our generation capacity21. Note that in purely local and semi-local representations, there is a separate pool of units dedicated to the representation of each role, so there is a tendency to suppose that the two capacities are equal. As long as a connectionist model deals with structures (like four-letter words) that are so small that the number of bindings involved is within the human parallel generation capacity, there is no harm done. But when connectionist models address the human representation of large structures (like entire scenes or discourses), it will be important to be able to maintain a large number of bindings even though the number that can be generated in parallel is much smaller. Further properties and extensions Continuous structures. It can be argued that underlying the connectionist approach is a fundamentally continuous formalization of computation 13. This would suggest that a natural connectionist representation of structure would apply at least as well to continuous structures as to discrete ones. It is therefore of some interest that the tensor product representation applies equally well to structures characterized by a continuum of roles: a "string" extending through continuous time, for example, as in continuous speech. In place of a sum over a discrete set of bindings, l:Ji 0ri we have an integral over a continuum of bindings: J,r(t)0r(t) dt This goes over exactly to the discrete case if the fillers are discrete step-functions of time. Continuous patterns. There is a second sense in which the tensor product representation extends naturally to the continuum. If the patterns representing fillers and/or roles are continuous curves rather than discrete sets of activities, the tensor product operation is still well-defmed. (Imagine Fig. 3 with the filler and role patterns being continuous peaked curves instead of discrete approximations; the binding pattern is then a continuous peaked two-dimensional surface.) In this case, the vectors rand/or r are members of infmite-dimensional function spaces; regarding them as patterns of activity over a set of processors would require an infmite number of processors. While this might pose some problems for computer simulation, the case where rand/of r are functions rather than finite-dimensional vectors is not particularly problematic analytically. And if a problem with a continuum of roles is being considered, it may be desirable to assume a continuum of linearly independent role vectors: this requires considering infinite-dimensional representations. Values as variables. Treating both values and variables symmetrically as done in the tensor product representation makes it possible for the same entity to simultaneously serve both as a value and as a variable. In symbolic computation it often happens that the value bound to one variable is itself a variable which in tum has a value bound to it In a semi-local representation, where variables are localized pools of units and values are patterns of activity in these pools, it is difficult to see how the same entity can simultaneously serve as both value and variable. In the tensor product representation, both values and variables are patterns of activity, and whether a pattern is serving as a "variable" or "value"-Qr both-might be merely a matter of descriptive preference. 738 Symbolic structures in associative memories. The mathematical simplicity of the tensor product representation makes it possible to characterize conditions under which a set of symbolic structures can be stored in an associative memory without interference. These conditions involve an interesting mixture of the numerical character of the associative memory and the discrete character of the stored data. Learning optimal role patterns by recirculation. While the use of distributed patterns to represent constituents in structures is well-known, the use of such patterns to represent roles in structures poses some new challenges. In some domains, features for roles are familiar or easy to imagine; eg .. features of semantic roles in a case-frame semantics. But it is worth considering the problem of distributed role representations in domain-independent terms as well. The patterns used to represent roles determine how information about a structure's fillers will be coded, and these role patterns have an effect on how much information can subsequently be extracted from the representation by connectionist processing. The challenge of making the most information available for such future extraction can be posed as follows. Assume enough apparatus has been provided to do all the variable binding in parallel in a network like that of Fig. 4. Then we can dedicate a set of role units to each role; the pattern for each role can be set up once and for all in one set of role units. Since the activity of the role units provide multipliers for filler values at the triangular junctions, we can treat these fixed role patterns as weights on the lines from the filler units to the binding units. The problem of finding good role patterns now becomes the problem of finding good weights for encoding the fillers into the binding units. Now suppose that a second set of connections is used to try to extract all the fillers from the representation of the structure in the binding units. Let the weights on this second set of connections be chosen to minimize the mean-squared differences between the extracted ftller patterns and the actual original filler patterns. Let a set of role vectors be called optimal if this mean-squared error is as small as possible. It turns out that optimal role vectors can be characterized fairly simply both algebraically and geometrically (with the help of results from Williams24). Furthermore, having imbedded the role vectors as weights in a connectionist net, it is possible for the network to learn optimal role vectors by a fairly simple learning algorithm. The algorithm is derived as a gradient descent in the mean-squared error, and is what G. E. Hinton and J. L. McClelland (unpublished communication) have called a recirculation algorithm: it works by circulating activity around a closed network loop and training on the difference between the activities at a given node on successive passes. Acknowledgements This research has been supported by NSF grants 00-8609599 and ECE-8617947, by the Sloan Foundation's computational neuroscience program, and by the Department of Computer Science and Institute of Cognitive Science at the University of Colorado at Boulder. 739 References 1. D. S. Touretzky & G. E. Hinton. Proceedings of the International Joint Conference on Artificial Intelligence, 238-243 (1985). 2. D. S. Touretzky. Proceedings of the 8th Conference of the Cognitive Science Society, 522-530 (1986). 3. P. Smolensky. Technical Report CU-CS-355-87, Department of Computer Science, University of Colorado at Boulder (1987). 4. J. L. McClelland & D. E. Rumelhart. Psychological Review 88, 375-407 (1981). 5. D. E. Rumelhart & J. L. McClelland. Psychological Review 89, 60-94 (1982). 6. M. Fanty. Technical Report 174, Department of Computer Science, University of Rochester (1985). 7. J. A. Feldman. The Behavioral and Brain Sciences 8,265-289 (1985). 8. J. L. McClelland & J. L. Elman. In J. L. McClelland, D. E. Rumelhart, & the PDP Research Group, Parallel distributed processing: Explorations in the microstructure of cognition. Vol. 2: Psychological and biological models. Cambridge, MA: MIT Press/Bradford Books, 58-121 (1986). 9. T. J. Sejnowski & C. R. Rosenberg. Complex Systems 1,145-168 (1987). 10. J. A. Feldman. Technical Report 189, Department of Computer Science, University of Rochester (1986). 11. J. A. Anderson & G. E. Hinton. In G. E. Hinton and J. A. Anderson, Eds., Parallel models of associative memory. Hillsdale, NJ: Erlbaum, 9-48 (1981). 12. G. E. Hinton, J. L. McClelland, & D. E. Rumelhart. In D. E. Rumelhart, J. L. McClelland, & the PDP Research Group, Parallel distributed processing: Explorations in the microstructure of cognition. Vol. 1: Foundations. Cambridge, MA: MIT Press/Bradford Books, 77-109 (1986). 13. P. Smolensky. The Behavioral and Brain Sciences 11(1) (in press). 14. P. Smolensky. In J. L. McClelland, D. E. Rumelhart, & the PDP Research Group, Parallel distributed processing: Explorations in the microstructure of cognition. Vol. 2: Psychological and biological models. Cambridge, MA: MIT Press/Bradford Books, 390-431 (1986). 15. G. E. Hinton. In Hinton, G.E. and Anderson, J.A., Eds., Parallel models of associative memory. Hillsdale,NJ: Erlbaum, 161-188 (1981). 16. M. S. Riley & P. Smolensky. Proceedings of the Sixth Annual Conference of the Cognitive Science Society, 286-292 (1984). 17. P. Smolensky. In D. E. Rumelhart, J. L. McOelland, & the PDP Research Group, Parallel distributed processing: Explorations in the microstructure of cognition. Vol. 1: Foundations. Cambridge, MA: MIT Press/Bradford Books, 194-281 (1986). 18. J. L. McClelland & A. H. Kawamoto. In J. L. McClelland, D. E. Rumelhart, & the PDP Research Group, Parallel distributed processing: Explorations in the microstructure of cognition. Vol. 2: Psychological and biological models. Cambridge, MA: MIT Press/Bradford Books, 272-326 (1986). 19. M. Derthick. Proceedings of the National Conference on Artificial Intelligence, 346-351 (1987). 20. M. Dertbick. Proceedings of the Annual Conference of the Cognitive Science Society, 131-142 (1987). 21. D. E. Rumelhart & J. L. McClelland. In J. L. McClelland, D. E. Rumelhart, & the PDP Research Group, Parallel distributed processing: Explorations in the microstructure of cognition. Vol. 2: Psychological and biological models. Cambridge, MA: MIT Press/Bradford Books, 216-271 (1986) 22. G. E. Hinton. Proceedings of the Seventh International Joint Conference on Artificial Intelligence, 683-685 (1981). 23. A. M. Treisman & H. Schmidt. Cognitive Psychology 14,107-141 (1982). 24. R. J. Williams. Technical Report 8501, Institute of Cognitive Science, University of California, San Diego (1985).
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HIERARCHICAL LEARNING CONTROL AN APPROACH WITH NEURON-LIKE ASSOCIATIVE MEMORIES E. Ersu ISRA Systemtechnik GmbH, Schofferstr. 15, D-6100 Darmstadt, FRG H. Tolle TH Darmstadt, Institut fur Regelungstechnik, Schlo~graben 1, D-6100 Darmstadt, FRG ABSTRACT 249 Advances in brain theory need two complementary approaches: Analytical investigations by in situ measurements and as well synthetic modelling supported by computer simulations to generate suggestive hypothesis on purposeful structures in the neural tissue. In this paper research of the second line is described: Starting from a neurophysiologically inspired model of stimulusresponse (S-R) and/or associative memorization and a psychologically motivated ministructure for basic control tasks, pre-conditions and conditions are studied for cooperation of such units in a hierarchical organisation, as can be assumed to be the general layout of macrostructures in the brain. I. INTRODUCTION Theoretic modelling in brain theory is a highly speculative subject. However, it is necessary since it seems very unlikely to get a clear picture of this very complicated device by just analyzing the available measurements on sound and/or damaged brain parts only. As in general physics, one has to realize, that there are different levels of modelling: in physics stretching from the atomary level over atom assemblies till up to general behavioural models like kinematics and mechanics, in brain theory stretching from chemical reactions over electrical spikes and neuronal cell assembly cooperation till general human behaviour. The research discussed in this paper is located just above the direct study of synaptic cooperation of neuronal cell assemblies as studied e. g. in /Amari 1988/. It takes into account the changes of synaptic weighting, without simulating the physical details of such changes, and makes use of a general imitation of learning situation (stimuli) - response connections for building up trainable basic control loops, which allow dynamic S-R memorization and which are themsel ves elements of some more complex behavioural loops. The general aim of this work is to make first steps in studying structures, preconditions and conditions for building up purposeful hierarchies and by this to generate hypothesis on reasons and @) American Institute of Physics 1988 250 meaning behind substructures in the brain like the columnar organization of the cerebral cortex (compare e. g. /Mountcastle 1978/). The paper is organized as follows: In Chapter II a short description is given of the basic elements for building up hierarchies, the learning control loop LERNAS and on the role of its subelement AMS, some ~ssociati ve memory ~stem inspired by neuronal network considerations. Chapter III starts from certain remarks on substructures in the brain and discusses the cooperation of LERNASelements in hierarchies as possible imitations of substructures. Chapter IV specifies the steps taken in this paper in the direction of Chapter III and Chapter V presents the results achieved by computer simulations. Finally an outlook will be given on further investigations. II. LERNAS AND AMS Since the formal neuron was introduced by /McCulloch and Pitts 1943/, various kinds of neural network models have been proposed, such as the percept ron by /Rosenblatt 1957/ the neuron equation of /Caianello 1961/, the cerebellar model articulation controller CMAC by /Albus 1972, 1975/ or the associative memory models by /Fukushima 1973/, /Kohonen 1977/ and / Amari 1977/. However, the abili ty of such systems to store information efficiently and to perform certain pattern recognition jobs is not adequate for survival of living creatures. So they can be only substructures in the overall brain organization; one may call them a microstructure. Purposeful acting means a goal driven coordination of sensory information and motor actions. Al though the human brain is a very complex far end solution of evolution, the authors speculated in 1978 that it might be a hierarchical combination of basic elements, which would perform in an elementary way like the human brain in total, especially since there is a high similarity in the basic needs as well as in the neuronal tissue of human beings and relati vely simple creatures. This led to the design of the learning control loop LERNAS in 1981 by one of the authors - /Ersu 1984/ on the basis of psychological findings. He transformed the statement of /Piaget 1970/, that the complete intelligent action needs three elements: "1) the question, which directs possible search actions, 2) the hypothesis, which anticipates eventual solutions, 3) the control, which selects the solution to be chosen" into the structure shown in Fig. 1, by identifying the "question" with an performance criterion for assessment of possible advantages/disadvantages of certain actions, the "hypothesis" with a predictive model of environment answers and the "control" with a control strategy which selects for known situations the best action, for unknown situations some explorative action (active learning). In detail, Fig. 1 has to be understood in the following way: The predictive model is built up in a step by step procedure from a characterization of the actual situation at the time instant k-T s 251 T sampling time) and the measured response of the unknown ens vironment at time instant (k+1)T . The actual situation consists of s measurements regarding the stimuli and responses of the environment at time instant keTs plus - as far as necessary for a unique characterization - of the situation-stimuli and responses at time instants (k-1)T , (k-2)T ... , provided bv the short term memory. To s s reduce learning effort, the associative memory system used to store the predictive model has the ability of local generalization, that means making use of the trained response value not only for the corresponding actual situation, but also in similar situations. The assessment module generates on the basis of a given goal - a wanted environment response with an adequate performance criterion an evaluation of possible actions through testing them with the predictive model, as far as this is already built up and gives meaningful answers. The result is stored in the control strategyAMS together with its quality: real optimal action for the actual situation or only relatively optimal action, if the testing reached the border of the known area in the predictive model of the environment. In the second case, the real action is changed in a sense of curiosity, so that by the action the known area of the predictive model is extended. By this, one reaches more and more i:he first case, in which the real optimal actions are known. Since the first guess for a good action in the optimization phase is given to the assessment module from the control strategy AMS - not indicated in Fig. 1 to avoid unnecessary complication finally the planning level gets superfluous and one gets very quick optimal reactions, the checking with the planning level being necessary and helpful only to find out, whether the environment has not changed, possibly. Again the associative memory system used for the control strategy is locally generalizing to reduce the necessary training effort. The AMS storage elements for the predictive model, and for optimized actions are a refinement and implementation for on-line application of the neuronal network model CMAC from J. Albus - see e. g. /Ersu, Militzer 1982/ -, but it could be any other locally generalizing neural network model and even a storage element based on pure mathematical considerations, as has been shown in /Militzer, Tolle 1986/. The important property to build up an excellent capability to handle different tasks in an environment known only by some sensory information the property which qualifies LERNAS as a possible basic structure (a "ministructure") in the nervous system of living creatures - has been proven by its application to the control of a number of technical processes, starting with empty memories for the predictive model and the control strategy storage. Details on this as well as on the mathematical equations describing LERNAS can be found in /Ersu, Mao 1983/, /Ersu, Tolle 1984/ and /Ersu, Militzer 1984/. 252 It should be mentioned that the concept of an explicit predictive environmental model - as used in LERNAS - is neither the only meaningful description of human job handling nor a necessary part of our basic learning element. It suffices to use a prediction whether a certain action is advantegeous to reach the actual goal or whether this is not the case. More information on such a basic element MINLERNAS, which may be used instead of LERNAS in general (however, with the penalty of some performance degradation) are given in /Ersa, Tolle 1988/. III. HIERARCHIES There are a number of reasons to believe, that the brain is built up as a hierarchy of control loops, the higher levels having more and more coordinative functions. A very simple example shows the necessity in certain cases. The legs of a jumping jack can move together, only. If one wants to move them separately, one has to cut the connection, has to build up a separate controller for each leg and a coordinating controller in a hierarchically higher level to restore the possibility of coordinated movements. Actually, one can find such an evolution in the historical development of certain animals. In a more complex sense a multilevel hierarchy exists in the extrapyramidal motor system. Fig. 2 from /Albus 1979/ specifies five levels of hierarchy for motor control. It can be speculated, that hierarchical organizations are not existing in the senso-motoric level only, but also in the levels of general abstractions and thinking. E. g. /Dorner 1974/ supports this idea. If one assumes out of these indications, that hierarchies are a fundamental element of brain structuring - the details and numbers of hierarchy-levels not being known - one has to look for certain substructures and groupings of substructures in the brain. In this connection one finds as a first subdivision the cortical layers, but then as another more detailed subdivision the columns, cell assemblies heavily connected in the axis vertical to cortical layers and sparsely connected horizontally. /Mountcastle 1978/ defines minicolumns, which comprise in some neural tissue roughly 100 in other neural tissue roughly 250 individual cells. In addition to these mini columns certain packages of minicolumns, consisting out of several hundreds of the minicolumns, can be located. They are called macrocolumns by /Mountcastle 1978/. Fig. 3 gives some abstraction, how such structures could be interpreted: each minicolumn is considered to be a ministructure of the type LERNAS, a number of LERNAS units - here shown in a ring structure instead of a filled up cylindrical structure - building up a macrocolumn. The signals between the LERNAS elements could be overlapping and cooperating. Minicolumns being elements of macrocolumns of a higher cortical layer - here layer j projecting to layer k - could initiate and/or coordinate this cooperation in a hierarchical sense. Such a complex system is difficult to simulate. One has to go into this direction in a step by step procedure. In a first step the 253 overlapping or crosstalk between the minicolumns may be suppressed and the number of ministructures LERNAS representing the minicolumns should be reduced heavily. This motivates Fig. 4 as a fundamental blockdiagram for research on cooperation of LERNAS elements. IV. TOPICS ADDRESSED From Fig. 4 only the lowest level of coordination (layer 1), that means the coordination of two subprocesses was implemented up to now - right half of Fig. 5. This has two reasons: Firstly, a number of fundamental questions can be posed and discussed with such a formulation already. Secondly, it is difficult to set up meaningful subprocesses and coordination goals for a higher order system. The problem discussed in the following can be understood as the coordination of two minicolumns as described in Chapter III, but also as the coordination of higher level subtasks, which may be detailed themselves by ministructures and/or systems like Fig. 4. This is indicated in the left half of Fig. 5. Important questions regarding hierarchies of learning control loops are: I. What seem to be meaningful interventions from the coordinator onto the lower level systems? II. Is parallel learning in both levels possible or requires a meaningful learning strategy that the control of subtasks has to be learned at first before the coordination can be learned? III. Normally one expects, that the lower level takes care of short term requirements and the upper level of long term strategies. Is that necessary or what happens if the upper level works on nearly the same time horizon as the lower levels? IV. Furtheron one expects, that the upper level may look after other goals than the lower level, e. g. the lower level tries to suppress disturbances effects since the upper level tries to minimize overall energy consumption. But can such different strategies work without oscillations or destabilization of the system? Question I can be discussed by some general arguments, for questions II-IV only indications of possible answers can be given from simulation results. This will be postponed to Chapter V. Fig. 6 shows three possible intervention schemes from the coordinator. By case a) an intervention into the structure or the parameters of 254 the sublevel (=local) controllers is meant. Since associative mappings like AMS have no parameters being directly responsible for the behaviour of the controller - as would be the case with a parametrized linear or non-linear differential equation being the description of a conventional controller - this does not make sense for the controller built up in LERNAS. However, one could consider the possibility to change parameters or even elements, that means structural terms of the performance criterion, which is responsible for the shaping of the controller. But this would require to learn anew, which takes a too long time span in general. By case b) a distribution of work load regarding control commands is meant. The possible idea could be, that the coordinator gives control inputs to hold the long range mean value required, since the local controllers take into account fast dynamic fluctuations only. However, this has the disadvantage that the control actions of the upper level have to be included into the inputs to the local controllers, extending the dimension of in-put space of these storage devices, since otherwise the process appears to be highly time variant for the local controllers, which is difficult to handle for LERNAS. So case c) seems to be the best solution. In this case the coordinator commands the set points of the local controllers, generating by this local subgoals for the lower level controllers. Since this requires no input space extension for the local controllers and is in full agreement with the working conditions of single LERNAS loops, it is a meaningful and effective approach. Fig. 7 shows the accordingly built up structure in detail. The control strategy of Fig. 1 is divided here in two parts the storage element (the controller C) and the active learning AL. The elements are explicitly characterized for the upper level only. The whole lower level is considered by the coordinator as a single pseudoprocess to be controlled (see Fig. 4). v. SIMULATION RESULTS For answering questions II and III the very simple non-linear process shown in Fig. 8 - detailing the subprocesses SPl, SP2 and their coupling in Fig. 7 - was used. For the comparison of bottom up and parallel learning suitably fixed PI-controllers were used for bottom up learning instead of LERNAS land LERNAS 2, simulating optimally trained local controllers. Fig. 9a shows the result due to which in the first run a certain time is required for achieving a good set point following through coordinator assistance. However, with the third repetition (4th run) a good performance is reached from the first set point change on already. For parallel learning all (and not only the coordinator AMS-memories) were empty in the beginning. Practically the same performance was achieved as in bottom up training - Fig. 9b -, indicating, that at least in simple problems, as considered here, parallel learning is a real possibi255 lity. However what is not illustrated here the coordinator sampling time must be sufficiently long, so that the local controllers can reach the defined subgoals at least qualitatively in this time span. For answering question III, in which respect a higher difference in the time horizon between local controller and coordinator changes the picture, a doubling of the sampling rate for the coordinator was implemented. Fig. 10 give the results. They can be interpreted as follows: Smaller sampling rates allow the coordinator to get more information about the pseudo-sub-processes, the global goal is reached faster. Larger sampling rates lead to a better overall performance when the goal is reached: there is a higher amount of averaging regarding informations about the pseudo-sub-processes. Up to now in both levels the goal or performance criterion was the minimization of differences between the actual plant output and the requested plant output. The influence of different coordinator goals - question IV was investigated by simulating a two stage waste water neutralization process. A detailed description of this process set up and the simulation results shall not be given here out of space reasons. It was found that: • in hierarchical systems satisfactory overall behaviour may be reached by well defined subgoals with clearly different coordinator goals. • since learning is goal driven, one has to accept that implicit wishes on closed loop behaviour are fulfilled by chance only. Therefore important requirements have to be included in the performance criteria explicitly. It should be remarked finally, that one has to keep in mind, that simulation results with one single process are indications of possible behaviour only, not excluding that in other cases a fundamentally different behaviour can be met. VI. OUTLOOK As has been mentioned already in Chapter III and IV, this work is one of many first steps of investigations regarding hierarchical organization in the brain, its preconditions and possible behaviour. Subjects of further research should be the self-organizing task distribution between the processing units of each layer, and the formation of inter layer projections in order to build up meta-tasks composed of a sequence of frequently occuring elementary tasks. These investigations will on the other hand show to what extent this kind of higher-learning functions can be achieved by a hierarchy of LERNAS-type structures which model more or less low-level basic learning behaviour. 256 VII. ACKNOWLEDGEMENTS The work presented has been supported partly by the Stiftung Volkswagenwerk. The detailed evaluations of Chapter IV and V have been performed by Dipl.-Ing. M. Zoll and Dipl.-Ing. S. Gehlen. We are very thankful for this assistance. Albus, J. S. Albus, J. S. Albus, J. S. Amari, S. 1. Amari, S. 1. Caianello, E. R. Dorner, D. Ersil, E. Ersu, E. Mao, X. VIII. REFERENCES Theoretical and Experimental Aspects of a Cerebellar Model, Ph.D. Thesis, Univ. of Maryland, 1972 A New Approach to Manipulator Control: The Cerebellar Model Articulation Controller (CMAC), Trans. ASmE series, G, 1975 A Model of the Brain for Robot Control Part 3: A Comparison of the Brain and Our Model, Byte, 1979 Neural Theory of Association and Concept Formation, BioI. Cybernetics, Vol. 26, 1977 Mathematical Theory of Self-Organization in Neural Nets, in: Organization of Neural Networks, Structures and Models, ed. by von Seelen, Shaw, Leinhos, VHC-Verlagsges. Weinheim, W.-Germany, 1988 Outline of a Theory of Thought Process and Thinking Machines, Journal of Theoretical Biology, Vol. 1, 1961 Problemlosen als Informationsverarbeitung Verlag H. Huber, 1974 On the Application of Associative Neural Network Models to Technical Control Problems, in: Localization and Orientation in Biology and Engineering, ed. by Varju, Schnitzler, Springer Verlag Berlin, W.-Germany, 1984 Control of pH by Use of a Self-Organizing Concept with Associative Memories, ACI'83, Kopenhagen (Denmark), 1983 Ersu, E. Militzer, J. Ersu, E. Militzer, J. Ersu, E. Tolle, H. Ersu, E. Tolle, H. Fukushima, K. Kohonen, T. McCulloch, W. S. Pitts, W. H. Militzer, J. Tolle, H. Mountcastle, V. B. Piaget, J. Rosenblatt, F. Software Implementation of a Neuron-Like Associative Memory System for Control Application, Proceedings of the 2nd lASTED Conference on Mini- and Microcomputer Applications, MIMI'82, Davos (Switzerland), 1982 Real-Time Implementation of an Associative Memory-Based Learning Control Scheme for NonLinear Multivariable Processes, Symposium "Applications of Multivariable System Techniques", Plymouth (UK), 1984 A New Concept for Learning Control Inspired by Brain Theory, Proceed. 9th IFAC World Congress, Budapest (Hungary), 1984 Learning Control Structures with Neuron-Like Associative Memory Systems, in: Organization of Neural Networks, Structures and Models, ed. by von Seelen, Shaw, Leinhos, VCH Verlagsgesellschaft Weinheim, W.-Germany, 1988 A Model of Associative Memory in the Brain BioI. Cybernetics, Vol. 12, 1973 Associative Memory, Springer Verlag Berlin, W.-Germany, 1977 A Logical Calculus of the Ideas, Immanent in Nervous Activity, Bull. Math. Biophys. 9, 1943 Vertiefungen zu einem Teilbereiche der menschlichen Intelligenz imitierenden Regelungsansatz Tagungsband-DGLR-Jahrestagung, Munchen, W.-Germany, 1986 An Organizing Principle for Cerebral Function: The Unit Module and the Distributed System, in: The Mindful Brain by G. M. Edelman, V. B. Mountcastle, The MIT-Press, Cambridge, USA, 1978 Psychologie der Intelligenz, Rascher Verlag, 4th printing, 1970 The Perceptron: A Perceiving and Recognizing Automation, Cornell Aeronautical Laboratory, Report No. 85-460-1, 1957 257 258 ~ LUKAS I PROCESS /'oDlL LEU.IM' AND GoA\. Os I YEll Actio. OPtllllZAllOR I 1-· FIGURES • 0 -r)'p~Clt-ClI"9r.'OO:'5 Dr !IIv-ro-,,"';"'l ~================-'-I\-s-ts-s",-en-"--' p'lrwyd r.tiol'lS pt .. ning t===~~~[=~~j ~'i",ilrd Idion '-'-1I =' =g=Oaf=U="il=b'=U=~+t>o=t>I control s'r"t91 t-'--__ 1==:;-;=====IC='iors==:tll Ar'S _ 0- '_0_._._ 0_ ._._ . _oJ rudio .. u"k"ow" ." .. r on"""' (ZZZ::Zjg Associative Situalion-Response Yapping (lDng Term Yemory) Fig. 1. Architectural element LERNAS PiESTITIAL ------'" NUCLEUS IETI CUI..AR --------';----:::FORMATION SUBTHAl.APnC NUCLEUS NUClEUS PiE COHH:I S Sl'RALIS INTERSTITIAL NUCLEUS Fig. 2. The hierarchy of aotor control that exists in the extrapyramidal motor system. Basic reflexes remain even if the brain stem is cut at A-A. Coordination of these reflexes for standing is possible if the cut is at B-B. The sequential coordination required for walking requires the area below c-c to be operable. Simple tasks can be executed if the region below D-D is intact. Lengthy tasks and complex goals require the cerebral cortex. (/Albus 1979/) Fig. 3. Generic scetch of macrocolumns - drawn as ring structures - from different cortical layers with LERNAS-subunits representing ainicolumns laytT n ~i 1 T li!fIT 2 L.ERNA52i '" '" l;yrr • I f L.£JMjij I ~ lDNlSl' l 1 JI'~ l J .1 '" l' It LDM&;i J l~ 1 ~ J [lENjil J .J, 1'1 ...... ~ 'f --r t J. l' ~.oemi ~.l J I Ir·caDk ~l} ,rig. C. LERNAS-hierarchy as a si.plified research .odel for cooperation of columnar .tructures 259 260 LERNAS 3 lERNAS 1 SUBPROCESS Fig. 5. Hierarchical work! control distribution Fig. 6. Methods of intervention from the coordinator flOOIL 113 PsEUDC-~ ...fI-.cU$ r------------------------, I I I I I I I I I I I I I I _______ _______________ J Fig. 7. Implementation of the hierarchical structure .i'II IIC ... DOD-llneer 1rl C"':) ~ Z 0:: ~ 'Wz Fig. 8. Hierarchical structure with non-linear multivariable test-process reference value / y y run) run) 500 1 DOC I!IOO toOC eoo IOOC reference value !IOO 1000 I soc Fig. 9. Learning on trained (a) (T = 2 coordinator level using and untrained (b) lower sec, Tioc = 0.5 sec) already levels coord T d=4 sec coor y (1st run) y (4th run) I~ T d=2sec coor 261 1 100 'DOC 'soc aooo. BIle _ 1000 100 lDOC ,_ _ BDCI _ Fig. 10. Coordinator learning behaviour using different coordinator borizons (TI = 0.5 sec) oc
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154 PRESYNApnC NEURAL INFORMAnON PROCESSING L. R. Carley Department of Electrical and Computer Engineering Carnegie Mellon University, Pittsburgh PA 15213 ABSTRACT The potential for presynaptic information processing within the arbor of a single axon will be discussed in this paper. Current knowledge about the activity dependence of the firing threshold, the conditions required for conduction failure, and the similarity of nodes along a single axon will be reviewed. An electronic circuit model for a site of low conduction safety in an axon will be presented. In response to single frequency stimulation the electronic circuit acts as a lowpass filter. I. INTRODUCTION The axon is often modeled as a wire which imposes a fixed delay on a propagating signal. Using this model, neural information processing is performed by synaptically sum m ing weighted contributions of the outputs from other neurons. However, substantial information processing may be performed in by the axon itself. Numerous researchers have observed periodic conruction failures at norma! physiological impulse activity rates (e.g., in cat, in frog 2, and in man ). The oscillatory nature of these conduction failures is a result of the dependence of the firing threshold on past impulse conduction activity. The simplest view of axonal (presynaptic) information processing is as a switch: the axon will either conduct an im pulse or not. The state of the switch depends on how past impulse activity modulates the firing threshold, which will result in conduction failure if firing threshold is bigger than the incoming impulse strength. In this way, the connectivity of a synaptic neural network could be modulated by past impulse activity at sites of conduction failure within the network. More sophisticated presynaptic neural information processing is possible when the axon has more than one terminus, implying the existence of branch points within the axon. Section II will present a general description of potential for presynaptic information processing. The after-effects of previous activity are able to vary the connectivity of the axonal arbor at sites of low conduction safety according to the temporal pattern of the impulse train at each site (Raymond and LeUvin, 1978; Raymond, 1979). In order to understand the inform ation processing potential of presynaptic networks it is necessary to study the after- effects of activity on the firing threshold. Each impulse is normally followed by a brief refractory period (about 10m s in frog sciatic nerve) of increased © American Institute of Phvl'if:<' 1 qR~ 155 threshold and a longer superexcitable period (about 1 s in frog sciatic nerve) during which the threshold is actually below its resting level. During prolonged periods of activity, there is a gradual increase in firing threshold which can persist long (> 1 hour in frog nerve) after cessation of im pulse activity (Raymond and Lettvin, 1978). In section III, the methods used to measure the firing threshold and the after-effects of activity will be presented. In addition to understanding how impulse activity modulates sites of low conduction safety, it is important to explore possible constraints on the distribution of sites of low conduction safety within the axon's arbor. Section IV presents results from a study of the distribution of the aftereffects of activity along an axon. Section V presents an electronic circuit model for a region of low conduction safety within an axonal arbor. It has been designed to have a firing threshold that depends on the past activity in a manner similar to the activity dependence measured for frog sciatic nerve. II. PRESYNAPTIC SIGNAL PROCESSING Conduction failure has been observed in many diffe~e~t organisms, including man, at normal physiological activity rates.1, , The aftereffects of activity can "modulate" conduction failures at a site of low conduction safety. One common place where the conduction safety is low is at branch points where an impedance mismatch occurs in the axon. In order to clarify the meaning of presynaptic information processing, a simple example is in order. Parnas reported that in crayfish a single axon separately activates the medial (DEA~~ and lateral (DEAL) branches of the deep abdominal extensor muscles.' At low stimulus frequencies (below 40-50 Hz) impulses travel down both branches; however, each impulse evokes much smaller contractions in DEAL than in DEAM resulting in contraction of DEAM without significant contraction of DEAL. At higher stim ulus frequencies conduction in the branch leading to D EAM fails and DEAL contracts without DEAM contracting. Both DEAL and DEAM can be stim ulated separately by stim ulus patterns more com plicated than a single frequency. The theory of "fallible trees", which has been discussed by Lettvin, McCulloch and Pitts, Raymond, and Waxman and Grossman among others, suggests that one axon which branches many times forms an information processing element with one input and many outputs. Thus, the after-effects of previous activity are able to vary the connectivity of the axonal arbor at regions of low conduction safety according to the temporal pattern of the impulse train in each branch. The transfer function of the fallible tree is determined by the distribution of sites of low conduction safety and the distribution of superexcitability and depressibility at those sites. Thus, a single axon with 1000 terminals can potentially be in 21000 different states as a function of the locations of sites of conduction failure within the axonal arbor. And, each site of low conduction safety is 156 modulated by the past impulse activity at that site. Fallible trees have a number of interesting properties. They can be used to cause different input frequencies to excite different axonal terminals. Also, fallible trees, starting at rest, will preserve timing information in the input signal; Le., starting from rest, all branches will respond to the first impulse. III. AFTER- EFFECTS OF ACTIVITY In this section, the firing threshold will be defined and an experimental method for its measurem ent will be described. In addition, the aftereffects of activity will be characterized and typical results of the characterization process will be given. The following method was used to measure the firing threshold. Whole nerves were placed in the experimental setup (shown in figure 1). The whole nerve fiber was stim ulated with a gross electrode. The response from a single axon was recorded using a suction microelectrode. Firing threshold was measured by applying test stimuli through the gross stimulating electrode and looking for a response in the suction m icroelectrode. Motordriven vernier micrometer Ag-AgCI electrode F ixed-duration variable-amplitude current stimulator •• t. , . A ~; lh 0·4 mm diameter Suction electrOdep' . t' . Refer~nce f--MOVES~ Single axon ~ suctIon \ electrode Whole nerve Figure 1. Drawing of the experimental recording chamber. Threshold Hunting, a process forschoosin g the test stimulus strength, was used to characterize the axons. It uses the following paradigm. A test stimulus which fails to elicit a conducting impulse causes a small increase the strength of subsequent test stimuli. A test stim ulus which 157 elicits an im pulse causes a small decrease in the strength of subsequent test stimuli. Conditioning Stimuli, ones large enough to guarantee firing an impulse, can be interspersed between test stimuli in order to achieve a controlled overall activity rate. Rapid variations in threshold following one or more conditioning impulses can be measured by slowly increasing the time delay between the conditioning stimuli and the test stimulus. Several phases follow each impulse. First, there is a refractory period of short duration (about 10ms in frog nerve) during which another impulse cannot be initiated. Following the refractory period the axon actually becomes more excitable than at rest for a period (ranging from 200ms to 1 s in frog nerve, see figure 2). The superexcitable period is measured by applying a conditioning stimulus and then delaying by a gradually increasing time delay and applying a test stimulus (see figure 3). There is only a slight increase in the peak of the superexcitable period following multiple im pulses? The superexcitability of an axon was characterized by the % decrease of the threshold from its resting level at the peak of the superexcitable period. • 5 5'(1) fo, P, 0.50 + :......-TO~ :_TO'Ald-~ . 1 T[ST 5t IMULU5 1 T [5T 5T IMULUS CONO I TlONING co NOI T 10NING o~! ____ ~~~I ----~~~I ~--17~~'---IOc~~Td INTERVAL 'm .. c) Figure 2. Typical superexcitable period in axon from frog sciatic nerve. :_FRAMC 1-;-rRAMC ?-:Figure 3. Stim ulus pattern used for measuring superexcitability. During a period of repetitive impulse conduction, the firing threshold may gradually increase. After the period of increased im pulse activity ends, the threshold gradually recovers from its maximum over the course of several minutes or more with complete return of the threshold to its resting level taking as long as an hour or two (in frog nerve) depending on the extent of the preceding im pulse activity. The depressibility of an axon can be characterized by the initial upward slope of the depression and the time 158 constant of the recovery phase (see figure 4). The pattern of conditioning and test stimuli used to generate the curve in figure 4 is shown in figure 5. Depression may be correlated with microanatomical changes which occur ira the glial cells in the nodal region during periods of increased activity. During periods of repetitive stim ulation the size and num ber of extracellular paranodal intramyelinic vacuoles increases causing changes in the paranodal geom etry. 200 120 40 Threshold \percenl.gt of rHling level) Cond.t.on.,,!! burst °o+-' --5~-tO--15--2-0--2+-5--:'30 l' 5 min >" T Time (min) r-~ On Off Test Time Figure 4. Typical depression in an axon from frog sciatic nerve. The average activity rate was 4 impulses/sec between the 5 min mark and the 10 min mark. Figure 5. Stim ulus pattern used for measuring depression. IV. CONSTRAINTS ON FALLIBLE TREES The basic fallible tree theo ry places no constraints on the distribution of sites of conduction failure among the branches of a single axon. In this section one possible constraint on the distribution of sites of conduction failure will be presented. Experiments have been performed in an attempt to determine if the extremely wide variations in superexcitability anS depressibility found between nodes from different axons in a single nerve (particularly for depressibility) also occur between nodes from the same axon. A study of the distribution of the after-effects of activity along an unbranching length of frog sciatic nerve isund only sm all variations in the after- effects along a single axon. Both superexcitability and depressibility were extremely consistent for nodes from along a single unbranching length of axon (see figures 6 and 7). This suggests that there may be a cell-wide regulatory system that maintains the depressibility and 159 superexcitability at com parable levels throug hout the extent of the axon. Thus, portions of a fallible tree which have the same axon diameter would be expected to have the same superexcitability and depressibility. 3.() 95 Superexcitability (%1 30 Figure 6. PDF of Superexcitability. The upper trace represents the PDF of the entire population of nodes studied and the two lower traces represent the separate populations of nodes from two different axons. 0-8 2-5 8 -0 25 80 Upward slope ("'/minl Figure 7. PDF of Depressibility. The upper trace represents the PDF of the entire population of nodes studied and the two lower traces represent the separate populations of nodes from two different axons. This study did not examine axons which branched, therefore it cannot be concluded that superexcitability and depressibility must remain constant throughout a fallible tree. For example, it is quite likely that the cell actually regulates quantities like pump- site density, not depressibility. In that case, daughter branches of smaller diameter might be expected to show consistently higher depressibility. Further research is needed to determine how the activity dependence of the threshold scales with axon diameter along a single axon before the consistency of the after-effects along an unbranching axon can be used as a constraint on presynaptic information processing networks. V. ELECTRICAL AXON CIRCUIT This section presents a simple electronic circuit which has been designed to have a firing threshold that depends on the past states of the output in a manner similar to the activity dependence measured for frog sciatic nerve. In response to constant frequency stimuli, the circuit acts as 160 a low pass filter whose corner frequency depends on the coefficients which determine the after-effects of activity. Figure B shows the circuit diagram for a switched capacitor circuit which approximates the after- effects of activity found in the frog sciatic nerve. The circuit employs a two phase nonoverlapping clock, e for the even clock and 0 for the odd clock, typical of switched capacitor circuits. It incorporates a basic model for superexcitability and depressibility. VTH represents the resting threshold of the axon. On each clock cycle the V'N is com pared with VTH+ Vo- Vs. The two capacitors and three switches at the bottom of figure B model the change in threshold caused by superexcitability. Note that each impulse resets the comparator's minus input to (1-cx.)VTH, which decays back to VTH on subsequent clock cycles with a time constant inversely proportional to Ps. This is a slight deviation from the actual physiological situation in which multiple conditioning im pulses will generate slightly more superexcitability than a single impulse? The two capacitors and two switches at the upper left of figure B model the depressibility of the axon. The current source represents a fixed increment in the firing threshold with every past impulse. The depression voltage decays back to 0 on subsequent clock cycles with a time constant inversely proportional to PO. Figure B. Circuit diagram for electrical circuit analog of nerve threshold. The electrical circuit exhibits response patterns similar to those of neurons that are conducting intermittently (see figure 9). During bursts of conduction, the depression voltage increases linearly until the comparator 161 fails to fire. The electrical axon then fails to fire until the depression voltage decays back to (1 +aOV)VTH' The connectivity between the input and output of the axon is defined to be the average fraction of impulses which are conducted. In terms of connectivity, the electrical axon model acts as a lowpass filter (see figure 10). riftiNG VD ' tll4 Vs YES NO .. ~ v S ,00 10 300 T,,.a: «Sl:CONUS I Figure 9. Typical waveform s for intermittent conduction. The upper trace indicates whether impulses are conducted or not. VD and Vs are the depression voltage and the superexcitable voltage respectively. rlUINC "'I1ACTI(lN , . \ •• •• o : o. :.I--~~----;-t-----;2r-------.c. INruT rR(:QVE~' Figure 10. Frequency response of electrical axon model. The connectivity is reflected by the fraction of impulses which are conducted out of a seq uence of 100.000 stimuli where the frequency is in stim uli/second. For a fixed stim ulus frequency. the average fraction of im pulses which are conducted by the electrical model can be predicted analytically. The expressions can be greatly simplified by making the assumption that VD increases and decreases in a linear fashion. Under that assumption. in terms of the variables indicated on the schematic diagram, where M is the number of clock cycles between input stimuli. which is inversely proportional to the input frequency. The frequency at which only half of the impulses are conducted is defined as the corner frequency of the low pass filter. The corner frequency is 162 f(P == 0.5) _...!. == log(1-~D) M aD log(1--) aOV Using the above equations, lowpass filters with any desired cutoff frequency can be designed. The analysis indicates that the corner frequency of the lowpass filter can be varied by changing the degree of conduction safety (aov) without changing either depressibility or superexcitability. This suggests that the existence of a cell- wide regulatory system maintaining the depressibility and superexcitability at comparable levels throughout the extent of the axon would not prevent the construction of a bank of low pass filters since their corner frequencies could still be varied by varying the degree of conduction safety (aov). VI. CONCLUSIONS Recent studies report that the primary effect of several common anesthetics is to abolish the activity dependence of the firing threshold without interfering with impulse conduction.11 This suggests that presynaptic processing may play an important role in human consciousness. This paper has explored some of the basic ideas of presynaptic information processing, especially the after- effects of activity and their modulation of impulse conduction at sites of low conduction safety. A switched capacitor circuit which sim ulates the activity dependent conduction block that occurs in axons has been designed and simulated. Simulation results are very similar to the intermittent conduction patterns measured experimentally in frog axons. One potential information processing possibility for the arbor of a single axon, suggested by the analysis of the electronic circuit, is to act as a filterbank; every terminal could act as a lowpass filter with a different corner frequency. BIBLIOGRAPHY [1] Barron D. H. and B. H. C. Matthews, Intermittent conduction in the spinal chord. J. Physiol. 85, p. 73-103 (1935). [2] Fuortes M. G. F., Action of strychnine on the "intermittent conduction" of impulses along dorsal columns of the spinal chord of frogs. J. Physiol. 112, p.42 (1950). [3] Culp W. and J. Ochoa, Nerves and Muscles as Abnormal Impulse Generators. (Oxford University Press, London, 1980). [4] Grossman V., I. Parnas, and M. E. Spira, Ionic mechanisms involved in differential conduction of action potentials at high frequency in a branching axon. J. Physiol. 295, p.307 - 322 (1978). [5] Parnas I., Differential block at high frequency of branches of a single axon innervating two muscles. J. Physiol. 35, p. 903-914, 1972. [6] Carley, L.R. and S.A. Raymond, Threshold Measurement: Applications to Excitable Membranes of Nerve and Muscle. J. Neurosci. Meth. 9, p. 309 - 333 (1983). [7] Raymond S. A. and J. V. Lettvin, After-effects of activity in peripheral axons as a clue to nervous coding. In Physiology and Pathobiology of Axons, S. G. Waxman (ed.), (Raven Press, New York, 1978), p. 203 - 225. [8] Wurtz C. C. and M. H. Ellisman, Alternations in the ultrastructure of peripheral nodes of Ranvier associated with repetitive action potential propagation. J. Neurosci. 6(11), 3133- 3143 (1986). [9] Raym ond S. A., Effects of nerve im pulses on threshold of frog sciatic nerve fibers. J. Physiol. 290,273- 303 (1979). [10] Carley, L.R. and S.A. Raymond, Com parison of the after- effects of impulse conduction on threshold at nodes of Ranvier along single frog Sciatic axons. J. Physiol. 386, p. 503 - 527 (1987). [11] Raymond S. A. and J. G. Thalhammer, Endogenous activitydependent mechanisms for reducing hyperexcitability ofaxons: Effects of anesthetics and CO2, In Inactivation of Hypersensistive Neurons, N. Chalazonitis and M. Gola, (eds.), (Alan R. Liss Inc., New Vork, 1987), p. 331-343. 163
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AN OPTIMIZATION NETWORK FOR MATRIX INVERSION Ju-Seog Jang, S~ Young Lee, and Sang-Yung Shin Korea Advanced Institute of Science and Technology, P.O. Box 150, Cheongryang, Seoul, Korea ABSTRACT Inverse matrix calculation can be considered as an optimization. We have demonstrated that this problem can be rapidly solved by highly interconnected simple neuron-like analog processors. A network for matrix inversion based on the concept of Hopfield's neural network was designed, and implemented with electronic hardware. With slight modifications, the network is readily applicable to solving a linear simultaneous equation efficiently. Notable features of this circuit are potential speed due to parallel processing, and robustness against variations of device parameters. INTRODUCTION Highly interconnected simple analog processors which mmnc a biological neural network are known to excel at certain collective computational tasks. For example, Hopfield and Tank designed a network to solve the traveling salesman problem which is of the np -complete class,l and also designed an AID converter of novel architecture2 based on the Hopfield's neural network model?' 4 The network could provide good or optimum solutions during an elapsed time of only a few characteristic time constants of the circuit. The essence of collective computation is the dissipative dynamics in which initial voltage configurations of neuron-like analog processors evolve simultaneously and rapidly to steady states that may be interpreted as optimal solutions. Hopfield has constructed the computational energy E (Liapunov function), and has shown that the energy function E of his network decreases in time when coupling coefficients are symmetric. At the steady state E becomes one of local minima. In this paper we consider the matrix inversion as an optimization problem, and apply the concept of the Hopfield neural network model to this problem. CONSTRUCTION OF THE ENERGY FUNCTIONS Consider a matrix equation AV=I, where A is an input n Xn matrix, V is the unknown inverse matrix, and I is the identity matrix. Following Hopfield we define n energy functions E Ie' k = 1, 2, ... , n, n n n E 1 = (1I2)[(~ A 1j Vj1 -1)2 + (~A2) Vj1 )2 + ... + (~Anj Vj1)2] )-1 j-1 n n E2 = (1/2)[(~A1)V)2l + (~A2)V)2-1)2 + )=1 )=1 © American Institute of Physics 1988 )-1 n + (~An)V}2)2] }-1 397 398 n n n En = (1/2)[(~ A1J VJn)2 + (~A2J Vjn )2 + ... + (~An) VJn _1)2] (1) j=l }=1 J-1 where AiJ and ViL.are the elements of ith row and jth column of matrix A and V, respectively. when A is a nonsingular matrix, the minimum value (=zero) of each energy function is unique and is located at a point in the corresponding hyperspace whose coordinates are { V u:, V 2k ' "', V nk }, k = 1, 2, "', n. At this minimum value of each energy function the values of V 11' V 12' ... , Vnn become the elements of the inverse matrix A -1. When A is a singular matrix the minimum value (in general, not zero) of each energy function is not unique and is located on a contour line of the minimum value. Thus, if we construct a model network in which initial voltage configurations of simple analog processors, called neurons, converge simultaneously and rapidly to the minimum energy point, we can say the network have found the optimum solution of matrix inversion problem. The optimum solution means that when A is a nonsingular matrix the result is the inverse matrix that we want to know, and when A is a singular matrix the result is a solution that is optimal in a least-square sense of Eq. (1). DESIGN OF THE NETWORK AND THE HOPFIELD MODEL Designing the network for matrix inversion, we use the Hopfield model without inherent loss terms, that is, --= dt a ---Ek(V 11' V 2k' ... , Vnk ) aVik i,k=1,2, ... ,n (2) where uik is the input voltage of ith neuron in the kth network, Vik is its output, and the function gik is the input-output relationship. But the neurons of this scheme operate in all the regions of gik differently from Hopfield's nonlinear 2state neurons of associative memory models.3• 4 From Eq. (1) and Eq. (2), we can define coupling coefficients Tij between ith and jth neurons and rewrite Eq. (2) as --= dt n - ~ TiJ V)k + Aki , j=l n TiJ = ~ AliAIJ = Tji ' 1=1 (3) It may be noted that Ti · is independent of k and only one set of hardware is needed for all k. The implemented network is shown in Fig. 1. The same set of n hardware with bias levels, ~ A Ji h), can be used to solve a linear simultaneous )=1 399 equation represented by Ax=b for a given vector b. INPUT OUTPUT Fig. 1. Implemented network for matrix inversion with externally controllable coupling coefficients. Nonlinearity between the input and the output of neurons is assumed to be distributed in the adder and the integrator. The application of the gradient Hopfield model to this problem gives the result that is similar to the steepest descent method.s But the nonlinearity between the input and the output of neurons is introduced. Its effect to the computational capability will be considered next. CHARACTERISTICS OF THE NETWORK For a simple case of 3 x3 input matrices the network is implemented with electronic hardware and its dynamic behavior is simulated by integration of the Eq. (3). For nonsingular input matrices, exact realization of Tij connection and bias Ali is an important factor for calculation accuracy, but the initial condition and other device parameters such as steepness, shape and uniformity of gil are not. Even a complex gik function shown in Fig. 2 can not affect the computational capability. Convergence time of the output state is determined by the characteristic time constant of the circuit. An example of experimental results is shown in Fig. 3. For singular input matrices, the converged output voltage configuration of the network is dependent upon the initial state and the shape of gil' 400 ,...-_____ Vm-t-___ --::==----r A ik > 1 = 1 < 1 Vm Ui\< Fig. 2. gile functions used in computer simulations where Aile is the steepness of sigmoid function tanh (Aile uile)' o input matrix output matrix [ 1 2 I] A = -I r 1 1 0-1 (cf) [ 0.50 -0.98 -0.49J V = 0.02 0.99 1.00 0.53 -0.98 - 1.50 [ 0.5 -I A-I = 0 1 0.5 -I Fig. 3. An example of experimental results -o.~] -1.5 0.5 ° COMPLEXITY ANALYSIS By counting operations we compare the neural net approach with other wellknown methods such as Triangular-decomposition and Gauss-Jordan elimination.6 (1) Triangular-decomposition or Gauss-Jordan elimination method takes 0 (8n 3/3) multiqlications/divisions and additions for large n Xn matrix inversion, and o (2n /3) multiplications/divisions and additions for solving the linear simultaneous equation Ax=b. 401 (2) The neural net approach takes the number of operations required to calculate Tij (nothing but matrix-matrix multiplication), that is, 0 (n 3/2) multiplications and additions for both matrix inversion and solving the linear simultaneous equation. And the time required for output stablization is about a few times the characteristic time constant of the network. The calculation of coupling coefficients can be directly executed without multiple iterations by a specially designed optical matrix-matrix multiplier,' while the calculation of bias values in solving a linear simultaneous equation can be done by an optical vector-matrix multiplier.8 Thus, this approach has a definite advantage in potential calculation speed due to global interconnection of simple parallel analog processors, though its calculation accuracy may be limited by the nature of analog computation. A large number of controllable Tij interconnections may be easily realized with optoelectronic devices.9 CONCLUSIONS We have designed and implemented a matrix inversion network based on the concept of the Hopfield's neural network model. 1bis network is composed of highly interconnected simple neuron-like analog processors which process the information in parallel. The effect of sigmoid or complex nonlinearities on the computational capability is unimportant in this problem. Steep sigmoid functions reduce only the convergence time of the network. When a nonsingular matrix is given as an input, the network converges spontaneously and rapidly to the correct inverse matrix regardless of initial conditions. When a singular matrix is given as an input, the network gives a stable optimum solution that depends upon initial conditions of the network. REFERENCES 1. J. J. Hopfield and D. W. Tank, BioI. Cybern. 52, 141 (1985). 2. D. W. Tank and J. J. Hopfield, IEEE Trans. Circ. Sys. CAS-33, 533 (1986). 3. J. J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982). 4. J. J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 81, 3088 (1984). 5. G. A. Bekey and W. J. Karplus, Hybrid Computation (Wiley, 1968), P. 244. 6. M. J. Maron, Numerical Analysis: A Practical Approach (Macmillan, 1982), p. 138. 7. H. Nakano and K. Hotate, Appl. Opt. 26, 917 (1987). 8. J. W. Goodman, A. R. Dias, and I. M. Woody, Opt. Lett. ~ 1 (1978). 9. J. W. Goodman, F. J. Leonberg, S-Y. Kung, and R. A. Athale, IEEE Proc. 72, 850 (1984).
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524 BASINS OF ATTRACTION FOR ELECTRONIC NEURAL NETWORKS C. M. Marcus R. M. Westervelt Division of Applied Sciences and Department of Physics Harvard University, Cambridge, MA 02138 ABSTRACT We have studied the basins of attraction for fixed point and oscillatory attractors in an electronic analog neural network. Basin measurement circuitry periodically opens the network feedback loop, loads raster-scanned initial conditions and examines the resulting attractor. Plotting the basins for fixed points (memories), we show that overloading an associative memory network leads to irregular basin shapes. The network also includes analog time delay circuitry, and we have shown that delay in symmetric networks can introduce basins for oscillatory attractors. Conditions leading to oscillation are related to the presence of frustration; reducing frustration by diluting the connections can stabilize a delay network. (1) INTRODUCTION The dynamical system formed from an interconnected network of nonlinear neuron-like elements can perform useful parallel computation l - 5 . Recent progress in controlling the dynamics has focussed on algorithms for encoding the location of fixed pOints 1,4 and on the stability of the flow to fixed points 3 • 5-8. An equally important aspect of the dynamics is the structure of the basins of attraction, which describe the location of all pOints in initial condition space which flow to a particular attractor10 . 22 . In a useful associative memory, an initial state should lead reliably to the "closest" memory. This requirement suggests that a well-behaved basin of attraction should evenly surround its attractor and have a smooth and regular shape. One dimensional basin maps plotting "pull in" probability against Hamming distance from an attract or do not reveal the shape of the basin in the high dimensional space of initial states9. 19 . Recently, a numerical study of a Hopfield network with discrete time and two-state neurons showed rough and irregular basin shapes in a two dimensional Hamming space, suggesting that the high dimensional basin has a complicated structure 10 . It is not known how the basin shapes change with the size of the network and the connection rule. We have investigated the basins of attraction in a network with continuous state dynamics by building an electronic neural network with eight variable gain sigmoid neurons and a three level (+,0,-) interconnection matrix. We have also built circuitry that can map out the basins of attraction in two dimensional slices of initial state space (Fig .1) . The network and the basin measurements are described in section 2. @ American Institute of Physics 1988 525 In section 3, we show that the network operates well as an associative memory and can retrieve up to four memories (eight fixed points) without developing spurious attractors, but that for storage of three or more memories, the basin shapes become irregular. In section 4, we consider the effects of time delay. Real network components cannot switch infinitely fast or propagate signals instantaneously, so that delay is an intrinsic part of any hardware implementation of a neural network. We have included a controllable CCD (charge coupled device) analog time delay in each neuron to investigate how time delay affects the dynamics of a neural network. We find that networks with symmetric interconnection matrices, which are guaranteed to converge to fixed points for no delay, show collective sustained oscillations when time delay is present. By discovering which configurations are maximally unstable to oscillation, and looking at how these configurations appear in networks, we are able to show that by diluting the interconnection matrix, one can reduce or eliminate the oscillations in neural networks with time delay. (2) NETWORK AND BASIN MEASUREMENT A block diagram of the network and basin measurement circuit is shown in fig.1. ¥c sigmoid amplifiers with digital comparator and oscillation detector desired memory Fi~.l Block diagram of the network and basin measurement system. The main feedback loop consists of non-linear amplifiers ("neurons", see fig.2) with capacitive inputs and a resistor matrix allowing interconnection strengths of -l/R, 0, +l/R (R = 100 kn). In all basin measurements, the input capacitance was 10 nF, giving a time constant of 1 ms. A charge coupled device (CCD) analog time delayll was built into each neuron, providing an adjustable delay per neuron over a range 0.4 8 ms. 526 50k 1/ Inverting output Fig.2 Electronic neuron. Non-linear gain provided by feedback diodes. Inset: Nonlinear behavior at several different values of gain. Analog switches allow the feedback path to be periodically disconnected and each neuron input charged to an initial voltage. The network is then reconnected and settles to the attractor associated with that set of initial conditions. Two of the initial voltages are raster scanned (on a time scale that is long compared to the load/run switching time) with function generators that are also connected to the X and Y axes of a storage scope. The beam of the scope is activated when the network settles into a de sired at tractor, producing an image of the basin for that attractor in a twodimensional slice of initial condition space. The "attractor of interest" can be one of the 2 8 fixed points or an oscillatory attractor. A simple example of this technique is the case of three neurons with symmetric non-inverting connection shown in fig.3. " G§§gBASIN FORt t t Fig.3 Basin " ", of " ", _ BASIN FOR ~ •• attraction for three 3-D .... with neurons symSTATE metric non-inverting SPACE • -1.0V coupling. Slices are . . in the plane of 1V initial voltages on neurons 1 and 2. The CIRCUIT: two fixed points are • all neurons saturated j~ positive or all negative. The data -1 V .~. are photographs of + the scope screen. -1 V 1V (3) BASINS FOR FIXED POINTS ASSOCIATIVE MEMORY Two dimensional slices of the eight dimensional initial condition space (for the full network) reveal important qualitative features about the high dimensional basins. Fig. 4 shows a typical slice for a network programmed with three memories according to a clipped Hebb rule1, 12: 527 (1 ) where ~ is an N-component memory vector of l's and -l's, and m is the number of memories. The memories were chosen to be orthogonal (~a .~~ = N 8a~) . 1V- r---r--------., 1, ·1, 1,·1, 1,·1, 1,·1 \ 0_\ '1,1,·1,·1, 1,1,1,1 1,1, ·1,·1, 1,1,·1,·1 1,1,1,1, ·1,·1,·1,·1 -1,1,-1,1, ·1,1,-1,1 .w._ •. w ..... "" .. -1 V- ""'=====:;:===~ .• I L.:.:.~ __ ~~~~_""'_ -IV 0 ~ IV MEMORIES: 1,1,1,1,-1,-1,-1,-1 1 ,-1,1 ,-1,1 ,-1,1 ,-1 1,1,-1,-1,1 ,1,-1,-1 Fi~. 4 A slice of initial condition space shows the basins of attraction for five of the six fixed points for three memories in eight-neuron Hopfield net. Learning rule was clipped Hebb (Eq.1). Neuron gain = 15. Because the Hebb rule (eq. 1) makes ~a and _~a stable attractors, a three-memory network will have six fixed point attractors. In fig.4, the basins for five of these attractors are visible, each produced with a different rastering pattern to make it distinctive. Several characteristic features should be noted: -All initial conditions lead to one of the memories (or inverses), no spurious attractors were seen for three or four memories. This is interesting in light of the well documented emergence of spurious attractors at miN -15% in larger networks with discrete time2,lS. -- The basins have smooth and continuous edges. -- The shapes of the basins as seen in this slice are irregular. Ideally, a slice with attractors at each of the corners should have rectangular basins, one basin in each quadrant of the slice and the location of the lines dividing quadrants determined by the initial conditions on the other neurons (the "unseen" dimensions). With three or more memories the actual basins do not resemble this ideal form. (4) TIME DELAY, FRUSTRATION AND SUSTAINED OSCILLATION Arguments defining conditions which guarantee convergence to fixed points 3 , 5,6 (based, for example, on the construction of a Liapunov function) generally assume instantaneous communication between elements of the network. In any hardware implementation, these assumptions break down due to the finite switching speed of amplifiers and the charging time of long interconnect lines. 13 It is the ratio of delay/RC which is important for stability, so keeping this ratio small limits how fast a neural network chip can be designed to run. Time delay is also relevant to biological neural nets where propagation and response times are comparable. 14 ,lS 528 Our particular interest in this section is how time delay can lead to sustained oscillation in networks which are known to be stable when there is no delay. We therefore restrict our attention to networks with symmetric interconnection matrices (Tlj = Tjl)' An obvious ingredient in producing oscillations in a delay network is feedback, or stated another way, a graph representing the connections in a network must contain loops. The simplest oscillatory structure made of delay elements is the ring oscillator (fig.Sa). Though not a symmetric configuration, the ring oscillator illustrates an important point: the ring will oscillate only when there is negative feedback at dc - that is, when the product of'interconnection around the loop is negative. Positive feedback at dc (loop product of connections > 0) will lead to saturation. Observing various symmetric configurations (e.g. fig.Sb) in the delayed-neuron network, we find that a negative product of connections around a loop is also a necessary condition for sustained oscillation in symmetric circuits. An important difference between the ring (fig.Sa) and the symmetric loop (fig.Sb) is that the period of oscillation for the ring is the total accumulated delay around the ring the larger the ring the longer the period. In contrast, for those symmetric configurations which have oscillatory attractors, the period of oscillation is roughly twice the delay, regardless of the size of the configuration or the value of delay. This indicates that for symmetric configurations the important feedback path is local, not around the loop. I·\:~~~illator (NEGATIVE ••• ,............ FEEDBACK) /;\\ ~bJmmetric 1/ \ \ (FI~~~TRATED) ................ otl ~ •••••. h.", • =lime delay neuron /' =non-inverting connection ,,:1' .. inverting .• ' connection Fir;;r . S (a) A ring oscillator: needs negative feedback at dc to oscillate. (b) Symmetrically connected triangle. This configuration is "frustrated" (defined in text), and has both oscillatory and fixed point attractors when neurons have delay . Configurations with loop connection product < 0 are important in the theory of spin glasses 16 , where such configurations are called "frustrated." Frustration in magnetic (spin) systems, gives a measure of "serious" bond disorder (disorder that cannot be removed by a change of variables) which can lead to a spin glass state. 16.17 Recent results based on the similarity between spin glasses and symmetric neural networks has shown that storage capacity limitations can be understood in terms of this bond disorder. 18 ,19 Restating our observation above: We only find stable oscillatory modes in symmetric networks with delay when there is frustration. A similar result for a sign-symmetric network (Tlj, Tjl both ~o or $0) with no delay is described by Hirsch. 6 We can set up the basin measurement system (fig.l) to plot the basin of attraction for the oscillatory mode. Fig.6 shows a slice of the oscillatory basin for a frustrated triangle of delay neurons. 1.5V o ·1.5V . . . . . . . " ,'.," • .. : ... f .... .' 'I ,:, '.'.' " .l,,· 'IO ' , ., 'i\" :,1 , . , . . ' ; .: ' I ·1.5V I o I 1.5V Fig.6 Basin for oscillatory attractor (cross-hatched region) in frustrated triangle of delay-neurons. Connections were all symmetric and inverting; other frustrated configurations (e.g. two non-inverting, one inverting, all symmetric) were similar. (6a): delay = O.48RC, inset shows trajectory to fixed point and oscillatory mode for two close-lying initial conditions. (6b): delay = O. 61RC, basin size increases. 529 A fully connected feedback associative network with more that one memory will contain frustration. As more memories are added, the amount of frustration will increases until memory retrieval disappears. But before this point of memory saturation is reached, delay could cause an oscillatory basin to open. In order to design out this possibility, one must understand how frustration, delay and global stability are related. A first step in determining the stability of a delay network is to consider which small configurations are most prone to oscillation, and then see how these "dangerous" configurations show up in the network. As described above, we only need to consider frustrated configurations. A frustrated configuration of neurons can be sparsely connected, as in a loop, or densely connected, with all neurons connected to all others, forming what is called in graph theory a "clique." Representing a network with inverting and non-inverting connections as a signed graph (edges carry + and -), we define a frustrated clique as a fully connected set of vertices (r vertices,' r (r-l) /2 edges) with all sets of three vertices in the clique forming frustrated triangles. Some examples of frustrated loops and cliques are shown in fig. 7. Notice that neurons connected with all inverting symmetric connections, a configuration that is useful as circuit, is a frustrated clique. a "winner-take-all" <> \·······1 \! ..... ·····~FRUSTRATED ,/ \! ! LOOPS .' . . . ........ ,~ / \ .... =inverting /",non-inverting ,.. .. symmetric connection symmetric connection , .~~----------------~--------------~ •...............• ..•.. ..' FRUSTRATED ~ ; . ::::.j..\.>. /i~;~/' CLIQUES : \">'" ,.:(/ ~~);;.::~:. (fully connected; : ! •..•... ~: ~:" ;;.!: .1'''. : all triangles ... ,' '." ;.~~r.~ ,'................ •...........• '" .. ' frustrated) FiQ.7 Examples of frustrated loops and frustrated cliques. In the graph representation vertices (black dots) are neurons (with delay) and undirected edges are symmetric connections. 530 We find that delayed neurons connected in a frustrated loop longer than three neurons do not show sustained oscillation for any value of delay (tested up to delay = 8RC). In contrast, when delayed neurons are connected in any frustrated clique configuration, we do find basins of attraction for sustained oscillation as well as fixed point attractors, and that the larger the frustrated clique, the more easily it oscillates in the following ways: (1) For a given value of delay/RC, the size of the oscillatory basin increases with r, the size of the frustrated clique (fig. 8). (2) The critical value of delay at which the volume of the oscillatory basin goes to zero decreases with increasing r (fig.9); For r=8 the critical delay is already less than 1/30 RC. /\ 1.1\ •..............• .: .............. =Fig.8 Size of basin for oscillatory mode increases with size of frustrated clique. The delay is 0.46RC per neuron in each picture. Slices are in the space of initial voltages on neurons 1 and 2, other initial voltages near 1 iG u . ~ u -4: 0:....... >co Q) "0 - .1 • • • • •• 1 size of frustrated clique (r) 10 zero. Fig.9 The critical valu,? of delay where the oscillatory mode vanishes . Measured by reducing delay until system leaves oscillatory attractor . Delay plotted in units of the characteristic time RioC, where Rio = (Lj 1 /Rij) -1=10Sn/ (r-1) and C=10nF, indicating that the critical delay decreases faster than 1/(r-1). Having identified frustrated cliques as the maximally unstable configuration of time delay neurons, we now ask how many cliques of a given size do we expect to find in a large network. A set of r vertices (neurons) can be fully connected by r(r-1)/2 edges of two types (+ or -) to form 2r (r-1)/2 different cliques. Of these, 2 (r-1) will be frustrated cliques. Fig .·10 shows all 2 (4-1) =8 cases for r=4. ~ A ,':11 : " : II i " ,', II . r.III' 1'1',1 ~ ............... ~ .---. ............... .-------'. Eig.10 All graphs of size r=4 that are frustrated cliques (fully connected, every triangle frustrated.) Solid lines = positive edges, dashed lines = negative edges. 531 For a randomly connected network, this result combined with results from random graph theory20 gives an expected number of frustrated cliques of size r in a network of size N, EN(r): N EN(r) = (r) c(r,p) c(r,p) = r(r-l) (r-2)/2 pr(r-l)/2 (2) (3) N where (r) is the binomial coefficient and c(r,p) is defined as the concentration of frustrated cliques. p is the connectance of the network, defined as the probability that any two neurons are connected. Eq.3 is the special case where + and edges (noninverting, inverting connections) are equally probable. We have also generalized this result to the case p(+)~p(-). Fig.11 shows the dramatic reduction in the concentration of all frustrated configurations in a diluted random network. For the general case (p(+)~p(-» we find that the negative connections affect the concentrations of frustrated cliques more strongly than the positive connections, as expected (Frustration requires negatives, not positives, see fig. 10) . 10°y---------~------~--~--r__r~~~_, connectance (p) 1 Fig.11 Concentration of frustrated cliques of size r=3,4,S,6 in an unbiased random network, from eq.3. Concentrations decrease rapidly as the network is diluted, especially for large cliques (note: log scale) . When the interconnections in a network are specified by a learning rule rather than at random, the expected numbers of any configuration will differ from the above results. We have compared the number of frustrated triangles in large three-valued (+1,0,-1) Hebb interconnection matrices (N=100,300,600) to the expected number in a random matrix of the same size and connectance. The Hebb matrix was constructed according to the rule: Tij = Zk (La=l,m ~ia ~ja) ; Tii = ° (4a) Zk(X) = +1 for x > k; 0 for -k $x $k; -1 for x < -k; (4b) m is the number of memories, Zkis a threshold function with cutoff k, and ~a is a random string of l's and -l's. The matrix constructed by eq.4 is roughly unbiased (equal number of positive and negative connections) and has a connectance p(k). Fig.12 shows the ratio of frustrated triangles in a diluted Hebb matrix to the expected number in a random graph with the same connectance for different numbers of 532 memories stored in the Hebb matrix. At all values of connectance, the Hebb matrix has fewer frustrated triangles than the random matrix by a ratio that is decreased by diluting the matrix or storing fewer memories. The curves do not seem to depend on the size of the matrix, N. This result suggests that diluting a Hebb matrix breaks up frustration even more efficiently than diluting a random matrix. ." • ratio 1TI=o15 N .. 300 CD 0.9 • ratio m-25 ~ FiQ'.12 The number of frustrated a ratio m-40 I'l! • ratio maS5 triangles in a (+,0,-) Hebb rule ·c 0.7 • ratio 1TI=o100 matrix (300x300) divided by the "C ~ expected number in a random ~ 0.5 signed graph with equal ::l .:: connectance. The different sets '0 0.3 of points are for different .2 T§ numbers of random memories in the 0.1 Hebb matrix. The lines are .1 connectance guides to the eye. The sensitive dependence of frustration on connectance suggests that oscillatory modes in a large neural network with delay can be eliminated by diluting the interconnection matrix. As an example, consider a unbiased random network with delay = RC/10. From fig.9, only frustrated cliques of size r=5 or larger have oscillatory basins for this value of delay; frustration in smaller configurations in the network cannot lead to sustained oscillation in the network. Diluting the connectance to 60% will reduce the concentration of frustrated cliques with r=5 by a factor of over 100 and r=6 by a factor of 2000. The reduction would be even greater for a clipped Hebb matrix. Results from spin glass theory21 suggest that diluting a clipped Hebb matrix can actually improve the storage capacity for moderated dilution, with a maximum in the capacity at a connectance of 61%. To the extent this treatment applies to an analog continuous-time network, we should expect that by diluting connections, oscillatory modes can be killed before memory capacity is compromised. We have confirmed the stabilizing effect of dilution in our network: For a fully connected eight neuron network programmed with three orthogonal memories according to eq.l, adding a delay of 0.4RC opens large basins for sustained oscillation. By randomly diluting the interconnections to p ...... 0.85, we were able to close the oscillatory basins and recover a useful associative memory. SUMMARY We have investigated the structure of fixed point and oscillatory basins of attraction in an electronic network of eight non-linear amplifiers with controllable time delay and a three value (+,0,-) interconnection matrix. For fixed point attractors, we find that the network performs well as an associative memory no spurious attractors were seen for up to four stored memories but for three or more memories, the shapes of the basins of attraction became irregular. 533 A network which is stable with no delay can have basins for oscillatory at tractors when time delay is present. For symmetric networks with time delay, we only observe sustained oscillation when there is frustration. Frustrated cliques (fully connected configurations with all triangles frustrated), and not loops, are most prone to oscillation, and the larger the frustrated clique, the more easily it oscillates. The number of the se "dangerous" configurations in a large network can be greatly reduced by diluting the connections. We have demonstrated that a network with a large basin for an oscillatory attractor can be stabilized by dilution. ACKNOWLEDGEMENTS We thank K.L.Babcock, S.W.Teitsworth, S.Strogatz and P.Horowitz for useful discussions. One of us (C.M.M) acknowledges support as an AT&T Bell Laboratories Scholar. This work was supported by JSEP contract no. N00014-84-K-0465. REFERENCES 1) J.S.Denker, Physica 22ll, 216 (1986). 2) J.J. Hopfield, Proc.Nat.Acad.Sci. ~, 2554 (1982). 3) J.J. Hopfield, Proc.Nat.Acad.Sci. al, 3008 (1984). 4) J.S. Denker, Ed. Neural Networks for Computing, AlP Conf. Proc. l.5..l. (1986). 5) M.A. Cohen, S. Grossberg, IEEE Trans. SMC-13, 815 (1983). 6) M.W.Hirsch, Convergence in Neural Nets, IEEE Conf.on Neural Networks, 1987. 7) K.L. Babcock, R.M. Westervelt, Physica 2Jll,464 (1986). 8) K.L. Babcock, R.M. Westervelt, Physica zau,305 (1987). 9) See, for example: D.B.Schwartz, et aI, Appl.Phys.Lett.,~ (16), 1110 (1987); or M.A.Silviotti,et aI, in Ref.4, pg.408. 10) J.D. Keeler in Ref.4, pg.259. 11) CCD analog delay: EG&G Reticon RD5106A. 12) D.O.Hebb, The Organization of Behavior, (J.Wiley, N.Y., 1949). 13) Delay in VLSI discussed in: A. Muhkerjee, Introduction to nMOS and CMOS VLSI System Design, (Prentice Hall, N.J.,1985). 14) U. an der Heiden, J.Math.Biology, ~, 345 (1979). 15) M.C. Mackey, U. an der Heiden, J.Math.Biology,~, 221 (1984). 16) Theory of spin glasses reviewed in: K. Binder, A.P. Young, Rev. Mod. Phys. ,.5,a (4),801, (1986). 17) E. Fradkin,B.A. Huberman,S.H. Shenker, Phys.Rev.lila (9),4789 (1978) . 18) D.J. Amit, H. Gutfreund, H. Sompolinski, Ann.Phys. ~, 30, (1987) and references therein. 19) J.L. van Hemmen, I. Morgenstern, Editors, Heidelberg Colloquium on Glassy Dynamics, Lecture Notes in Physics~, (SpringerVerlag, Heidelberg, 1987). 20) P. Erdos, A. Renyi, Pub. Math. Inst. Hung .Acad. Sci., .5.,17, (1960). 21) I.Morgenstern in Ref.19, pg.399;H.Sompolinski in Ref.19, pg.485. 22) J. Guckenheimer, P.Holmes, Nonlinear Oscillations,Dynamical Systems and Bifurcations of Vector Fields (Springer,N.Y.1983).
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564 PROGRAMMABLE SYNAPTIC CHIP FOR ELECTRONIC NEURAL NETWORKS A. Moopenn, H. Langenbacher, A.P. Thakoor, and S.K. Khanna Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91009 ABSTRACT A binary synaptic matrix chip has been developed for electronic neural networks. The matrix chip contains a programmable 32X32 array of "long channel" NMOSFET binary connection elements implemented in a 3-um bulk CMOS process. Since the neurons are kept offchip, the synaptic chip serves as a "cascadable" building block for a multi-chip synaptic network as large as 512X512 in size. As an alternative to the programmable NMOSFET (long channel) connection elements, tailored thin film resistors are deposited, in series with FET switches, on some CMOS test chips, to obtain the weak synaptic connections. Although deposition and patterning of the resistors require additional processing steps, they promise substantial savings in silcon area. The performance of a synaptic chip in a 32neuron breadboard system in an associative memory test application is discussed. INTRODUCTION The highly parallel and distributive architecture of neural networks offers potential advantages in fault-tolerant and high speed associative information processing. For the past few years, there has been a growing interest in developing electronic hardware to investigate the computational capabilities and application potential of neural networks as well as their dynamics and collective propertiesl - 5 • In an electronic hardware implementation of neural networks6 • 7 r the neurons (analog processing units) are represented by threshold amplifiers and the synapses linking the neurons by a resistive connection network. The synaptic strengths between neurons (the electrical resistance of the connections) represent the stored information or the computing function of the neural network. Because of the massive interconectivity of the neurons and the large number of the interconnects required with the increasing number of neurons, implementation of a synaptic network using current LSI/VLSI technology can become very difficult. A synaptic network based on a multi-chip architecture would lessen this difficulty. He have designed, fabricated, and successfully tested CMOS-based programmable synaptic chips which could serve as basic " cascadabl e" building blocks for a multi-chip electronic neural network. The synaptic chips feature complete programmability of 1024, (32X32) binary synapses. Since the neurons are kept offchip, the synaptic chips can be connected in parallel, to obtain multiple grey levels of the connection strengths, as well as ® American Institute of Physics 1988 565 "cascaded" to form larger synaptic arrays for an expansion to a 512neuron system in a feedback or feed-forward architecture. As a research tool, such a system would offer a significant speed improvement over conventional software-based neural network simulations since convergence times for the parallel hardware system would be significantly smaller. In this paper, we describe the basic design and operation of synaptic CMOS chips incorporating MOSFET's as binary connection elements. The design and fabrication of synaptic test chips with tailored thin film resistors as ballast resistors for controlling power dissipation are also described. Finally, we describe a synaptic chip-based 32-neuron breadboard system in a feedback configuration and discuss its performance in an associative memory test application. BINARY SYNAPTIC CMOS CHIP WITH MOSFET CONNECTION ELEMENTS There are two important design requirements for a binary connection element in a high density synaptic chip. The first requirement is that the connection in the ON state should be "weak" to ensure low overall power dissipation. The required degree of "weakness" of the ON connection largely depends on the synapse density of the chip. If, for example, a synapse density larger than 1000 per chip is desired, a dynamic resistance of the ON connection should be greater than ~100 X-ohms. The second requirement is that to obtain grey scale synapses with up to four bits of precision from binary connections, the consistency of the ON state connection resistance must be better than +/-5 percent, to ensure proper threshold operation of the neurons. Both of the requirements are generally difficult to satisfy simultaneously in conventional VLSI CMOS technology. For example, doped-polysilicon resistors could be used to provide the weak connections, but they are difficult to fabricate with a resistance uniformity of better than 5 percent. We have used NMOSFET's as connection elements in a multi-chip synaptic network. By designing the NMOSFET's with long channel, both the required high uniformity and high ON state resistance have been obtained. A block diagram of a binary synaptic test chip incorporating NMOSFET's as programmable connection elements is shown in Fig. 1. A photomicrograph of the chip is shown in Fig. 2. The synaptic chip was fabricated through MOSIS (MOS Implementation Service) in a 3-micron, bulk CMOS, two-level metal, P-well technology. The chip contains 1024 synaptic cells arranged in a 32X32 matrix configuration. Each cell consists of a long channel NMOSFET connected in series with another NMOSFET serving as a simple ON/OFF switch. The state of the FET switch is controlled by the output of a latch which can be externally addressed via the ROW/COL address decoders. The 32 analog input lines (from the neuron outputs) and 32 analog output lines (to the neuron inputs) allow a number of such chips to be connected together to form larger connection matrices with up to 4-bit planes. The long channel NMOSFET can function as either a purely resistive or a constant current source connection element, depending 566 /I6-A9 FROM NEURON OUTPUTS 1 ••• 32 vG ______ ~~~----~~----_=~ ~-.--a. .... . . .::-Q C~ AOOR OECOOER Ur-····················11 \ C~ • . \ S • .\ . . 0 \ ROW~V~ • ·\ RST R • SETI L -----d I RST --------I 6=. . . . . =a AO-M TO NEURON 32 INPUTS Figure 1. Block diagram of a 32X32 binary synnaptic chip with long channel NMOSFETs as connection elements . . ~' ... ,,) .,. , " Figure 2. Photomicrographs of a 32X32 binary connection CMOS chip. The blowup on the right shows several synaptic cells; the "S"-shape structures are the long-channel NMOSFETs. on whether analog or binary output neurons are used. As a resistive connection. the NMOSFET's must operate in the linear region of the transistor's drain I-V characteristics. In the linear region. the channel resistance is approximately given byB Ro N = (11K) (LIN) (VG VT H ) - 1 • 567 Here, K is a proportionality constant which depends on process parameters, Land Ware the channel length and width respectively, VG is the gate voltage, and VTH is the threshold voltage. The transistor acts as a linear resistor provided the voltage across the channel is much less than the difference of the gate and threshold voltages, and thus dictates the operating voltage range of the connection. The NMOSFET's presently used in our synaptic chip design have a channel length of 244 microns and width of 12 microns. At a gate voltage of 5 volts, a channel resistance of about 200 Kohms was obtained over an operating voltage range of 1.5 volts. The consistency of the transistor I-V characteristics has been verified to be within +/-3 percent in a single chip and +/-5 percent for chips from different fabrication runs. In the latter case, the transistor characteristics in the linear region can be further matched to within +/-3% by the fine adjustment of their common gate bias. With two-state neurons, current source connections may be used by operating the transistor in the saturation mode. Provided the voltage across the channel is greater than (VG VTH), the transistor behaves almost as a constant current source with the saturation current given approximately byB ION = K (W/L) (VG - VTH)2 . With the appropriate selection of L, W, and VG, it is possible to obtain ON-state currents which vary by two orders of magnitude in values. Figure 3 shows a set of measured I-V curves for a NMOSFET with the channel dimensions, L= 244 microns and W=12 microns and applied gate voltages from 2 to 4.5 volts. To ensure constant current source operation, the neuron's ON-state output should be greater than 3.5 volts. A consistency of the ON-state currents to within +/-5 percent has similarly been observed in a set of chip samples. With current source connections therefore, quantized grey scale synapses with up to 16 grey levels (4 bits) can be realized using a network of binary weighted current sources. Figure 3. I-V characteristics of an NMOSFET connection element. Channel dimension: L=244 urn, W=12um For proper operation of the NMOSFET connections, the analog output lines (to neuron inputs) should always be held close to ground potential. Moreover, the voltages at the analog input lines must be at or above ground potential. Since the current normally 568 flows from the analog input to the output, the NMOSFET's may be used as either all excitatory or inhibitory type connections. However, the complementary connection function can be realized using long channel PMOSFET's in series with PMOSFET switches. For a PMOSFET connection, the voltage of an analog input line would be at or below ground. Furthermore, due to the difference in the mobilites of electrons and holes in the channel, a PMOSFET used as a resistive connection has a channel resistance about twice as large as an NMOSFET with the same channel dimension. This fact results in a subtantial reduction in the size of PMOSFET needed. THIN FILM RESISTOR CONNECTIONS The use of MOSFET's as connection elements in a CMOS synaptic matrix chip has the major advantage that the complete device can be readily fabricated in a conventional CMOS production run. However, the main disadvantages are the large area (required for the long channel) for the MOSFET's connections and their non-symmetrical inhibitory/excitatory functional characteristics. The large overall gate area not only substantially limits the number of synapses that can be fabricated on a single chip, but the transistors are more susceptible to processing defects which can lead to excessive gate leakage and thus reduce chip yield considerably. An alternate approach is simply to use resistors in place of MOSFET's. We have investigated one such approach where thin film resistors are deposited on top of the passivation layer of CMOS-processed chips as an additional special processing step to the normal CMOS fabrication run. With an appropriate choice of resistive materials, a dense array of resistive connections with highly uniform resistance of up to 10 M-ohms appears feasible. Several candidate materials, including a cermet based on platinum/aluminum oxide, and amorphous semiconductor/metal alloys such as a-Ge:Cu and a-Ge:Al, have been examined for their applicability as thin film resistor connections. These materials are of particular interest since their resistivity can easily be tailored in the desired semiconducting range of 1-10 ohm-cm by controlling the metal content'. The a-Ge/metal films are deposited by thermal evaporation of presynthesized alloys of the desired composition in high vacuum, whereas platinum/aluminum oxide films are deposited by co-sputtering from platinum and aluminum oxide targets in a high purity argon and oxygen gas mixture. Room temperature resistivities in the 0.1 to 100 ohm-cm range have been obtained by varying the metal content in these materials. Other factors which would also determine their suitability include their device processing and material compatibilities and their stability with time, temperature, and extended application of normal operating electric current. The temperature coefficient of resistance (TCR) of these materials at room temperature has been measured to be in the 2000 to 6000 ppm range. Because of their relatively high TCR's, the need for weak connections to reduce the effect of localized heating is especially important here. The a-Ge/metal alloy films are observed to be relatively stable with exposure to air for temperatures below 130o C. 569 The platinum/aluminum oxide film stabilize with time after annealing in air for several hours at 130o C. Sample test arrays of thin film resistors based on the described materials have been fabricated to test their consistency. The resistors, with a nominal resistance of 1 M-ohm, were deposited on a glass substrate in a 40X40 array over a O.4cm by O.4cm area. Variation in the measured resistance in these test arrays has been found to be from +/- 2-5 percent for all three materials. Smaller test arrays of a-Ge:Cu thin film resistors on CMOS test chips have also been fabricated. A photo-micrograph of a CMOS synaptic test chip containing a 4X4 array of a-Ge:Cu thin film resistors is shown in Fig. 4. Windows in the passivation layer of silicon nitride (SiN) were opened in the final processing step of a normal CMOS fabrication run to provide access to the aluminum metal for electrical contacts. A layer of resistive material was deposited and patterned by lift-off. A layer of buffer metal of platinum or nickel was then deposited by RF sputtering and also patterned by lift-off. The buffer metal pads serve as a conducting bridges for connecting the aluminum electrodes to the thin film resistors. In addition to providing a reliable ohmic contact to the aluminum and resistor, it also provides conformal step coverage over the silicon nitride window edge. The resistor elements on the test chip are 100 micron long, 10 micron wide with a thickness of about 1500 angstroms and a nominal resistance of 250 K-ohms. Resistance variations from 10-20 percent have been observed in several such test arrays. The unusually large variation is largely due to the surface roughness of the chip passivation layer. As one possible solution, a thin spinFigure 4. Photomicrographs of· a CMOS synaptic test chip with a 4X4 array of a-Ge:Cu thin film resistors. The nominal resistance was 250 K-ohms. 570 on coating of an insulating material such as polyimide to smooth out the surface of the passivation layer prior to depositing the resistors is under investigation. SYNAPTIC CHIP-BASED 32-NEURON BREADBOARD SYSTEM A 32-neuron breadboard system utilizing an array of discrete neuron electronics has been fabricated to evaluate the operation of 32X32 binary synaptic CMOS chips with NMOSFET connection elements. Each neuron consists of an operational amplifier configured as a current to voltage converter (with virtual ground input) followed by a fixed-gain voltage difference amplifier. The overall time constant of the neurons is approximately 10 microseconds. The neuron array is interfaced directly to the synaptic chip in a full feedback configuration. The system also contains prompt electronics consisting of a programmable array of RC discharging circuits with a relaxation time of approximately 5 microseconds. The prompt hardware allows the neuron states to be initialized by precharging the selected capacitors in the RC circuits. A microcomputer interfaced to the breadboard system is used for programming the synaptic matrix chip, controlling the prompt electronics, and reading the neuron outputs. The stability of the breadboard system is tested in an associative mellory feedback configuration,b. A dozen random dilutecoded binary vectors are stored using the following simplified outer-product storage scheme: f -1 Ti j = 1 0 ~ s s if L Vi Vj = 0 S otherwise. In this scheme, the feedback matrix consists of only inhibitory (1lor open (0) connections. The neurons are set to be normally ON and are driven OFF when inhibited by another neuron via the feedback matrix. The system exhibits excellent stability and associative recall performance. Convergence to a nearest stored memory in Hamming distance is always observed for any given input cue. Figure 5 shows some typical neuron output traces for a given test prompt and a set of stored memories. The top traces show the response of two neurons that are initially set ON; the bottom traces for two other neurons initially set OFF. Convergence times of 10-50 microseconds have been observed, depending on the prompt conditions, but are primarily governed by the speed of the neurons. CONCLUSIONS Synaptic CMOS chips containing 1024 programmable binary synapses in a 32X32 array have been designed, fabricated, and tested. These synaptic chips are designed to serve as basic building blocks for large multi-chip synaptic networks. The use of long channel MOSFET's as either resistive or current source connection elements meets the "weak" connection and consistency 571 Figure 5. Typical neuron response curves for a test prompt input. (Horiz scale: 10 microseconds per div) requirements. Alternately, CMOS-based synaptic test chips with specially deposited thin film high-valued resistors, in series with FET switches, offer an attractive approach to high density programmable synaptic chips. A 32-neuron breadboard system incorporating a 32X32 NMOSFET synaptic chip and a feedback configuration exhibits excellent stability and associative recall performance as an associative memory. Using discrete neuron array, convergence times of 10-50 microseconds have been demonstrated. With optimization of the input/output wiring layout and the use of high speed neuron electronics, convergence times can certainly be reduced to less than a microsecond. ACKNOWLEDGEMENTS This work was performed by the Jet Propulsion Laboratory, California Institute of Technology, and was sponsored by the Joint Tactical Fusion Program Office, through an agreement with the National Aeronautics and Space Administration. The authors would like to thank John Lambe for his invaluable suggestions, T. Duong for his assistance in the breadboard hardware development, J. Lamb and S. Thakoor for their help in the thin film resistor deposition, and R. Nixon and S. Chang for their assistance in the chip layout design. REFERENCES 1. J. Lambe, A. Moopenn, and A.P. Thakoor, Proc. AIAA/ACM/NASA/IEEE Computers in Aerospace V, 160 (1985) 2. A.P. Thakoor, J.L. Lamb, A. Moopenn, and S.K. Khanna, MRS Proc. 95, 627 (1987) 3. W. Hubbard, D. Schwartz, J. Denker, H.P. Graf, R. Howard, L. Jackel, B. Straughn, and D. Tennant, AIP Conf. Proc. 151, 227 (1986) 4. M.A. Sivilotti, M.R. Emerling, and C. Mead, AIP Conf. Proc. 151, 408 (1986) 5. J.P. Sage, K. Thompson, and R.S. Withers, AIP Conf. Proc. 151, 572 381 (19861 6. 3.3. Hopfield, Proc. Nat. Acad. SCi., 81, 3088 (1984) 7. 3.3. Hopfield, Proc. Nat. Acad. Sci., 79, 2554 (1982) 8, S.M. Sze, "Semiconductor Devices-Physics and Technology," (Wiley, New York, 1985) p.205 9. 3.L. Lamb, A.P. Thakoor, A. Moopenn, and S.K. Khanna, 3. Vac. Sci. Tech., A 5(4), 1407 (1987)
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622 LEARNING A COLOR ALGORITHM FROM EXAMPLES Anya C. Hurlbert and Tomaso A. Poggio Artificial Intelligence Laboratory and Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA ABSTRACT A lightness algorithm that separates surface reflectance from illumination in a Mondrian world is synthesized automatically from a set of examples, pairs of input (image irradiance) and desired output (surface reflectance). The algorithm, which resembles a new lightness algorithm recently proposed by Land, is approximately equivalent to filtering the image through a center-surround receptive field in individual chromatic channels. The synthesizing technique, optimal linear estimation, requires only one assumption, that the operator that transforms input into output is linear. This assumption is true for a certain class of early vision algorithms that may therefore be synthesized in a similar way from examples. Other methods of synthesizing algorithms from examples, or "learning", such as backpropagation, do not yield a significantly different or better lightness algorithm in the Mondrian world. The linear estimation and backpropagation techniques both produce simultaneous brightness contrast effects. The problems that a visual system must solve in decoding two-dimensional images into three-dimensional scenes (inverse optics problems) are difficult: the information supplied by an image is not sufficient by itself to specify a unique scene. To reduce the number of possible interpretations of images, visual systems, whether artificial or biological, must make use of natural constraints, assumptions about the physical properties of surfaces and lights. Computational vision scientists have derived effective solutions for some inverse optics problems (such as computing depth from binocular disparity) by determining the appropriate natural constraints and embedding them in algorithms. How might a visual system discover and exploit natural constraints on its own? We address a simpler question: Given only a set of examples of input images and desired output solutions, can a visual system synthesize. or "learn", the algorithm that converts input to output? We find that an algorithm for computing color in a restricted world can be constructed from examples using standard techniques of optimal linear estimation. The computation of color is a prime example of the difficult problems of inverse optics. We do not merely discriminate betwN'n different wavelengths of light; we assign @ American Institute of Physics 1988 623 roughly constant colors to objects even though the light signals they send to our eyes change as the illumination varies across space and chromatic spectrum. The computational goal underlying color constancy seems to be to extract the invariant surface spectral reflectance properties from the image irradiance, in which reflectance and iI-" lumination are mixed 1 • Lightness algorithms 2-8, pioneered by Land, assume that the color of an object can be specified by its lightness, or relative surface reflectance, in each of three independent chromatic channels, and that lightness is computed in the same way in each channel. Computing color is thereby reduced to extracting surface reflectance from the image irradiance in a single chromatic channel. The image irra.diance, s', is proportional to the product of the illumination intensity e' and the surface reflectance r' in that channel: s' (x, y) = r' (x, y )e' (x, y). (1 ) This form of the image intensity equation is true for a Lambertian reflectance model, in which the irradiance s' has no specular components, and for appropriately chosen color channels 9. Taking the logarithm of both sides converts it to a sum: s(x, y) = rex, y) + e(x,y), where s = loges'), r = log(r') and e = log(e'). (2) Given s(x,y) alone, the problem of solving Eq. 2 for r(x,y) is underconstrained. Lightness algorithms constrain the problem by restricting their domain to a world of Mondrians, two-dimensional surfaces covered with patches of random colors2 and by exploiting two constraints in that world: (i) r'(x,y) is unifonn within patches but has sharp discontinuities at edges between patches and (ii) e' (x, y) varies smoothly across the Mondrian. Under these constraints, lightness algorithms can recover a good approximation to r( x, y) and so can recover lightness triplets that label roughly constant colors 10. We ask whether it is possible to synthesize from examples an algorithm that ex· tracts reflectance from image irradiance. and whether the synthesized algorithm will resemble existing lightness algorithms derived from an explicit analysis of the constraints. We make one assumption, that the operator that transforms irradiance into reflectance is linear. Under that assumption, motivated by considerations discussed later, we use optimal linear estimation techniques to synthesize an operator from examples. The examples are pairs of images: an input image of a Mondrian under illumination that varies smoothly across space and its desired output image that displays the reflectance of the Mondrian without the illumination. The technique finds the linear estimator that best maps input into desired output. in the least squares sense. For computational convenience we use one-dimensional "training vectors" that represent vertical scan lines across the ~londrian images (Fig. 1). We generate many 624 1S0t · -~ · ~ 100 , '. , ~~~--------------~~--~--~~ SO 100 110 100 llO 100 a lilt., Input d.t. i;[:2:=:0 :~kfhJfEirQ b a SO 101 ISO ZOO UI )01 0 I I 100 ISO 100 110 100 p/Jte' I"'" f~l o 50 100 UO ZOO ZSO )00 p,xe' .1 100 ISO 100 110 lot p'." OlltPllt lIlll.l .. ll.a Fig. 1. (a) The input data, a one-dimensional vector 320 pixels long. Its random Mondrian reflectance pattern is superimposed on a linear illumination gradient with a random slope and offset. (b) shows the corresponding output solution, on the left the illumination and on the right reBectance. We used 1500 such pairs of inputoutput examples (each different from the others) to train the operator shown in Fig. 2. (c) shows the result obtained by the estimated operator when it acts on the input data (a), not part of the training set. On the left is the illumination and on the right the reflectance, to be compared with (b). This result is fairly typical: in some cases the prediction is even better, in others it is worse. c different input vectors s by adding together different random T and e vectors, according to Eq. 2. Each vector r represents a pattern of step changes across space, corresponding to one column of a reHectance image. The step changes occur at random pixels and are of random amplitude between set minimum and maximum values. Each vector t represents a smooth gradient across space with a random offset and slope, correspondin~ to one column of an illumination image. We th~n arrange the training vectors sand r as the columns of two matrices Sand R, resp~ti· .. ely. Our goal is then to compute the optimal solution L of LS = R where L is a linear operator represented as a matrix. 625 It is well known that the solution of this equation that is optimal in the least squares sense is ( 4) where S+ is the Moore-Penrose pseudoinverse 11. We compute the pseudoinverse by overconstraining the problem - using many more training vectors than there are number of pixels in each vector - and using the straightforward formula that applies in the overconstrained case 12: S+ = ST(SST)-l. The operator L computed in this way recovers a good approximation to the correct output vector r when given a new s, not part of the training set, as input (Fig. Ic). A second operator, estimated in the same way, recovers the illumination e. Acting on a random two-dimensional Mondrian L also yields a satisfactory approximation to the correct output image. Our estimation scheme successfully synthesizes an algorithm that performs the lightness computation in a Mondrian world. What is the algorithm and what is its relationship to other lightness algorithms? To answer these questions we examine the structure of the matrix L. We assume that, although the operator is not a convolution operator, it should approximate one far from the boundaries of the image. That is, in its central part, the operator should be space-invariant, performing the same action on each point in the image. Each row in the central part of L should therefore be the same as the row above but displaced by one element to the right. Inspection of the matrix confirmes this expectation. To find the form of L in its center, we thus average the rows there, first shifting them appropriately. The result, shown in Fig. 2, is a space-invariant filter with a narrow positive peak and a broad, shallow, negative surround. Interestingly, the filter our scheme synthesizes is very similar to Land's most recent retinex operator 5, which divides the image irradiance at each pixel by a weighted average of the irradiance at all pixels in a large surround and takes the logarithm of that result to yield lightness 13. The lightness triplets computed by the retinex operator agree well with human perception in a Mondrian world. The retinex operator and our matrix L both differ from Land's earlier retinex algorithms, which require a non-linear thresholding step to eliminate smooth gradients of illumination. The shape of the filter in Fig. 2, particularly of its large surround, is also suggestive of the "nonclassical" receptive fields that have been found in V 4, a cortical area implicated in mechanisms underlying color constancy 14-17. The form of the space-invariant filter is similar to that derived in our earlier formal analysis of the lightness problem 8. It is qualitatively the same as that which results from the direct application of regularization methods exploiting the spatial constraints on reflectance and illumination described above 9.18.19. The Fourier transform of the filter of Fig. 2 is approximately a bandpass filter that cuts out low frequencies due 626 ~ 0 (.) ~ C' C s::. .2' ~ a -80 -80 0 Pi xe Is ----------o Pixels +80 Fig. 2. The space-invariant part of the estimated operator, obtained by shifting and averaging the rows of a 160-pixel-wide central square of the matrix L, trained on a set of 1500 examples with linear illumination gradients (see Fig. 1). When logarithmic illumination gradients are used, a qualitatively similar receptive field is obtained. In a separate experiment we use a training set of one-dimensional Mondrians with either linear illumination gradients or slowly varying sinusoidal illumination components with random wavelength, phase and amplitude. T he resulting filter is shown in the inset. The surrounds of both filters extend beyond the range we can estimate reliably, the range we show here. to slow gradients of illumination and preserves intennediate frequencies due to step changes in reflectance. In contrast, the operator that recovers the illumination, e. takes the form of a low-pass filter. \Ve stress that the entire operator L is not a space-invariant filter. In this context, it is clear that the shape of the estimated operator should vary with the type of illumination gradient in the training set. We synthesize a second operator using a new set of examples that contain equal numbers of vectors with random, sinusoidally varying illumination components and VE"(tors with random, linear illumination gradients. Whereas the first operator, synthE.>Sized from examples with strictly linear illumination gradients, has a broad negative surround that remains virtually constant throughout its extent, the new operator's surround (Fig. 2, inset) has a smaller ext(,111 627 and decays smoothly towards zero from its peak negative value in its center. We also apply the operator in Fig. 2 to new input vectors in which the density and amplitude of the step changes of reflectance differ greatly from those on which the operator is trained. The operator performs well, for example, on an input vector representing one column of an image of a small patch of one reflectance against a uniform background of a different reflectance, the entire image under a linear illumination gradient. This result is consistent with psychophysical experiments that show that color constancy of a patch holds when its Mondrian background is replaced by an equivalent grey background 20. The operator also produces simultaneous brightness contrast, as expected from the shape and sign of its surround. The output reflectance it computes for a patch of fixed input reflectance decreases linearly with increasing average irradiance of the input test vector in which the patch appears. Similarly, to us, a dark patch appears darker when against a light background than against a dark one. This result takes one step towards explaining such illusions as the Koffka Ring 21. A uniform gray annulus against a bipartite background (Fig. 3a) appears to split into two halves of different lightnesses when the midline between the light and dark halves of the background is drawn across the annulus (Fig. 3b). The estimated operator acting on the Koffka Ring of Fig. 3b reproduces our perception by assigning a lower output reflectance to the left half of the annulus (which appears darker to us) than to the right half 22. Yet the operator gives this brightness contrast effect whether or not the midline is drawn across the annulus (Fig. 3c). Becau~e the opf'rator can perform only a linear transformation between the input and output images, it is not surprising that the addition of the midline in the input evokes so little change in the output. These results demonstrate that the linear operator alone cannot compute lightness in all worlds and suggest that an additional operator might be necessary to mark and guide it within bounded regions. Our estimation procedure is motivated by our previous observation 9.23,18 that standard regularization algorithms 19 in early vision define linear mappings between input and output and therefore can be estimated associatively under certain condi· tions. The technique of optimal linear estimation that we use is closely related to optimal Bayesian estimation 9. If we were to assume from the start that the optimal linear operator is space-invariant, we could considerably simplify (and streamline) the computation by using standard correlation te<:hniques 9.24. How does our estimation technique compare with other methods of "learning" a lightness algorithm? We can compute the r~ularized pseudoinverse using gradient descent on a "neural" network 25 with linf'ar units. Since the pseudoinverse is lhf" unique best linear approximation in the L1 norm. a gradient descent method that 628 minimizes the square error between the actual output and desired output of a fully connected linear network is guaranteed to converge, albeit slowly. Thus gradient descent in weight space converges to the same result as our first technique, the global minimum. a c .n 0 sa .It .sa _ ut _ input data pixel ... ~~ u , :i II ... , . . . ~:== e. sa ~ ~ _ ut _ output reflectance - with edge ~'~ :i'I~~ ... I' _.. ~. . -.. ' =.. I ( , e. " ~ ~ _ ut _ output reflectance - without edge b Fig. 3. (a) Koffka Ring. (b) Koftka Ring with midline drawn across annulus. (c) Horizontal scan lines across Koffka Ring. Top: Scan line starting at arrow in (b). Middle: Scan line at corresponding location in the output of linear operator acting on (b). Bottom: Scan line at same location in the output of operator acting on (a). 629 We also compare the linear estimation technique with a "backpropagation" network: gradient descent on a 2-layer network with sigmoid units 25 (32 inputs, 32 "hidden units", and 32 linear outputs), using training vectors 32 pixels long. The network requires an order of magnitude more time to converge to a stable configuration than does the linear estimator for the same set of 32-pixel examples. The network's performance is slightly, yet consistently, better, measured as the root-mean-square error in output, averaged over sets of at least 2000 new input vectors. Interestingly, the backpropagation network and the linear estimator err in the same way on the same input vectors. It is possible that the backpropagation network may show considerable inprovement over the linear estimator in a world more complex than the Mondrian one. We are presently examining its performance on images with real-world features such as shading, shadows, and highlights26. We do not think that our results mean that color constancy may be learned during a critical period by biological organisms. It seems more reasonable to consider them simply as a demonstration on a toy world that in the course of evolution a visual system may recover and exploit natural constraints hidden in the physics of the world. The significance of our results lies in the facts that a simple statistical technique may be used to synthesize a lightness algorithm from examples; that the technique does as well as other techniques such as backpropagation; and that a similar technique may be used for other problems in early vision. Furthermore, the synthesized operator resembles both Land's psychophysically-tested retinex operator and a neuronal nonclassical receptive field. The operator's properties suggest that simultaneous color (or brightness) contrast might be the result of the visual system's attempt to discount illumination gradients 27 REFERENCES AND NOTES 1. Since we do not have perfect color constancy, our visual system must not extract reflectance exactly. The limits on color constancy might reveal limits on the underlying computation. 2. E.H. Land, Am. Sci. 52,247 (1964). 3. E.H. Land and J.J. McCann, J. Opt. Soc. Am. 61, 1 {1971}. 4. E.H. Land, in Central and Peripheral Mechanisms of Colour Vision, T. Ottoson and S. Zeki, Eds., (Macmillan, New York, 1985), pp. 5-17. 5. E.H. Land, Proc. Nat. Acad. Sci. USA 83, 3078 (1986). 6. B.K.P. Hom, Computer Graphics and Image Processing 3, 277 (1974). 630 7. A. Blake, in Central and Peripheral Mechanisms of Colour Vision, T. Ottoson and S. Zeki, Eds., (Macmillan, New York, 1985), pp. 45-59. 8. A. Hurlbert, J. Opt. Soc. Am. A 3,1684 (1986). 9. A. Hurlbert and T. Poggio, ArtificiaLIntelligence Laboratory Memo 909, (M.LT., Cambridge, MA, 1987). 10. r'{x,y) can be recovered at best only to within a constant, since Eq. 1 is invariant under the transformation of r' int.o ar' and e' into a-ie', where a is a constant. 11. A. Albert, Regression and the Moore-Penrose Pseudoinllerse, (Academic Press, New York, 1972). 12. The pseudoinverse, and therefore L, may also be computed by recursive techniques that improve its form as more data become availablell . 13. Our synthesized filter is not exactly identical with Land's: the filter of Fig. 2 subtracts from the value at each point the average value of the logarithm of irradiance at all pixels, rather than the logarithm of the average values. The estimated operator is therefore linear in the logarithms, whereas Land's is not. The numerical difference between the outputs of the two filters is small in most cases (Land, personal communication), and both agree well with psychophysical results. 14. R. Desimone, S.J. Schein, J. Moran and L.G. Ungerleider, Vision Res. 25, 441 (1985). 15. H.M. Wild, S.R. Butler, D. Carden and J.J. Kulikowski, Nature (London) 313, 133 (1985). 16. S.M. Zeki, Neuroscience 9, 741 (1983). 17. S.M. Zeki, Neuroscience 9, 767 (1983). 18. T. Poggio, et. al, in Proceedings Image Understanding Workshop, L. Baumann, Ed., (Science Applications International Corporation, McLean, VA, 1985), pp.' 25-39. 19. T. Poggio, V. Torre and C. Koch, Nature (London) 317,314 (1985). 20. A. Valberg and B. Lange-Malecki, Investigative Ophthalmology and Visual Science Supplement 28, 92 (1987). 21. K. Koffka, Principles of Gestalt Psychology, (Harcourt, Brace and Co., New York, 1935). 22. Note that the operator achieves this effect by subtracting a non-existent illumination gradient from the input signal. 23. T. Poggio and A. Hurlbert, Artificial Intelligence Laboratory Working Paper 264, (M.LT., Cambridge, MA, 1984). 24. Estimation of the operator on two-dimensional examples is possible, but computationally very expensive if done in the same way. The present computer simulations require several hours when run on standard serial computers. The two-dimensional case 631 will need much more time (our one-dimensional estimation scheme runs orders of magnitude faster on a CM-1 Connection Machine System with 16K-processors). 25. D. E. Rumelhart, G.E. Hinton and R.J. Williams, Nature (London) 323, 533 (1986 ). 26. A. Hurlbert, The Computation of Color, Ph.D. Thesis, M.l. T., Cambridge, MA, in preparation. 2i. We are grateful to E. Land, E. Hildreth, .J. Little, F. Wilczek and D. Hillis for reading the draft and for useful discussions. A. Rottenberg developed the routines for matrix operations that we used on the Connection Machine. T. Breuel wrote the backpropagation simulator.
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602 GENERALIZATION OF BACKPROPAGATION TO RECURRENT AND HIGHER ORDER NEURAL NETWORKS Fernando J. Pineda Applied Physics Laboratory, Johns Hopkins University Johns Hopkins Rd., Laurel MD 20707 Abstract A general method for deriving backpropagation algorithms for networks with recurrent and higher order networks is introduced. The propagation of activation in these networks is determined by dissipative differential equations. The error signal is backpropagated by integrating an associated differential equation. The method is introduced by applying it to the recurrent generalization of the feedforward backpropagation network. The method is extended to the case of higher order networks and to a constrained dynamical system for training a content addressable memory. The essential feature of the adaptive algorithms is that adaptive equation has a simple outer product form. Preliminary experiments suggest that learning can occur very rapidly in networks with recurrent connections. The continuous formalism makes the new approach more suitable for implementation in VLSI. Introduction One interesting class of neural networks, typified by the Hopfield neural networks (1,2) or the networks studied by Amari(3,4) are dynamical systems with three salient properties. First, they posses very many degrees of freedom, second their dynamics are nonlinear and third, their dynamics are dissipative. Systems with these properties can have complicated attractor structures and can exhibit computational abilities. The identification of attractors with computational objects, e.g. memories at d rules, is one of the foundations of the neural network paradigm. In this paradigl n, programming becomes an excercise in manipulating attractors. A learning algorithm is a rule or dynamical equation which changes the locations of fixed points to encode information. One way of doing this is to minimize, by gradient descent, some function of the system parameters. This general approach is reviewed by Amari(4) and forms the basis of many learning algorithms. The formalism described here is a specific case of this general approach. The purpose of this paper is to introduce a fonnalism for obtaining adaptive dynamical systems which are based on backpropagation(5,6,7). These dynamical systems are expressed as systems of coupled first order differential equations. The formalism will be illustrated by deriving adaptive equations for a recurrent network with first order neurons, a recurrent network with higher order neurons and finally a recurrent first order associative memory. Example 1: Recurrent backpropagation with first order units Consider a dynamical system whose state vector x evolves according to the following set of coupled differential equations ® American Institute ofPhvsics 1988 603 dx·/dt = -x' + g'(LW"X') + I· I 1 1. IJ J I J (1) where i=l, ... ,N. The functions g' are assumed to be differentiable and may have different forms for various populations of neurons. In this paper we shall make no other requirements on gi' In the neural network literature it is common to take these functions to be Sigmoid shaped functions. A commonly used form is the logistic function, (2) This form is biologically motivated since it attempts to account for the refractory phase of real neurons. However, it is important to stress that there is nothing in the mathematical content of this paper which requires this form -- any differentiable function will suffice in the formalism presented in this paper. For example, a choice which may be of use in signal processing is sin(~). A necessary condition for the learning algorithms discussed here to exist is that the system posesses stable isolated attractors, i.e. fixed points. The attractor structure of (1) is the same as the more commonly used equation du/dt = -ui +~Wijg(Uj) + Ki' (3) J Because (1) and (3) are related by a simple linear transformation. Therefore results concerning the stability of (3) are applicable to (1). Amari(3) studied the dynamics of equation (3) in networks with random conections. He found that collective variables corresponding to the mean activation and its second moment must exhibit either stable or bistable behaviour. More recently, Hopfield(2) has shown how to construct content addressable memories from symmetrically connected networks with this same dynamical equation. The symmetric connections in the network gaurantee global stability. The solution of equation (1) is also globally asymptotically stable if w can be transformed into a lower triangular matrix by row and column exchange operations. This is because in such a case the network is a simply a feedforward network and the output can be expressed as an explicit function of the input. No Liapunov function exists for arbitrary weights as can be demonstrated by constructing a set of weights which leads to oscillation. In practice, it is found that oscillations are not a problem and that the system converges to fixed points unless special weights are chosen. Therefore it shall be assumed, for the purposes of deriving the backpropagation equations, that the system ultimately settles down to a fixed point. Consider a system of N neurons, or units, whose dynamics is determined by equation (1). Of all the units in the network we will arbitrarily define some subset of them (A) as input units and some other subset of them (0) as output units. Units which are neither members of A nor 0 are denoted hidden units. A unit may be simultaneously an input unit and an output unit. The external environment influences the system through the source term, I. If a unit is an input unit, the corresponding component of I is nonzero. To make this more precise it is useful to introduce a notational convention. Suppose that <I> represent some subset of units in the network then the function 8i<I> is defined by 1 if i-th unit is a member of <I> 8'm= { 1'V 0 th o erwise In terms of this function, the components of the I vector are given by (4) (5) 604 where ~i is detennined by the external environment. Our goal will be to fmd a local algorithm which adjusts the weight matrix w so that a given initial state XO = x(to)' and a given input I result in a fixed point, xoo= x(too), whose components have a desired set of values Ti along the output units. This will be accomplished by minimizing a function E which tneasures the distance between the desired fixed point and the actual fixed point i.e., 1 N E =- :E Ji2 (6) 2 i=l where J. (T. - xoo. ) e'n I I I I.u. (7) E depends on the weight matrix w through the fixed point Xoo(w). A learning algorithm drives the fixed points towards the manifolds which satisfy xi 00 = Ti on the output units. One way of accomplishing this with dynamics is to let the system evolve in the weight space along trajectories which are antiparallel to the gradient of E. In . other words, dE dWi/dt = T\ -dw .. IJ (8) where T\ is a numerical constant which defines the (slow) time scale on which w changes. T\ must be small so that x is always essentially at steady state, i.e. x(t) == xoo. It is important to stress that the choice of gradient descent for the learning dynamics is by no means unique, nor is it necessarily the best choice. Other learning dynamics which employ second order time derivatives (e.g. the momentum method(5» or which employ second order space derivatives (e.g. second order backpropagation(8» may be more useful in particular applications. However, equation (8) does have the virtue of being the simplest dynamics which minimizes E. On performing the differentiations in equation (8), one immediately obtains dxoo k dwrs/dt = T\ 1: Jk a k wrs (9) The derivative of xoo k with respect to w rs is obtained by first noting that the fixed points of equation (1) satisfy the nonlinear algebraic equation Xoo. = g·(:Ewooxoo.) + J. (10) I I. IJ J I' J differentiating both sides of this equation with respect to Wrs and finally solving for dxooId dWrs' The result is dXook = (L-1)kr gr'(Ur)xoo s (11) dWrs where gr' is the derivative of gr and where the matrix L is given by (12) Bii is the Kroneker B function ( BU= 1 if i=j, otherwise Bij = 0). On substituting (11) into (9) one obtains the remarkablY simple form where dWrsldt = 11 YrXoo s Yr = gr'(ur) LJk(L -1)kr k= 605 (13) (14) Equations (13) and (14) specify a fonnallearning rule. Unfortunately, equation (14) requires a matrix inversion to calculate the error signals Yk' Direct matrix inversions are necessarily nonlocal calculations and therefore this learning algorithm is not suitable for implementation as a neural network. Fortunately, a local method for calculating Yr can be obtained by the introduction of an associated dynamical system. To obtain this dynamical system fIrst rewrite equation (14) as LLrk (Yr / gr'(ur)} = Jk . (15) r Then multiply both sides by fk'(uk)' substitute the explicit form for L and finally sum over r. The result is o = -Yk + gk'(uk){ LWrkYr + Jk} . (16) r One now makes the observation that the solutions of this linear equation are the fIxed points of the dynamical system given by dYk/dt = - Yk +gk'(uk){LWrkYr + Jk} . (17) r This last step is not unique, equation (16) could be transformed in various ways leading to related differential equations, cf. Pineda(9). It is not difficult to show that the frrst order fInite difference approximation (with a time step ~t = 1) of equations (1), (13) and (17) has the same form as the conventional backpropagation algorithm. Equations (1), (13) and (17) completely specify the dynamics for an adaptive neural network, provided that (1) and (17) converge to stable fixed points and provided that both quantities on the right hand side of equation (13) are the steady state solutions of (1) and (17). It was pointed out by Almeida(10) that the local stability of (1) is a sufficient condition for the local stability of (17). To prove this it suffices to linearize equation (1) about a stable fixed point. The resulting linearized equation depends on the same matrix L whose transpose appears in the derivation of equation (17), cf. equation (15). But Land LT have the same eigenValues, hence it follows that the fIXed points of (17) must also be locally stable if the fIxed points of (1) are locally stable. Learning multiple associations It is important to stress that up to this point the entire discussionhas assumed that I and T are constant in time, thus no mechanism has been obtained for learning multiple input/output associations. Two methods for training the network to learn multiple associations are now discussed. These methods lead to qualitatively different learning behaviour. Suppose that each input/output pair is labeled by a pattern label n, i.e. {In ,Tn}. Then the energy function which is minimized in the above discussion must also depend on this label since it is an implicit function of the In ,Tn pairs. In order to learn multiple input/output associations it is necessary to minimize all the E[n] simultaniously. In otherwords the function to minimize is (18) 606 where the sum is over all input/output associations. From (18) it follows that the gradient for Etotal is simply the sum of the gradients for each association, hence the corresponding gradient descent equation has the form, dWijldt = 11 L yOOi[a] xOOia] . (19) a In numerical simulations, each time step of (19) requires relaxing (1) and (17) for each pattern and accumulating the gradient over all the patterns. This fonn of the algorithm is deterministic and is guaranteed to converge because, by construction, Etotal is a Liapunov function for equation (19). However, the system may get stuck m a local minimum. This method is similar to the master/slave approach of Lapedes and Farber(1l). Their adaptive equation, which plays the same role as equation (19), also has a gradient form, although it is not strictly descent along the gradient. For a randomly or fully connected network it can be shown that tbe number of oper~tions required per weight update in the master/slave fonnalis~ is proportional to N where N is the number of units. This is because there are O(N ) update equations and each equation requires O(N) operations (assuming some precomputation). On the other hand, in the backpropagation formalism each update equation re~uires only 0(1) operations because of their trivial outer product form. Also O(N ) operations are required t~ precompute XOO and yoo. The result is that each weight update requires only O(N ) operations. It is not possible to conclude from this argument that one or the other approach will be more efficient in a particular application because there are other factors to consider such as the number of patterns and the number of time steps required for x and y to converge. A detailed comparison of the two methods is in preparation. A second approach to learning multiple patterns is to use (13) and to change the patterns randomly on each time step. The system therefore receives a sequence of random impulses each of which attempts to minimize E[ ex] for a single pattern. One can then defme L(w) to be the mean E[a] (averaged over the distribution of patterns). L(w) = <E [w, la,Ta ]> (20) Amari(4) has pointed out that if the sequence of random patterns is stationary and if L(w) has a unique minimum then the theory of stochastic approximation guarantees that the solution of (13) wet) will converge to the minimum point '!min of L(w) to within a small fluctuating tenn which vanishes as 11 tends to zero. hVlaently 11 is analogous to the temperature parameter in simulated annealing. This second approach generally converges more slowly than the first, but it will ultimately converge (in a statistical sense) to the global minimum. In principle the fixed points, to which the solutions of (1) and (17) eventually converge, depend on the initial states. Indeed, Amari's(3) results imply that equation (1) is bistable for certain choices of weights. Therefore the presentation of multiple patterns might seem problematical since in both approaches the final state of the previous pattern becomes the initial state of the new pattern. The safest approach is to reinitialize the network to the same initial state each time a new pattern is presented. e.g. xi(t~ = 0.5 for all i. In practice the system learns robustly even if the initial conditIons are chosen randomly. Example 2: Recurrent higher order networks It is straightforward to apply the technique of the previous section to a dynamical system with higher order units. Higher order systems have been studied by Sejnowski (12) and Lee et al.(13). Higher order networks may have definite advantages 607 over networks with first order units alone A detailed discussion of the backpropagation fonnalism applied to higher order networks is beyond the scope of this paper. Instead, the adaptive equations for a network with purely n-th order units will be presented as an example of the fonnalism. To this end consider a dynamical system of the fonn dx·/dt - -x' + g'(lI!) + I· I 1 1-:1 1 (21) where (22) and where there are n+ 1 indices and the summations are over all indices except i. The superscript on the weight tensor indicates the order of the correlation. Note that an additional nonlinear function f has been added to illustrate a further generalization. Both f and g must be differentiable and may be chosen to be sigmoids. It is not difficult, although somewhat tedious, to repeat the steps of the previous example to derive the adaptive equations for this system. The objective function in this case is the same as was used in the fIrst example, i.e. equation (6). The n-th order gradient descent equation has the fonn (23) Equation (23) illustrates the major feature of backpropagation which distinguishes it from other gradient descent algorithms or similar algorithms which make use of a gradient. Namely, that the gradient of the objective function has a very trivial outer product fonn. y (n)oo is the steady state solution of dy(n)k/dt = - y(n)k + gk'(uk) {fk'(xk)Ly(n)rkY (n)r + Jk}. (24) r The matrix v(n) plays the role of w in the previous example, however v(n) now depends on the state of the network according to y(n)ij = L'" L s<n)ijk"'l ( f(xk) ... f(xI)} (25) k I where is s(n) a tensor which is symmetric with respect to the exchange of the second index and all the indices to the right, i.e. S(n).. - w(n) + w(n) + ... + w(n) IJk"1 ijk"'l ikj"'l ijl"'k . (26) Finally, it should be noted that: 1) If the polynomial ui is not homogenous, the adaptive equations are more complicated and involve cross tenns between the various orders and that: 2) The local stability of the n-th order backpropagation equations now depends on the eigenvalues of the matrix L .. = 0" - g.'(u·) f.'(x·) y(n) .. IJ IJ 1 1 1 1 IJ' (27) As before, if the forward propagation converges so will the backward propagation. Example 3: Adaptive content addressable memory In this section the adaptive equations for a content addressable memory (CAM) are derived as a fmal illustration of the generality of the formalism. Perhaps 608 the best known (and best studied) examples of dYnamical systems which exhibit CAM behaviour are the systems discussed by Hopfield(l). Hopfield used a nonadaptive method for programming the symmetric weight matrix. More recently Lapedes and Farber<ll) have demonstrated how to contruct a master dynamical system which can be used to train the weights of a slave system which has the Hopfield fonn. This slave system then performs the CAM operation. The resulting weights are not symmetric. The learning proceedure presented in this section is most closely related to the method of Lapedes and Farber in that a master network is used to adjust the weights of a slave network. In constrast to the afforementioned formalism, which requires a very large associated weight matrix for the master network, both the master and slave networks of the following approach make use of the same weight matrix. The CAM under consideration is based on equation (1). However, the interpretation of the dynamics will be somewhat different from the first section. The main difference is that the dynamics in the learning phase is constrained. The constrained dynamical system is denoted the master network. The unconstrained system is denoted the slave network. The units in the network are divided into only two sets: the set of visible units (V) and the set of internal or hidden units (H). There will be no distinction made between input and output units. Thus, I will generally be zero unless an input bias is needed in some application. The dynamical system will be used as an autoassociative memory, thus the memory recall is performed by starting the network at a particular initial state which represents partial information about a stored memory. More precisely, suppose that there exists a subset K of the visible units whose states are known to have values Ti' Then the initial state of the network is (28) where the bi are arbitrary. The CAM relaxes to the previously stored memory whose basin of attraction contains this partial state. Memories are stored by a master network whose topology is exactly the same as the slave network, but whose dynamics is somewhat modified. The state vector z of the master network evolves according to the equation N d~/dt = -~ + gi(LwikZk) + Ii (29) k=l where Z is defmed by Z, = T· E)·V + z· E)'H 1 1 1 1 1 • (30) The components of Z along the visible units are just the target value specified by T. This equation is useful as a master equation because if the weights can be chosen so that the zi of the visible units relax to the target values Ti,: then a fixed point of (29) is also a fixed point of (1). It can be concluded therefore, that by training the weights of the master network one is also training the weights of the slave network. Note that the form of Z implies that equation (29) can be rewritten as (31) where 9i = - LWikTk . (32) keY From equations (31) and (32) it is clear that the dynamics of the master system is driven by the thresholds which depend on the targets. where To derive the adaptive equations consider the objective function 1 N 2 Emaster = 2" 1: Ii (33) i=l (34) 609 It is straightforward to apply the steps discussed in previous sections to EJ1Iaster' This results in adaptive equations for the weights. The mathematical details Will be omitted since they are essentially the same as before, the gradient descent equation is dWi/dt = 11yoo iZOOj where yOO is the steady state solution of (35) dyk"dt = - Yk +g'k(vkHeiHLwrkYr + Ik} (36) r where vi i ~ikZoo k . (37) Equations (31), and (35)-37) define the dynamics of the master network. To train the slave network to be an autoassociative memory it is necessary to use the stored memories as the initial states of the master network, i.e. z·(t ) = T· e·V + b· eiH 1 0 1 1 1 (39) where bi is an arbitrary value as before. The previous discussions concerning the stability of the three equations (1), (13) and (17) apply to equations (31) (35) and (36) as well. It is also possible to derive the adaptive equations for a higher order associative network, but this will not be done here. Only preliminary computer simulations have been performed with this algorithm to verify their validity, but more extensive experiments are in progress. The fIrst simulation was with a fully connected network with 10 visible units and 5 hidden units. The training set consisted of four random binary vectors with the magnitudes of the vectors adjusted so that 0.1 ~ Ti S; 0.9. The equations were approximated by first order fmite difference equations with ~t = 1 and 11 = 1. The training was performed with the detenninistic method for learning multiple associations. Figure 1. shows Etotal as a function of the number of updates for both the master and slave networks. Etota! for the slave exhibits discontinous behaviour because the trajectory through the weight space causes x(to) to cut across the basins of attraction for the fixed points of equation (1). The number of updates required for the network to learn the patterns is relatively modest and can be reduced further by increasing 11. This suggests that learning can occur very rapidly in this type of network. Discussion The algorithms presented here by no means exhaust the class of possible adaptive algorithms which can be obtained with this formalism. Nor is the choice of gradient descent a crucial feature in this formalism. The key idea is that it is possible to express the gradient of an objective function as the outer product of vectors which can be calculated by dynamical systems. This outer produc2,form is also responsible for the fact that the gradient can be calculated with only O(N ) operations in a fully connected or randomly connected network. In fact the number of operations per 610 weight update is proportional to the number of connections in the network. The methods used here will generalize to calculate higher order derivatives of the objective function as well. The fact that the algorithms are expressed as differential equations suggests that they may be implemented in analog electronic or optical hardware. 2.00 .....--------------, 1.00 --"", .. ~ 20 40 60 80 100 Updates ~ Master -.Slave figure 1. Etota! as a function of the the number of updates. References (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) I. J. HopfieJd. Neural Networks as Physical Systems with Emergent Collective Computational Abilities. Proc. Nat. Acad. Sci. USA. Bio.79. 2554-2558. (1982) 1. I. Hopfield. Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci. USA. Bio . .8l, 3088-3092. (1984) Shun-Ichi Amari. IEEE Trans. on Systems Man and Cybernetics. 2.643-657. (1972) Shun-Ichi Amari. in Systems Neuroscience. ed. Jacqueline Metzler. Academic press. (1977) D. E. Rumelhart. G. E. Hinton and R.I. Williams. in Parallel Distributed Processing. edited by D. E. Rumelhart and 1. L. McClelland. M.LT. press. (1986) David B. Parker. Learning-Logic, Invention Report. S81-64. File 1. Office of Technology Licensing. Stanford University. October. 1982 Y. LeChun. Proceedings of Cognitiva. 85. p. 599. (1985) David B. Parker. Second Order Backpropagation: Implementing an Optimal O(n) Approximation to Newton's Method as an Artificial Neural Network. submitted to Computer. (1987) Fernando J. Pineda. Generalization ofbackpropagation to recurrent neural networks, Phys. Rev. Lett .• l.8. 2229-2232. (1987) Luis B. Almeida. in the Proceedings of the IEEE First Annual International Conference on Neural Networks. San Diego. California. June 1987. edited by 611 M. Caudil and C. Butler (to be published This is a discrete version of the algorithm presented as the fIrst example (11) Alan Lapedes and Robert Farber, A self-optimizing, nonsymmetrical neural net for content addressable memory and pattern recognition, Physica, D22, 247-259, (1986), see also, Programming a Massively Parallel, Computation Universal System: Static Behaviour, in Neural Networks for Computing Snowbird, UT 1986, AIP Conference Proceedings, 151, (1986), edited by John S. Denker (12) Terrence J. Sejnowski, Higher-order Boltzmann Machines, Draft preprint obtained from author (13) Y.C. Lee, Gary Doolen, H.H. Chen, G.Z. Sun, Tom Maxwell, H.Y. Lee and C. Lee Giles, Machine Learning using a higher order correlation network, Physica D22, 276-306, (1986)
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442 Abstract: How Neural Nets Work Alan Lapedes Robert Farber Theoretical Division Los Alamos National Laboratory Los Alamos, NM 87545 There is presently great interest in the abilities of neural networks to mimic "qualitative reasoning" by manipulating neural incodings of symbols. Less work has been performed on using neural networks to process floating point numbers and it is sometimes stated that neural networks are somehow inherently inaccurate and therefore best suited for "fuzzy" qualitative reasoning. Nevertheless, the potential speed of massively parallel operations make neural net "number crunching" an interesting topic to explore. In this paper we discuss some of our work in which we demonstrate that for certain applications neural networks can achieve significantly higher numerical accuracy than more conventional techniques. In particular, prediction of future values of a chaotic time series can be performed with exceptionally high accuracy. We analyze how a neural net is able to do this , and in the process show that a large class of functions from Rn. ~ Rffl may be accurately approximated by a backpropagation neural net with just two "hidden" layers. The network uses this functional approximation to perform either interpolation (signal processing applications) or extrapolation (symbol processing applicationsJ. Neural nets therefore use quite familiar methods to perform. their tasks. The geometrical viewpoint advocated here seems to be a useful approach to analyzing neural network operation and relates neural networks to well studied topics in functional approximation. 1. Introduction Although a great deal of interest has been displayed in neural network's capabilities to perform a kind of qualitative reasoning, relatively little work has been done on the ability of neural networks to process floating point numbers in a massively parallel fashion. Clearly, this is an important ability. In this paper we discuss some of our work in this area and show the relation between numerical, and symbolic processing. We will concentrate on the the subject of accurate prediction in a time series. Accurate prediction has applications in many areas of signal processing. It is also a useful, and fascinating ability, when dealing with natural, physical systems. Given some .data from the past history of a system, can one accurately predict what it will do in the future? Many conventional signal processing tests, such as correlation function analysis, cannot distinguish deterministic chaotic behavior from from stochastic noise. Particularly difficult systems to predict are those that are nonlinear and chaotic. Chaos has a technical definition based on nonlinear, dynamical systems theory, but intuitivly means that the system is deterministic but "random," in a rather similar manner to deterministic, pseudo random number generators used on conventional computers. Examples of chaotic systems in nature include turbulence in fluids (D. Ruelle, 1971; H. Swinney, 1978), chemical reactions (K. Tomita, 1979), lasers (H. Haken, 1975), plasma physics (D. Russel, 1980) to name but a few. Typically, chaotic systems also display the full range of nonlinear behavior (fixed points, limit cycles etc.) when parameters are varied, and therefore provide a good testbed in which to investigate techniques of nonlinear signal processing. Clearly, if one can uncover the underlying, deterministic algorithm from a chaotic time series, then one may be able to predict the future time series quite accurately, © American Institute of Physics 1988 443 In this paper we review and extend our work (Lapedes and Farber ,1987) on predicting the behavior of a particular dynamical system, the Glass-Mackey equation. We feel that the method will be fairly general, and use the GlassMackey equation solely for illustrative purposes. The Glass-Mackey equation has a strange attractor with fractal dimension controlled by a constant parameter appearing in the differential equation. We present results on a neural network's ability to predict this system at two values of this parameter, one value corresponding to the onset of chaos, and the other value deeply in the chaotic regime. We also present the results of more conventional predictive methods and show that a neural net is able to achieve significantly better numerical accuracy. This particular system was chosen because of D. Farmer's and J. Sidorowich's (D. Farmer, J. Sidorowich, 1987) use of it in developing a new, non-neural net method for predicting chaos. The accuracy of this non-neural net method, and the neural net method, are roughly equivalent, with various advantages or disadvantages accruing to one method or the other depending on one's point of view. We are happy to acknowledge many valuable discussions with Farmer and Sidorowich that has led to further improvements in each method. We also show that a neural net never needs more than two hidden layers to solve most problems. This statement arises from a more general argument that a neural net can approximate functions from Rn. -+ Rm with only two hidden layers, and that the accuracy of the approximation is controlled by the number of neurons in each layer. The argument assumes that the global minimum to the backpropagation minimization problem may be found, or that a local minima very close in value to the global minimum may be found. This seems to be the case in the examples we considered, and in many examples considered by other researchers, but is never guaranteed. The conclusion of an upper bound of two hidden layers is related to a similar conclusion of R. Lipman (R. Lipman, 1987) who has previously analyzed the number of hidden layers needed to form arbitrary decision regions for symbolic processing problems. Related issues are discussed by J. Denker (J. Denker et.al. 1987) It is easy to extend the argument to draw similar conclusions about an upper bound of two hidden layers for symbol processing and to place signal processing, and symbol processing in a common theoretical framework. 2. Backpropagation Backpropagation is a learning algorithm for neural networks that seeks to find weights, T ij, such that given an input pattern from a training set of pairs of Input/Output patterns, the network will produce the Output of the training set given the Input. Having learned this mapping between I and 0 for the training set, one then applies a new, previously unseen Input, and takes the Output as the "conclusion" drawn by the neural net based on having learned fundamental relationships between Input and Output from the training set. A popular configuration for backpropagation is a totally feedforward net (Figure 1) where Input feeds up through "hidden layers" to an Output layer. 444 OUTPUT Figure 1. A feedforward neural net. Arrows schematically indicate full feedforward connectivity Each neuron forms a weighted sum of the inputs from previous layers to which it is connected, adds a threshold value, and produces a nonlinear function of this sum as its output value. This output value serves as input to the future layers to which the neuron is connected, and the process is repeated. Ultimately a value is produced for the outputs of the neurons in the Output layer. Thus, each neuron performs: (1) where Tii are continuous valued, positive or negative weights, 9. is a constant, and g(x) is a nonlinear function that is often chosen to be of a sigmoidal form. For example, one may choose 1 g(z) = 2" (1 + tanhz) (2) where tanh is the hyperbolic tangent, although the exact formula of the sigmoid is irrelevant to the results. If t!") are the target output values for the pth Input pattern then ones trains the network by minimizing E = L L (t~P) - o!P)) 2 (3) p i where t~p) is the target output values (taken from the training set) and O~pl is the output of the network when the pth Input pattern of the training set is presented on the Input layer. i indexes the number of neurons in the Output layer. An iterative procedure is used to minimize S. For example, the commonly used steepest descents procedure is implemented by changing Tii and S, by AT'i and AS, where aE ~T. .. = --'E '1 aT. .. '1 445 (4a) ( 4b) This implies that ~E < 0 and hence E will decrease to a local minimum. Use o~ the chain .rule and definition of some intermediate quantities allows the followmg expressIons for ~Tij to be obtained (Rumelhart, 1987): ~Tij = L E6lp)o~.p) p where if i is labeling a neuron in the Output layer; and 6Jp) = O!p) (1 o~p») LTij 6;p) j (Sa) (Sb) (6) (7) if i labels a neuron in the hidden layers. Therefore one computes 6Jp) for the Output layer first, then uses Eqn. (7) to computer 6ip ) for the hidden layers, and finally uses Eqn. (S) to make an adjustment to the weights. We remark that the steepest descents procedure in common use is extremely slow in simulation, and that a better minimization procedure, such as the classic conjugate gradient procedure (W. Press, 1986), can offer quite significant speedups. Many applications use bit representations (0,1) for symbols, and attempt to have a neural net learn fundamental relationships between the symbols. This procedure has been successfully used in converting text to speech (T. Sejnowski, 1986) and in determining whether a given fragment of DNA codes for a protein or not (A. Lapedes, R. Farber, 1987). There is no fundamental reason, however, to use integer's as values for Input and Output. If the Inputs and Outputs are instead a collection of floating point numbers, then the network, after training, yields a specific continuous function in n variables (for n inputs) involving g(x) (Le. hyperbolic tanh's) that provides a type of nonlinear, least mean square interpolant formula for the discrete set of data points in the training set. Use of this formula a = 1(11, 1", ... 1'1) when given a new input not in the training set, is then either interpolation or extrapolation. Since the Output values, when assumed to be floating point numbers may have a dynamic range great than 10,1\, one may modify the g(x) on the Output layer to be a linear function, instead of sigmoidal, so as to encompass the larger dynamic range. Dynamic range of the Input values is not so critical, however we have found that numerical problems may be avoided by scaling the Inputs (and 446 also the Outputs) to [0,1], training the network, and then rescaling the Ti;, (J, to encompass the original dynamic range. The point is that scale changes in I and 0 may, for feedforward networks, always be absorbed in the T ijJ (J, and vice versa. We use this procedure (backpropagation, conjugate gradient, linear outputs and scaling) in the following section to predict points in a chaotic time series. 3. Prediction Let us consider situations in Nature where a system is described by nonlinear differential equations. This is faily generic. We choose a particular nonlinear equation that has an infinite dimensional phase space, so that it is similar to other infinite dimensional systems such as partial differential equations. A differential equation with an infinite dimensional phase space (i.e. an infinite number of values are necessary to describe the initial condition) is a delay, differential equation. We choose to consider the time series generated by the Glass-Mackey equation: X= az(t - 1') b t 1 + Z 10 (t _ 1') Z ( ) (8) This is a nonlinear differential, delay equation with an initial condition specified by an initial function defined over a strip of width l' (hence the infinite dimensional phase space i.e. initial functions, not initial constants are required). Choosing this function to be a constant function, and a = .2, b = .1, and l' = 17 yields a time series, x(t), (obtained by integrating Eqn. (8)), that is chaotic with a fractal attractor of dimension 2.1. Increasing l' to 30 yields more complicated evolution and a fractal dimension of 3.5. The time series for 500 time steps for 1'=30 (time in units of 1') is plotted in Figure 2. The nonlinear evolution of the system collapses the infinite dimensional phase space down to a low (approximately 2 or 3 dimensional) fractal, attracting set. Similar chaotic systems are not uncommon in Nature. Figure 2. Example time series at tau ~ 30. 447 The goal is to take a set of values of xO at discrete times in some time window containing times less than t, and use the values to accurately predict x(t + P), where P is some prediction time step into the future. One may fix P, collect statistics on accuracy for many prediction times t (by sliding the window along the time series), and then increase P and again collect statistics on accuracy. This one may observe how an average index of accuracy changes as P is increased. In terms of Figure 2 we will select various prediction time steps, P, that correspond to attempting to predict within a "bump," to predicting a couple of "bumps" ahead. The fundamental nature of chaos dictates that prediction accuracy will decrease as P is increased. This is due to inescapable inaccuracies of finite precision in specifying the x( t) at discrete times in the past that are used for predicting the future. Thus, all predictive methods will degrade as P is increased - the question is "How rapidly does the error increase with P?" We will demonstrate that the neural net method can be orders of magnitude more accurate than conventional methods at large prediction time steps, P. Our goal is to use backpropagation, and a neural net, to construct a function O(t + P) = f (11(t), 12(t - A) ... lm(t - mA)) (9) where O(t + P) is the output of a single neuron in the Output layer, and 11 ~ 1m are input neurons that take on values z(t), z(t - A) ... z(t - rnA), where A is a time delay. O(t + P) takes on the value x(t + P). We chose the network configuation of Figure 1. We construct a training set by selecting a set of input values: (10) 1m = x(tp - rnA) with associated output values 0 = x(tp + P), for a collection of discrete times that are labelled by tp. Typically we used 500 I/O pairs in the training set so that p ranged from 1~ 500. Thus we have a collection of 500 sets of {lip), l~p), ... , 1::); O(p)} to use in training the neural net. This procedure of using delayed sampled values of x{t) can be implemented by using tapped delay lines, just as is normally done in linear signal processing applications, (B. Widrow, 1985). Our prediction procedure is a straightforward nonlinear extension of the linear Widrow Hoff algorithm. After training is completed, prediction is performed on a new set of times, tp, not in the training set i.e. for p = 500. We have not yet specified what m or A should be, nor given any indication why a formula like Eqn. (9) should work at all. An important theorem of Takens (Takens, 1981) states that for flows evolving to compact attracting manifolds of dimension d.A" that a functional relation like Eqn. (9) does exist, and that m lies in the range d.A, < m + 1 < 2d.A, + 1. We therefore choose m = 4, for T = 30. Takens provides no information on A and we chose A = 6 for both cases. We found that a few different choices of m and A can affect accuracy by a factor of 2 a somewhat significant but not overwhelming sensitivity, in view of the fact that neural nets tend to be orders of magnitude more accurate than other methods. Takens theorem gives no information on the form of fO in Eqn. (9). It therefore 448 is necessary to show that neural nets provide a robust approximating procedure for continuous fO, which we do in the following section. It is interesting to note that attempts to predict future values of a time series using past values of x(t) from a tapped delay line is a common procedUre in signal processing, and yet there is little, if any, reference to results of nonlinear dynamical systems theory showing why any such attempt is reasonable. After trainin, the neural net as described above, we used it to predict 500 new values of x(tJ in the future and computed the average accuracy for these points. The accuracy is defined to be the average root mean square error, divided by a constant scale factor, which we took to be the standard deviation of the data. It is necessary to remove the scale dependence of the data and dividing by the standard deviation of the data provides a scale to use. Thus the resulting "index of accuracy" is insensitive to the dynamic range of x( t). As just described, if one wanted to use a neural net to continuously predict x(t) values at, say, 6 time steps past the last observed value (i.e. wanted to construct a net predicting x( t + 6)) then one would train one network, at P = 6, to do this. If one wanted to always predict 12 time steps past the last observed x( t) then a separate, P = 12, net would have to be trained. We, in fact, trained separate networks for P ranging between 6 and 100 in steps of 6. The index of accuracy for these networks (as obtained by computing the index of accuracy in the prediction phase) is plotted as curve D in Figure 3. There is however an alternate way to predict. If one wished to predict, say, x(t + 12) using a P = 6 net, then one can iterate the P = 6 net. That is, one uses the P = 6 net to predict the x(t +6) values, and then feeds x(t +6) back into the input line to predict x(t + 12) using the predicted x(t + 6) value instead of the observed x(t + 6) value. in fact, one can't use the observed x(t +6) value, because it hasn't been observed yet - the rule of the game is to use only data occurring at time t and before, to predict x( t + 12). This procedure corresponds to iterating the map given by Eqn. (9) to perform prediction at multiples of P. Of course, the delays, ~, must be chosen commensurate with P. This iterative method of prediction has potential dangers. Because (in our example of iterating the P = 6 map) the predicted x(t + 6) is always made with some error, then this error is compounded in iteration, because predicted, and not observed values, are used on the input lines. However, one may predict more accurately for smaller P, so it may be the case that choosing a very accurate small P prediction, and iterating, can ultimately achieve higher accuracy at the larger P's of interest. This tUrns out to be true, and the iterated net method is plotted as curve E in Figure 3. It is the best procedure to use. Curves A,B,C are alternative methods (iterated polynomial, Widrow-Hoff, and non-iterated polynomial respectively. More information on these conventional methods is in (Lapedes and Farber, 1987) ). A C 1 1 I, /' ~ , : .8 !I: " :J I / \f I / : .' . , I / I I I .. ,,: I .6 I I ~ I I ~ I ~ I = I I I .4 , I , I , .2 I o o 4. Why It Works B D P1-~~ictlon ~~. P (T.U3~ 30) Figure 3. 449 E 400 Consider writing out explicitly Eqn. (9) for a two hidden layer network where the output is assumed to be a linear neuron. We consider Input connects to Hidden Layer 1, Hidden Layer 1 to Hidden Layer 2, and Hidden Layer 2 to Output, Therefore: Recall that the output neurons a linear computing element so that only two gOs occur in formula (11), due to the two nonlinear hidden layers. For ease in later analysis, let us rewrite this formula as where Ot = L TtJcg (SU Mle + Ole) + Ot IetH 2 (12a) (12b) 450 The T's and (Ps are specific numbers specified by the training algorithm, so that after training is finished one has a relatively complicated formula (12a, 12b) that expresses the Output value as a specific, known, function of the Input values: Ot == 1(117 12," .lm). A functional relation of this form, when there is only one output, may be viewed as surface in m + 1 dimensional space, in exactly the same manner one interprets the formula z == f(x,y) as a two dimensional surface in three ' dimensional space. The general structure of fO as determined by Eqn. (12a, 12b) is in fact quite simple. From Eqn. (12b) we see that one first forms a sum of gO functions (where gO is s sigmoidal function) and then from Eqn. (12a) one (orms yet another sum involving gO functions. It may at first be thought that this special, simple form of fO restricts the type of surface that may be represented by Ot = f(Ii)' This initial tl.ought is wrong - the special form of Eqn. (12) is actually a general representation for quite arbitrary surfaces. To prove that Eqn. (12) is a reasonable representation for surfaces we first point out that surfaces may be approximated by adding up a series of "bumps" that are appropriately placed. An example of this occurs in familiar Fourier analysis, where wave trains of suitable frequency and amplitude are added together to approximate curves (or surfaces). Each half period of each wave of fixed wavelength is a "bump," and one adds all the bumps together to form the approximant. Let us noW see how Eqn. (12) may be interpreted as adding together bumps of specified heights and positions. First consider SUMk which is a sum of g( ) functions. In Figure (4) we plot an example of such a gO function for the case of two inputs. Figure 4. A sigmoidal surface. 451 The orientation of this sigmoidal surface is determined by T sit the position by 8;'1 and height by T"'i. Now consider another gO function that occurs in SUM",. The 8;, of the second gO function is chosen to displace it from the first, the Tii is chosen so that it has the same orientation as the first, and T "'i is chosen to have opposite sign to the first. These two g( ) functions occur in SUM"" and so to determine their contribution to SUM", we sum them together and plot the result in Fi ure 5. The result is a ridged surface. Figure 5. A ridge. Since our goal is to obtain localized bumps we select another pair of gO functions in SUMk, add them together to get a ridged surface perpendicular to the first ridged surface, and then add the two perpendicular ridged surfaces together to see the contribution to SUMk. The result is plotted in Figure (6). Figure 6. A pseudo-bump . 452 We see that this almost worked, in so much as one obtains a local maxima by this procedure. However there are also saddle-like configurations at the corners which corrupt the bump we were trying to obtain. Note that one way to fix this is to take g(SUMk + Ok) which will, if Ole is chosen appropriately, depress the local minima and saddles to zero while simultaneously sending the central maximum towards 1. The result is plotted in Figure (7) and is the sought after b~~ ____________________________________________ ___ Figure 7. A bump. Furthermore, note that the necessary gO function is supplied by Eqn. (12). Therefore Eqn. (12) is a procedure to obtain localized bumps of arbitrary height and position. For two inputs, the kth bump is obtained by using four gO functions from SUMk (two gO functions for each ridged surface and two ridged surfaces per bump) and then taking gO of the result in Eqn. (12a). The height of the kth bump is determined by T tJe in Eqn. (12a) and the k bumps are added together by that equation as well. The general network architecture which corresponds to the above procedure of adding two gO functions together to form a ridge, two perpendicular ridges together to form a pseudo-bump, and the final gO to form the final bump is represented in Figure (8). To obtain any number ot bumps one adds more neurons to the hidden layers by repeatedly using the connectivity of Figure (8) as a template (Le. four neurons per bump in Hidden Layer 1, and one neuron per bump in HiClden Layer 2). 453 Figure 8. Connectivity needed to obtain one bump. Add four more neurons to Hidden layer 1, and one more neuron to Hidden Layer 2, for each additional bump. One never needs more than two layers, or any other type of connectivity than that already schematically specified by Figure (8). The accuracy of the approximation depends on the number of bumps, whIch in turn is specified, by the number of neurons per layer. This result is easily generalized to higher dimensions (more than two Inputs) where one needs 2m hiddens in the first hidden layer, and one hidden neuron in the second layer for each bump. The argument given above also extends to the situation where one is pro-cessing symbolic information with a neural net. In this situation, the Input information is coded into bits (say Os and Is) and similarly for the Output. Or, the Inputs may still be real valued numbers, in which case the binary output is attempting to group the real valued Inputs into separate classes. To make the Output values tend toward 0 and lone takes a third and final gO on the output layer, i.e. each output neuron is represented by g(Ot) where Ot is given in Eqn. (11) . Recall that up until now we have used hnear neurons on the output layer. In typical backpropagation examples, one never actually achieves a hard 0 or 1 on the output layers but achieves instead some value between 0.0 and 1.0. Then typically any value over 0.5 is called 1, and values under 0.5 are called O. This "postprocessing" step is not really outside the framework of the network formalism, because it may be performed by merely increasing the slope of the sigmoidal function on the Output layer. Therefore the only effect of the third and final gO function used on the Output layer in symbolic information processing is to pass a hyperplane through the surface we have just been discussing. This plane cuts the surface, forming "decision regions," in which high values are called 1 and low values are called O. Thus we see that the heart of the problem is to be able to form surfaces in a general manner, which is then cut by a hyperplane into general decision regions. We are therefore able to conclude that the network architecture consisting of just two hidden layers is sufficient for learning any symbol processing training set. For Boolean symbol mappings one need not use the second hidden layer to remove the saddles on the bump (c.f. Fig. 6). The saddles are lower than the central maximum so one may choose a threshold on the output layer to cut the bump at a point over the saddles to yield the correct decision region. Whether this representation is a reasonable one for subsequently achieving good prediction on a prediction set, as opposed to "memorizing" a training set, is an issue that we address below. 454 We also note that use of Sigma IIi units (Rummelhart, 1986) or high order correlation nets (Y.-C. Lee, 1987) is an attempt to construct a surface by a general polynomial expansion, which is then cut by a hyperplane into decision regions, as in the above. Therefore the essential element of all these neural net learning algorithms are identical (Le. surface construction), only the particular method of parameterizing the surface varies from one algorithm to another. This geometrical viewpoint, which provides a unifying framework for many neural net algorithms, may provide a useful framework in which to attempt construction of new algorithms. Adding together bumps to approximate surfaces is a reasonable procedure to use when dealing with real valued inputs. It ties in to general approximation theory (c.f. Fourier series, or better yet, B splines), and can be quite successful as we have seen. Clearly some economy is gained by giving the neural net bumps to start with, instead of having the neural net form its own bumps from sigmoids. One way to do this would be to use multidimensional Gaussian functions with adjustable parameters. The situation is somewhat different when processing symbolic (binary valued) data. When input symbols are encoded into N bit bit-strings then one has well defined input values in an N dimensional input space. As shown above, one can learn the training set of input patterns by appropriately forming and placing bump surfaces over this space. This is an effective method for memorizing the training set, but a very poor method for obtaining correct predictions on new input data. The point is that, in contrast to real valued inputs that come from, say, a chaotic time series, the input points in symbolic processing problems are widely separated and the bumps do not add together to form smooth surfaces. Furthermore, each input bit string is a corner of an 2N vertex hypercube, and there is no sense in which one corner of a hypercube is surrounded by the other corners. Thus the commonly used input representation for symbolic processing problems requires that the neural net extrapolate the surface to make a new prediction for a new input pattern (i.e. new corner of the hypercube) and not interpolate, as is commonly the case for real valued inputs. Extrapolation is a farmore dangerous procedure than interpolation, and in view of the separated bumps of the training set one might expect on the basis of this argument that neural nets would fail dismally at symbol processing. This is not the case. The solution to this apparent conundrum, of course, is that although it is sufficient for a neural net to learn a symbol processing training set by forming bumps it is not necessary for it to operate in this manner. The simplest example of this occurs in the XOR problem. One can implement the input/output mapping for this problem by duplicating the hidden layer architecture of Figure (8) appropiately for two bumps ( i.e. 8 hid dens in layer 1, 2 hid dens in layer 2). As discussed above, for Boolean mappings, one can even eliminate the second hidden layer. However the architecture of Figure (9) will also suffice. OUTPUT Figure 9. Connectivity for XOR HIDDEN INPUT 455 Plotting the output of this network, Figure(9), as a function of the two inputs yields a ridge orientated to run between (0,1) and (1,0) Figure(lO). Thus a neural net may learn a symbolic training set without using bumps, and a high dimensional version of this process takes place in more complex symbol processing tasks.Ridge/ravine representations of the training data are considerably more efficient than bumps (less hidden neurons and weights) and the extended nature of the surface allows reasonable predictions i.e. extrapolations. 5. Conclusion. Figure 10 XOR surface (1, 1) Neural nets, in contrast to popular misconception, are capable of quite accurate number crunching, with an accuracy for the prediction problem we considered that exceeds conventional methods by orders of magnitude. Neural nets work by constructing surfaces in a high dimensional space, and their operation when performing signal processing tasks on real valued inputs, is closely related to standard methods of functional ,,-pproximation. One does not need more than two hidden layers for processing real valued input data, and the accuracy of the approximation is controlled by the number of neurons per layer, and not the number of layers. We emphasize that although two layers of hidden neurons are sufficient they may not be efficient. Multilayer architectures may provide very efficient networks (in the sense of number of neurons and number of weights) that can perform accurately and with minimal cost. Effective prediction for symbolic input data is achieved by a slightly different method than that used for real value inputs. Instead of forming localized bumps (which would accurately represent the training data but would not predict well on new inputs) the network can use ridge/ravine like surfaces (and generalizations thereof) to efficiently represent the scattered input data. While neural nets generally perform prediction by interpolation for real valued data, they must perform extrapolation for symbolic data if the usual bit representations are used. An outstanding problem is why do tanh representations seem to extrapolate well in symbol processing problema? How do other functional bases do? How does the representation for symbolic inputs affect the ability to extra~ olate? This geometrical viewpoint provides a unifyimt framework for examimr: 456 many neural net algorithms, for suggesting questions about neural net operation, and for relating current neural net approaches to conventional methods. Acknowledgment. We thank Y. C. Lee, J. D. Farmer, and J. Sidorovich for a number of valuable discussions. References C. Barnes, C. Burks, R. Farber, A. Lapedes, K. Sirotkin, "Pattern Recognition by Neural Nets in Genetic Databases", manuscript in preparation J. Denker et. al.," Automatic Learning, Rule Extraction,and Generalization", ATT, Bell Laboratories preprint, 1987 D. Farmer, J.Sidorowich, Phys.Rev. Lett., 59(8), p. 845,1987 H. Haken, Phys. Lett. A53, p77 (1975) A. Lapedes, R. Farber "Nonlinear Signal Processing Using Neural Networks: Prediction and System Modelling", LA-UR87-2662,1987 Y.C. Lee, Physica 22D,(1986) R. Lippman, IEEE ASAP magazine,p.4, 1987 D. Ruelle, F. Takens, Comm. Math. Phys. 20, p167 (1971) D. Rummelhart, J. McClelland in "Parallel Distributed Processing" Vol. 1, M.I.T. Press Cambridge, MA (1986) D. Russel et al., Phys. Rev. Lett. 45, pU75 (1980) T. Sejnowski et al., "Net Talk: A Parallel Network that Learns to Read Aloud," Johns Hopkins Univ. preprint (1986) H. Swinney et al., Physics Today 31 (8), p41 (1978) F. Takens, "Detecting Strange Attractor in Turbulence," Lecture Notes in Mathematics, D. Rand, L. Young (editors), Springer Berlin, p366 (1981) K. Tomita et aI., J. Stat. Phys. 21, p65 (1979)
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Neural Network Implementation Approaches for the Connection Machine Nathan H. Brown, Jr. MRJlPerkin Elmer, 10467 White Granite Dr. (Suite 304), Oakton, Va. 22124 ABSlRACf 127 The SIMD parallelism of the Connection Machine (eM) allows the construction of neural network simulations by the use of simple data and control structures. Two approaches are described which allow parallel computation of a model's nonlinear functions, parallel modification of a model's weights, and parallel propagation of a model's activation and error. Each approach also allows a model's interconnect structure to be physically dynamic. A Hopfield model is implemented with each approach at six sizes over the same number of CM processors to provide a performance comparison. INTRODUCflON Simulations of neural network models on digital computers perform various computations by applying linear or nonlinear functions, defined in a program, to weighted sums of integer or real numbers retrieved and stored by array reference. The numerical values are model dependent parameters like time averaged spiking frequency (activation), synaptic efficacy (weight), the error in error back propagation models, and computational temperature in thermodynamic models. The interconnect structure of a particular model is implied by indexing relationships between arrays defined in a program. On the Connection Machine (CM), these relationships are expressed in hardware processors interconnected by a 16-dimensional hypercube communication network. Mappings are constructed to defme higher dimensional interconnectivity between processors on top of the fundamental geometry of the communication network. Parallel transfers are defined over these mappings. These mappings may be dynamic. CM parallel operations transform array indexing from a temporal succession of references to memory to a single temporal reference to spatially distributed processors. Two alternative approaches to implementing neural network simulations on the CM are described. Both approaches use "data parallelism" 1 provided by the *Lisp virtual machine. Data and control structures associated with each approach and performance data for a Hopfield model implemented with each approach are presented. DATA STRUCTURES The functional components of a neural network model implemented in *Lisp are stored in a uniform parallel variable (pvar) data structure on the CM. The data structure may be viewed as columns of pvars. Columns are given to all CM virtual processors. Each CM physical processor may support 16 virtual processors. In the fust approach described, CM processors are used to represent the edge set of a models graph structure. In the second approach described, each processor can represent a unit, an outgoing link, or an incoming link in a model's structure. Movement of activation (or error) through a model's interconnect structure is simulated by moving numeric values © American Institute of Physics 1988 128 over the eM's hypercube. Many such movements can result from the execution of a single CM macroinstruction. The CM transparently handles message buffering and collision resolution. However, some care is required on the part of the user to insure that message traffic is distributed over enough processors so that messages don't stack up at certain processors, forcing the CM to sequentially handle large numbers of buffered messages. Each approach requires serial transfers of model parameters and states over the communication channel between the host and the CM at certain times in a simulation. The first approach, "the edge list approach," distributes the edge list of a network graph to the eM, one edge per CM processor. Interconnect weights for each edge are stored in the memory of the processors. An array on the host machine stores the current activation for all units. This approach may be considered to represent abstract synapses on the eM. The interconnect structure of a model is described by product sets on an ordered pair of identification (id) numbers, rid and sid. The rid is the id of units receiving activation and sid the id of units sending activation. Each id is a unique integer. In a hierarchical network, the ids of input units are never in the set of rids and the ids of output units are never in the set of sids. Various set relations (e.g. inverse, reflexive, symmetric, etc.) defined over id ranges can be used as a high level representation of a network's interconnect structure. These relations can be translated into pvar columns. The limits to the interconnect complexity of a simulated model are the virtual processor memory limits of the CM configuration used and the stack space ~uired by functions used to compute the weighted sums of activation. Fig. 1 shows a R -> R2 -> R4 interconnect structure and its edge list representation on the CM. 6 7 8 9 :z 3 eM PROCESSOR 0 1 2 3 4 5 6 7 8 9 1 0111213 :~ (~·,';)H f H H H f ff if SAcr ( 8 j ) 1 2 3 1 2 3 4 5 4 5 4 5 4 5 Fig. 1. Edge List Representation of a R3_> R2 -> R4 Interconnect Structure This representation can use as few as six pvars for a model with Hebbian adaptation: rid (i), sid (j), interconnect weight (wij), ract (ai), sact (aj), and learn rate (11)· Error back propagation requires the addition of: error (ei), old interconnect weight (wij(t-l», and the momentum term (ex). The receiver and sender unit identification pvars are described above. The interconnect weight pvar stores the weight for the interconnect. The activation pvar, sact, stores the current activation, aj' transfered to the unit specified by rid from the unit specified by sid. The activation pvar, ract, stores the current weighted activation ajwij- The error pvar stores the error for the unit specified by the sid. A variety of proclaims (e.g. integer, floating point, boolean, and field) exist in *Lisp to define the type and size ofpvars. Proclaims conserve memory and speed up execution. Using a small number of pvars limits the 129 amount of memory used in each CM processor so that maximum virtualization of the hardware processors can be realized. Any neural model can be specified in this fashion. Sigma-pi models require multiple input activation pvars be specified. Some edges may have a different number of input activation pvars than others. To maintain the uniform data structure of this approach a tag pvar has to be used to determine which input activation pvars are in use on a particular edge. The edge list approach allows the structure of a simulated model to "physically" change because edges may be added (up to the virtual processor limit), or deleted at any time without affecting the operation of the control structure. Edges may also be placed in any processor because the subselection (on rid or sid) operation performed before a particular update operation insures that all processors (edges) with the desired units are selected for the update. The second simulation approach, "the composite approach," uses a more complicated data structure where units, incoming links, and outgoing links are represented. Update routines for this approach use parallel segmented scans to form the weighted sum of input activation. Parallel segmented scans allow a MIMD like computation of the weighted sums for many units at once. Pvar columns have unique values for unit, incoming link, and outgoing link representations. The data structures for input units, hidden units, and output units are composed of sets of the three pvar column types. Fig. 2 shows the representation for the same model as in Fig. 1 implemented with the composite approach. 2 3 4 5 6 7 8 9 o 1 2 3 4 5 6 7 8 9 101112 1314151617181920212223242526272829303132333435 c o rr, f~ ~\'~~Ii~ +----+ ~~+-t+*~ IO~ O.~~ II I( II I( loll ~ t -~. ~ c--•. c--•. I Fig. 2. Composite Representation of a R3 -> R2 -> R4 Interconnect Structure In Fig. 2, CM processors acting as units, outgoing links, and incoming links are represented respectively by circles, triangles, and squares. CM cube address pointers used to direct the parallel transfer of activation are shown by arrows below the structure. These pointers defme the model interconnect mapping. Multiple sets of these pointers may be stored in seperate pvars. Segmented scans are represented by operation-arrow icons above the structure. A basic composite approach pvar set for a model with Hebbian adaptation is: forward B, forward A, forward transfer address, interconnect weight (Wij), act-l (ai), act-2 (aj), threshold, learn rate (Tl), current unit id (i), attached unit id U), level, and column type. Back progagation of error requires the addition of: backward B, backward A, backward transfer address, error (ei), previous interconnect weight (Wij(t-l», and the momentum tenn (ex). The forward and backward boolean pvars control the segmented scanning operations over unit constructs. Pvar A of each type controls the plus scanning and pvar B of each type controls the copy scanning. The forward transfer pvar stores cube addresses for 130 forward (ascending cube address) parallel transfer of activation. The backward transfer pvar stores cube addresses for backward (descending cube address) parallel transfer of error. The interconnect weight, activation, and error pvars have the same functions as in the edge list approach. The current unit id stores the current unit's id number. The attached unit id stores the id number of an attached unit. This is the edge list of the network's graph. The contents of these pvars only have meaning in link pvar columns. The level pvar stores the level of a unit in a hierarchical network. The type pvar stores a unique arbitrary tag for the pvar column type. These last three pvars are used to subselect processor ranges to reduce the number of processors involved in an operation. Again, edges and units can be added or deleted. Processor memories for deleted units are zeroed out. A new structure can be placed in any unused processors. The level, column type, current unit id, and attached unit id values must be consistent with the desired model interconnectivity. The number of CM virtual processors required to represent a given model on the CM differs for each approach. Given N units and N(N-1) non-zero interconnects (e.g. a symmetric model), the edge list approach simply distributes N(N-1) edges to N(N-1) CM virtual processors. The composite approach requires two virtual processors for each interconnect and one virtual processor for each unit or N +2 N (N -1) CM virtual processors total. The difference between the number of processors required by the two approaches is N2. Table I shows the processor and CM virtualization requirements for each approach over a range of model sizes. TABLE I Model Sizes and CM Processors Required Run No. Grid Size Number of Units Edge List Quart CM Virt. Procs. Virt. LeveL N(N-1) 1 82 64 4032 8192 0 2 92 81 6480 8192 0 3 112 121 14520 16384 0 4 132 169 28392 32768 2 5 162 256 65280 65536 4 6 192 361 129960 131072 8 Run No. Grid Size Number of Units Composite Quart CM Virt. Procs. Virt. LeveL N+2N(N-1) 7 82 64 8128 8192 0 8 92 81 13041 16384 0 9 112 121 29161 32768 2 10 132 169 56953 65536 4 11 162 256 130816 131072 8 12 192 361 260281 262144 16 131 CONTROL STRUCTURES The control code for neural network simulations (in *Lisp or C*) is stored and executed sequentially on a host computer (e.g. Symbolics 36xx and V AX 86xx) connected to the CM by a high speed communication line. Neural network simulations executed in *Lisp use a small subset of the total instruction set: processor selection reset (*all), processor selection (*when), parallel content assignment (*set), global summation (*sum), parallel multiplication (*!! ), parallel summation (+! I), parallel exponentiation (exp! I), the parallel global memory references (*pset) and (pref! I), and the parallel segmented scans (copy!! and +!!). Selecting CM processors puts them in a "list of active processors" (loap) where their contents may be arithmetically manipulated in parallel. Copies of the list of active processors may be made and used at any time. A subset of the processors in the loap may be "subselected" at any time, reducing the loap contents. The processor selection reset clears the current selected set by setting all processors as selected. Parallel content assignment allows pvars in the currently selected processor set to be assinged allowed values in one step. Global summation executes a tree reduction sum across the CM processors by grid or cube address for particular pvars. Parallel multiplications and additions multiply and add pvars for all selected CM processors in one step. The parallel exponential applies the function, eX, to the contents of a specified pvar, x, over all selected processors. Parallel segmented scans apply two functions, copy!! and +!!, to subsets ofCM processors by scanning across grid or cube addresses. Scanning may be forward or backward (Le. by ascending or descending cube address order, respectively). Figs. 3 and 4 show the edge list approach kernels required for Hebbian learning for a R2 -> R2 model. The loop construct in Fig. 3 drives the activation update (1) operation. The usual loop to compute each weighted sum for a particular unit has been replaced by four parallel operations: a selection reset (*all), a subselection of all the processors for which the particular unit is a receiver of activation (*when (=!! rid (!! (1+ u»», a parallel multiplication (*!! weight sact), and a tree reduction sum (*sum ... ). Activation is spread for a particular unit, to all others it is connected to, by: storing the newly computed activation in an array on the host, then subselecting the processors where the particular unit is a sender of activation (*when (=!! sid (!! (1 + u»», and broadcasting the array value on the host to those processors. (dotimes (u 4) (*all (*when (=!! rid (!! (1+ u») (setf (aref activation u) (some-nonlinearity (*sum (*!! weight sact»» (*set ract (!! (aref activation u») (*all (*when (=!! sid (!! (1+ u») (*set sact (!! (aref activation u»»» Fig. 3. Activation Update Kernel for the Edge Lst Approach. Fig. 4 shows the Hebbian weight update kernel 132 (2) (*all (*set weight (*!! learn-rate ract sact»» Fig. 4. Hebbian Weight Modification Kernel for the Edge List Approach The edge list activation update kernel is essentially serial because the steps involved can only be applied to one unit at a time. The weight modification is parallel. For error back propagation a seperate loop for computing the errors for the units on each layer of a model is required. Activation update and error back propagation also require transfers to and from arrays on the host on every iteration step incurring a concomitant overhead. Other common computations used for neural networks can be computed in parallel using the edge list approach. Fig. 5 shows the code kernel for parallel computation of Lyapunov engergy equations (3) where i= 1 to number of units (N). (+ (* -.5 (*sum (*!! weight ract sact») (*sum (*!! input sact») Fig. 5. Kernel for Computation of the Lyapunov Energy Equation Although an input pvar, input, is defined for all edges, it is only non-zero for those edges associated with input units. Fig. 6 shows the pvar structure for parallel computation of a Hopfield weight prescription, with segmented scanning to produce the weights in one step, W· · -l:S I(2ar·-I)(2ar·-I) IJ r= 1 J (4) where wii=O, Wij=Wjh and r=I to the number of patterns, S, to be stored. seg t n n t n n ract vII V21 ... VSI vII V2I ... VSI .. . sact V I2 v22' .. VS2 v13 v23 ... VS3 .. . weight wI2 w13 Fig. 6. Pvar Structure for Parallel Computation QfHopfield Weight Prescription Fig. 7 shows the *Lisp kernel used on the pvar structure in Fig. 6. (set weight (scan '+!! (*!! (-!! (*!! ract (!! 2» (!! 1» (-!! (*!! sact (!! 2» (!! 1»» :segment-pvar seg :inc1ude-self t) Fig. 7. Parallel Computation of Hopfield Weight Prescription 133 The inefficiencies of the edge list activation update are solved by the updating method used in the composite approach. Fig. 8 shows the *Lisp kernel for activation update using the composite approach. Fig. 9 shows the *Lisp kernel for the Hebbian learning operation in the composite approach. (*a1l (*when (=!! level (!! 1» (*set act (scan!! act-I 'copy!! :segment-pvar forwardb :include-self t» (*set act (*!! act-l weight» (*when (=!! type (!! 2» (*pset :overwrite act-l act-I ftransfer») (*when (=!! level (!! 2» (*all (*set act (scan!! act-l '+!! :segment-pvar forwarda :include-self t» (*when (=!! type (!! 1» (some-nonlinearity!! act-I»» Fig. 8. Activation Update Kernel for the Composite Approach (*set act-l (scan!! act-I 'copy!! :segment-pvar forwardb :include-self t» (*when (=!! type (!! 2» (*set act-2 (pref!! act-I btransfer») (*set weight (+!! weight (*!! learn-rate act-l act-2»») Fig. 9. Hebbian Weight Update Kernel for the Composite Approach It is immediately obvious that no looping is invloved. Any number of interconnects may be updated by the proper subselection. However, the more subselection is used the less efficient the computation becomes because less processors are invloved. COMPLEXITY ANALYSIS The performance results presented in the next section can be largely anticipated from an analysis of the space and time requirements of the CM implementation approaches. For simplicity I use a Rn -> Rn model with Hebbian adaptation. The oder of magnitude requirements for activation and weight updating are compared for both CM implementation approaches and a basic serial matrix arithmetic approach. F~r the given model the space requirements on a conventional serial machine are 2n+n locations or O(n2). The growth of the space requirement is dominated by the nxn weight matrix. defining the system interconnect structure. The edge list appro~ch uses six pvars for each processor and uses nxn processors for the mapping, or 6n locations or O(n2). The composite approach uses 11 pvars. There are 2n processors for units and 2n2 proces~ors for interconnects in the given model. The composite approach uses 11(2n+2n ) locations or O(n2). The CM implementations take up roughly the same space as the serial implementation, but the space for the serial implementation is composed of passive memory whereas the space for the CM implementations is composed of interconnected processors with memory . The time analysis for the approaches compares the time order of magnitudes to compute the activation update (1) and the Hebbian weight update (2). On a serial 134 machine, the n weighted sums computed for the ac~vation update require n2 multiplicationsffd n(n-l) additions. There are 2n -n operations or time order of magnitude O(n ~ The time order of magnitude for the weight matrix update is O(n2) since there are n weight matrix elements. The edge list approach forms n weighted sums by performing a parallel product of all of the weights and activations in the model, (*!! weight sact), and then a tree reduction sum, (*sum ... ), of the products for the n uni~ (see Fig. 4). There are 1 +n(nlog2n) operations or time order of magnitude O(n ). This is the same order of magnitude as obtained on a serial machine. Further, the performance of the activation update is a function of the number of interconnects to be processed. The composite approach forms n weighted sums in nine steps (see Fig. 8): five .selection operations; the segmented copy scan before the parallel multiplication; the parallel multiplication; the parallel transfer of the products; and the segmented plus scan, which forms the n sums in one step. This gives the composite activation update a time order of magnitude O( 1). Performance is independent of the number of interconnects processed. The next section shows that this is not quite true. The n2 weights in the model can be updated in three parallel steps using the edge list approach (see Fig. 4). The n2 weights in the model can be updated in eight parallel steps using the composite approach (see Fig. 9). In either case, the weight update operation has a time order of magnitude 0(1). The time complexity results obtained for the composite approach apply to computation of the Lyaponov energy equation (3) and the Hopfield weighting prescription (4), given that pvar structures which can be scanned (see Figs. 1 and 6) are used. The same operations performed serially are time order of magnitude 0(n2). The above operations all incur a one time overhead cost for generating the addresses in the pointer pvars, used for parallel transfers, and arranging the values in segments for scanning. What the above analysis shows is that time complexity is traded for space complexity. The goal of CM programming is to use as many processors as possible at every step. PERFORMANCE COMPARISON Simulations of a Hopfield spin-glass model2 were run for six different model sizes over the same number (16,384) of physical CM processors to provide a performance comparison between implementation approaches. The Hopfield network was chosen for the performance comparison because of its simple and well known convergence dynamics and because it uses a small set of pvars which allows a wide range of network sizes (degrees of virtualization) to be run. Twelve treaments are run. Six with the edge list approach and six with the composite approach. Table 3-1 shows the model sizes run for each treatment. Each treatment was run at the virtualization level just necessary to accomodate the number of processors required for each simulation. Two exemplar patterns are stored. Five test patterns are matched against the two exemplars. Two test patterns have their centers removed, two have a row and column removed, and one is a random pattern. Each exemplar was hand picked and tested to insure that it did not produce cross-talk. The number of rows and columns in the exemplars and patterns increase as the size of the networks for the treatments increases. 135 Since the performance of the CM is at issue, rather than the performance of the network model used, a simple model and a simple pattern set were chosen to minimize consideration of the influence of model dynamics on performance. Performance is presented by plotting execution speed versus model size. Size is measured by the number of interconnects in a model. The execution speed metric is interconnects updated per second, N*(N-l )/t, where N is the number of units in a model and t is the time used to update the activations for all of the units in a model. All of the units were updated three times for each pattern. Convergence was determined by the output activation remaining stable over the fmal two updates. The value of t for a treatment is the average of 15 samples of t. Fig. 10 shows the activation update cycle time for both approaches. Fig. 11 shows the interconnect update speed plots for both approaches. The edge list approach is plotted in black. The composite approach is plotted in white. The performance shown excludes overhead for interpretation of the *Lisp instructions. The model size categories for each plot correspond to the model sizes and levels of eM virtualization shown in Table I. i.p.s. Activation Update Cycle Time vs Model Size 1 .6 1.4 1.2 sees O.B 0.6 0.4 0.2 • • • • o OO ___ ~~~ __ ~O~ __ ~O~ __ .O __ ~ 1 2 3 4 5 6 Model Size Fig. 10. Activation Update Cycle Times Interconnect Update Speed Comparison Edge Ust Approach vs. Composite Approach 2000000} 1500000 0 0 0 1000000 0 0 500000t 0 • • • • o· • 1 2 3 4 5 6 Model Size Fig. 11. Edge List Interconnect Update Speeds Fig. 11 shows an order of magnitude performance difference between the approaches and a roll off in performance for each approach as a function of the number of virtual processors supported by each physical processor. The performance tum around is at 4x virtualization for the edge list approach and 2x virtualization for the composite approach. 136 CONCLUSIONS Representing the interconnect structure of neural network models with mappings defined over the set of fine grain processors provided by the CM architecture provides good performance for a modest programming effort utilizing only a small subset of the instructions provided by *Lisp. Further, the perfonnance will continue to scale up linearly as long as not more than 2x virtualization is required. While the complexity analysis of the composite activation update suggests that its performance should be independent of the number of interconnects to be processed, the perfonnance results show that the performance is indirectly dependent on the number of interconnects to be processed because the level of virtualization required (after the physical processors are exhausted) is dependent on the number of interconnects to be processed and virtualization decreases performance linearly. The complexity analysis of the edge list activation update shows that its perfonnance should be roughly the same as serial implementations on comparable machines. The results suggest that the composite approach is to be prefered over the edge list approach but not be used at a virtualization level higher than 2x. The mechanism of the composite activation update suggest that hierarchical networks simulated in this fashion will compare in perfonnance to single layer networks because the parallel transfers provide a type of pipeline for activation for synchronously updated hierarchical networks while providing simultaneous activation transfers for asynchronously updated single layer networks. Researchers at Thinking Machines Corporation and the M.I.T. AI Laboratory in Cambridge Mass. use a similar approach for an implementation of NETtalk. Their approach overlaps the weights of connected units and simultaneously pipelines activation forward and error backward.3 Perfonnance better than that presented can be gained by translation of the control code from interpreted *Lisp to PARIS and use of the CM2. In addition to not being interpreted, PARIS allows explicit control over important registers that aren't accessable through *Lisp. The CM2 will offer a number of new features which will enhance perfonnance of neural network simulations: a *Lisp compiler, larger processor memory (64K), and floating point processors. The complier and floating point processors will increase execution speeds while the larger processor memories will provide a larger number of virtual processors at the performance tum around points allowing higher perfonnance through higher CM utilization. REFERENCES 1. "Introduction to Data Level Parallelism," Thinking Machines Technical Report 86.14, (April 1986). 2. Hopfield, J. J., "Neural networks and physical systems with emergent collective computational abilities," Proc. Natl. Acad. Sci., Vol. 79, (April 1982), pp. 2554-2558. 3. Blelloch, G. and Rosenberg, C. Network Learning on the Connection Machine, M.I.T. Technical Report, 1987.
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On the Power of Neural Networks for Solving Hard Problems J ehoshua Bruck Joseph W. Goodman Information Systems Laboratory Departmen t of Electrical Engineering Stanford University Stanford, CA 94305 Abstract This paper deals with a neural network model in which each neuron performs a threshold logic function. An important property of the model is that it always converges to a stable state when operating in a serial mode [2,5]. This property is the basis of the potential applications of the model such as associative memory devices and combinatorial optimization [3,6]. One of the motivations for use of the model for solving hard combinatorial problems is the fact that it can be implemented by optical devices and thus operate at a higher speed than conventional electronics. The main theme in this work is to investigate the power of the model for solving NP-hard problems [4,8], and to understand the relation between speed of operation and the size of a neural network. In particular, it will be shown that for any NP-hard problem the existence of a polynomial size network that solves it implies that NP=co-NP. Also, for Traveling Salesman Problem (TSP), even a polynomial size network that gets an €-approximate solution does not exist unless P=NP. The above results are of great practical interest, because right now it is possible to build neural networks which will operate fast but are limited in the number of neurons. 1 Background 137 The neural network model is a discrete time system that can be represented by a weighted and undirected graph. There is a weight attached to each edge of the graph and a threshold value attached to each node (neuron) of the graph. © American Institute of Physics 1988 138 The order of the network is the number of nodes in the corresponding graph. Let N be a neural network of order n; then N is uniquely defined by (W, T) where: • W is an n X n symmetric matrix, Wii is equal to the weight attached to edge (i, j) . • T is a vector of dimension n, Ti denotes the threshold attached to node i. Every node (neuron) can be in one of two possible states, either 1 or -1. The state of node i at time t is denoted by Vi(t). The state of the neural network at time t is the vector V(t). The next state of a node is computed by: Vi(t + 1) = sgn(H,(t)) = { ~1 ~t~;2i~ 0 (1) where n Hi(t) = L WiiVj(t) - Ti i=l The next state of the network, i.e. V(t + 1), is computed from the current state by performing the evaluation (1) at a subset of the nodes of the network, to be denoted by S. The modes of operation are determined by the method by which the set S is selected in each time interval. If the computation is performed at a single node in any time interval, i.e. 1 S 1= 1, then we will say that the network is operating in a serial mode; if 1 S 1= n then we will say that that the network is operating in a fully parallel mode. All the other cases, i.e. 1 <I S 1< n will be called parallel modes of operation. The set S can be chosen at random or according to some deterministic rule. A state V(t) is called stable iff V(t) = sgn(WV(t) - T), i.e. there is no change in the state of the network no matter what the mode of operation is. One of the most important properties of the model is the fact that it always converges to a stable state while operating in a serial mode. The main idea in the proof of the convergence property is to define a so called energy function and to show that this energy function is nondecreasing when the state of the network changes. The energy function is: (2) An important note is that originally the energy function was defined such that it is nonincreasing [5]; we changed it such that it will comply with some known graph problems (e.g. Min Cut). A neural network will always get to a stable state which corresponds to a local maximum in the energy function. This suggests the use of the network as a 139 device for performing a local search algorithm for finding a maximal value of the energy function [6]. Thus, the network will perform a local search by operating in a random and serial mode. It is also known [2,9] that maximization of E associated with a given network N in which T = 0 is equivalent to finding the Minimum Cut in N. Actually, many hard problems can be formulated as maximization of a quadratic form (e.g. TSP [6)) and thus can be mapped to a neural network. . 2 The Main Results The set of stable states is the set of possible final solutions that one will get using the above approach. These final solutions correspond to local maxima of the energy function but do not necessarily correspond to global optima of the corresponding problem. The main question is: suppose we allow the network to operate for a very long time until it converges; can we do better than just getting some local optimum? i.e., is it possible to design a network which will always find the exact solution (or some guaranteed approximation) of the problem? Definition: Let X be an instance of problem. Then 1 X 1 denotes the size of X, that is, the number of bits required to represent X. For example, for X being an instance of TSP, 1 X I is the number of bits needed to represent the matrix of the distances between cities. Definition: Let N be a neural network. Then 1 N 1 denotes the size of the network N. Namely, the number of bits needed to represent Wand T. Let us start by defining the desired setup for using the neural network as a model for solving hard problems. Consider an optimization problem L, we would like to have for every instance X of L a neural network N x with the following properties: • Every local maximum of the energy function associated with N x corresponds to a global optimum of X . • The network N x is small, that is, I N x 1 is bounded by some polynomial in 1 X I. Moreover, we would like to have an algorithm, to be denoted by AL , which given an instance X E L, generates the description for N x in polynomial (in I X I) time. Now, we will define the desired setup for using the neural network as a model for finding approximate solutions for hard problems. Definition: Let Eglo be the global maximum of the energy function. Let Eloc 140 be a local maximum of the energy function. We will say that a local maximum is an f-approximate of the global iff: Eglo Eloc --:;.--- < f Eglo The setup for finding approximate solutions is similar to the one for finding exact solutions. For fo > 0 being some fixed number. We would like to have a network N x~ in which every local maximum is an f-approximate of the global and that the global corresponds to an optimum of X. The network N x€ should be small, namely, 1 N x~ 1 should be bounded by a polynomial in 1 X I. Also, we would like to have an algorithm AL~, such that, given an instance X E L, it generates the description for N x€ in polynomial (in 1 X I) time. Note that in both the exact case and the approximate case we do not put any restriction on the time it takes the network to converge to a solution (it can be exponential) . A t this point the reader should convince himself that the above description is what he imagined as the setup for using the neural network model for solving hard problems, because that is what the following definition is about. Definition: We will say that a neural network for solving (or finding an fapproximation of) a problem L exists if the algorithm AL (or ALJ which generates the description of Nx (or Nx~) exists. The main results in the paper are summarized by the following two propositions. The first one deals with exact solutions of NP-hard problems while the second deals with approximate solutions to TSP. Proposition 1 Let L be an NP-hard problem. Then the existence of a neural network for solving L implies that NP = co-NP. Proposition 2 Let f > 0 be some fixed number. The existence of a neural network for finding an f-approximate solution to TSP implies that P=NP. Both (P=NP) and (NP=co-NP) are believed to be false statements, hence, we can not use the model in the way we imagine. The key observation for proving the above propositions is the fact that a single iteration in a neural network takes time which is bounded by a polynomial in the size of the instance of the corresponding problem. The proofs of the above two propositions follow directly from known results in complexity theory and should not be considered as new results in complexity theory. 141 3 The Proofs Proof of Proposition 1: The proof follows from the definition of the classes NP and co-NP, and Lemma 1. The definitions and the lemma appear in Chapters 15 and 16 in [8] and also in Chapters 2 and 7 in [4]. Lemma 1 If the complement of an NP-complete problem is in NP, then NP=co-NP. Let L be an NP-hard problem. Suppose there exists a neural network that solves L. Let 1 be an NP-complete problem. By definition, 1 can be polynomialy reduced to L. Thus, for every instance X E 1, we have a neural network such that from any of its global maxima we can efficiently recognize whether X is a 'yes' or a 'no' instance of 1. We claim that we have a nondeterministic polynomial time algorithm to decide that a given instance X E 1 is a 'no' instance. Here is how we do it: for X E 1 we construct the neural network that solves it by using the reduction to L. We then check every state of the network to see if it is a local maximum (that is done in polynomial time). In case it is a local maximum, we check if the instance is a 'yes' or a 'no' instance (this is also done in polynomial time). Thus, we have a nondeterministic polynomial time algorithm to recognize any 'no' instance of 1. Thus, the complement of the problem 1 is in NP. But 1 is an NP-complete problem, hence, from Lemma 1 it follows that NP=co-NP. 0 Proof of Proposition 2: The result is a corollary of the results in [7], the reader can refer to it for a more complete presentation. The proof uses the fact that the Restricted Hamiltonian Circuit (RHC) is an NP-complete problem. Definiton of RHC: Given a graph G = (V, E) and a Hamiltonian path in G. The question is whether there is a Hamiltonian circuit in G? It is proven in [7] that RHC is NP-complete. Suppose there exists a polynomial size neural network for finding an f-approximate solution to TSP. Then it can be shown that an instance X E RHC can be reduced to an instance X E TSP, such that in the network Nx£ the following holds: if the Hamiltonian path that is given in X corresponds to a local maximum in N x£ then X is a 'no' instance; else, if it does not correspond to a local maximum in N x£ then X is a 'yes' instance. Note that we can check for locality in polynomial time. Hence, the existence of N xe for all X E TSP implies that we have a polynomial time algorithm for RHC. 0 142 4 Concluding Remarks 1. In Proposition 1 we let I W I and I T I be arbitrary but bounded by a polynomial in the size of a given instance of a problem. If we assume that I W I and I T I are fixed for all instances then a similar result to Proposition 1 can be proved without using complexity theory; this result appears in [1]. 2. The network which corresponds to TSP, as suggested in [6], can not solve the TSP with guaranteed quality. However, one should note that all the analysis in this paper is a worst case type of analysis. So, it might be that there exist networks that have good behavior on the average. 3. Proposition 1 is general to all NP-hard problems while Proposition 2 is specific to TSP. Both propositions hold for any type of networks in which an iteration takes polynomial time. 4. Clearly, every network has an algorithm which is equivalent to it, but an algorithm does not necessarily have a corresponding network. Thus, if we do not know of an algorithmic solution to a problem we also will not be able to find a network which solves the problem. If one believes that the neural network model is a good model (e.g. it is amenable to implementation with optics), one should develop techniques to program the network to perform an algorithm that is known to have some guaranteed good behavior. Acknowledgement: Support of the U.S. Air Force Office of Scientific Research is gratefully acknowledged. References [1] Y. Abu Mostafa, Neural Networks for Computing? in Neural Networks for Computing, edited by J. Denker (AlP Conference Proceedings no. 151, 1986). [2] J. Bruck and J. Sanz, A Study on Neural Networks, IBM Tech Rep, RJ 5403, 1986. To appear in International Journal of Intelligent Systems, 1988. [3] J. Bruck and J. W. Goodman, A Generalized Convergence Theorem for Neural Networks and its Applications in Combinatorial Optimization, IEEE First ICNN, San-Diego, June 1987. [4] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, 1979. 143 [5] J. J. Hopfield, Neural Networks and Physical Systems with Emergent Collective Computational Abilities, Proc. Nat. Acad. Sci .. USA, Vol. 79, pp. 2554-2558, 1982. [6] J. J. Hopfield and D. W. Tank, Neural Computations of Decisions in Optimization Problems, BioI. Cybern. 52, pp. 141-152, 1985. [7] C. H. Papadimitriou and K. Steiglitz, On the Complexity of Local Search for the Traveling Salesman Problem, SIAM J. on Comp., Vol. 6, No.1, pp. 76-83, 1977. [8] C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algo:rithms and Complexity, Prentice-Hall, Inc., 1982. [9] J. C. Picard and H. D. Ratliff, Minimum Cuts and Related Problems, Networks, Vol 5, pp. 357-370, 1974.
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402 HOW THE PROCESSING CATFISH TRACKS ITS PREY: AN INTERACTIVE "PIPELINED" SYSTEM MAY DIRECT FORAGING VIA RETlCULOSPINAL NEURONS. Jagmeet S. Kanwal Dept. of Cellular & Structural Biology, Univ. of Colorado, Sch. of Medicine, 4200 East, Ninth Ave., Denver, CO 80262. ABSTRACT Ictalurid catfish use a highly developed gustatory system to localize, track and acquire food from their aquatic environment. The neural organization of the gustatory system illustrates well the importance of the four fundamental ingredients (representation, architecture, search and knowledge) of an "intelligent" system. In addition, the "pipelined" design of architecture illustrates how a goal-directed system effectively utilizes interactive feedback from its environment. Anatomical analysis of neural networks involved in target-tracking indicated that reticular neurons within the medullary region of the brainstem, mediate connections between the gustatory (sensory) inputs and the motor outputs of the spinal cord. Electrophysiological analysis suggested that these neurons integrate selective spatio-temporal patterns of sensory input transduced through a rapidly adapting-type peripheral filter (responding tonically only to a continuously increasing stimulus concentration). The connectivity and response patterns of reticular cells and the nature of the peripheral taste response suggest a unique "gustation-seeking" fUnction of reticulospinal cells, which may enable a catfish to continuously track a stimulus source once its directionality has been computed. INTRODUCTION Food search is an example of a broad class of behaviors generally classified as goal-directed behaviors. Goal-directed behavior is frequently exhibited by animals, humans and some machines. Although a preprogrammed, hard-wired machine may achieve a particular goal in a relatively short time, the general and heuristic nature of complex goal-directed tasks, however, is best exhibited by animals and best studied in some of the less advanced animal species, such as fishes, where anatomical, electrophysiological and behavioral analyses can be performed relatively accurately and easily. Food search, which may lead to food acquisition and ingestion, is critical for the survival of an organism and, therefore, only highly successful systems are selected during the evolution of a species. The act of food search may be classified into two distinct phases, (i) orientation, and (ii) tracking (navigation and homing). In the channel catfish (the animal model utilized for this study), locomotion (swimming) is primarily controlled by the large forked caudal fin, which also mediates turning and directional swimming. @ American Institute of Physics 1988 403 Both these forms of movement, which constitute the essential movements of target-tracking, involve control of the hypaxial/epiaxial muscles of the flank. The alternate contraction of these muscles causes caudal fin undulations. Each cycle of the caudal fin undulation provides either a symmetrical or an asymmetrical bilateral thrust. The former provides a net thrust forward, along the longitudinal axis of the fish causing it to move ahead, while the latter biases the direction of movement towards the right or left side of the fish. HRP injection site 1 /recording site *A ................................... ........................... ......................... NEUROBIOLOGY ......................................... . fEEDING BEHAVIOR MUSCLE SET MOTOR POOLS PREMOTOR NEURONS GUSTATORY INPUTS ....................................... .............................................................................................................................. food Search 1 Pick Up ,~ Selective Ingestion flank and Tail Fi n Muscles flank Mu:sculature Ja'W Muscles Caudal Reticular facial lobe Spinal Cord > ..... "...... f tl V I orma on > ...... _ ............. . : I : .... u.II .... UIlIl ..... "_ ....................... ,., . .. UIl. U Il ........... _WI ......... 111111 ... ' . Rostral Reticular facial lobe Spinal Cord >.............. f t· I orma Ion > ........ _ ............. . facial and/or > ........................................... _ ... _J.. ... _ ..... __ • Trigeminal Motor Nucleus Oral and Vagal Motor Vagal lobe I ntrl nsic Vagal lobe Pharyngeal Nuclei Musculature I nterneurons Fig. 1. Schematic representation of possible pathways for the gustatory modulation of foraging in the catfish. 404 Ictalurid catfishes possess a well developed gustatory system and use it to locate and acquire food from their aquatic environmentl,2,~ Behavioral evidence also indicates that ictalurid catfishes can detect small intensity (stimulus concentration) differences across their barbels (interbarbel intensity differences), and may use this or other extraoral taste information to compute the directionality in space and track a gustatory stimulus source 1 In other words, based upon the analysis of locomotion, it may be inferred that during food search, the gustatory sense of the catfish influences the duration and degree of asymmetrical or symmetrical undulations of the caudal fin, besides controlling reflex turns of the head and flank. Since directional swimming is ultimately dependent upon movement of the large caudal fin it may be postulated that, if the gustatory system is to coordinate food tracking, gustato-spinal connections exist upto the level of the caudal fin of the catfish (fig. 1). The objectives of this study were (i) to reconsider the functional organization of the gustatory system within the costraints of the four fundamental ingredients (representation, architecture, search and knowledge) of a naturally or artificially "intelligent" agent, (ii) to test the existence of the postulated gustato-spinal connections, and (iii) to delineate as far as possible, using neuroanatomical and electrophysiological techniques, the neural mechanism/s involved in the control of goal-directed (foraging) behavior. ORGANIZATIONAL CONSIDERATIONS I. REPRESENTATION Representation refers to the translation of a particular task into information structures and information processes and determines to a great extent the efficiency and efficacy with which a solution to the task can be generate<i4. The elaborate and highly sensitive taste system of an ictalurid catfish consists of an extensive array of chemo- and mechanosensory receptors distributed over most of the extraoral as well as oral regions of the epithelium2, 5. Peripherally, branches of the facial nerve (which innervates all extraoral taste buds) re~ond to a wide range of stimulus (amino acids) concentrationEP, 7 , l:Si..e. from 10-% to 10-3 M. The taste activity however, adapts rapidly (phasic response) to ongoing stimulation of the same concentration (Fig. 2) and responds tonically only to continuously increasing concentrations of stimuli, such as L-arginine and L-alanine. Fig.:t. Integrated, facial taste recordings to continuous application of amino acids to the palate and nasal barbel showing the phasic nature of the taste responses of the ramus palatinus (rp) and ramus ophthalmicus superficialis (ros), respectively. rp ros 405 Gustatory information from the extraoral and oral epithelium is "pipelined" into two separate subsystems, facial and glossopharyngeal-vagal, respectively. Each sUbsystem processes a subset of the incoming information (extraoral or oral) and coordinates a different component of food acquisition. Food search is accomplished by the extraoral subsystem, while selective ingestion is accomplished by the oral subsystem 2 (Fig. 3). The extraoral gustatory information terminates in the facial lobe where it is represented as a well-defin~d topographic map 9,10 , while the oral information terminates in the adjacent vagal lobe where it is represented as a relatively diffuse map 11. II. ARCHITECTURE The information represented in an information structure eventually requires an operating frame (architecture) within which to select and carry out the various processes. In ictalurid catfish, partially processed information from the primary gustatory centers (facial and vagal lobes) in the medullary region of the brainstem converges along ascending and descending pathways (Fig. 4). One of the centers in the ascending pathways is the secondary gustatory nucleus in the isthmic region which is connected to the corresponding nucleus of the opposite side via a large commissurel2 ,13. Facial and vagal gustatory information crosses over to the opposite side via this commissure thus making it possible for neurons to extract information about interbarbel or interflank intensity differences. Although neurons in this region are known to have large receptive fields14, the exact function of this large commissural nucleus is not yet clearly established. It is quite clear, however, that gustatory information is at first "pipelined" into separate regions where it is processed in parallel15 before converging onto neurons in the ascending (isthmic) and descending (reticular) processors as well as other regions within the medulla. The "pipelined" architecture underscores the need for differential processing of subsets of sensory inputs which are consequently integrated to coordinate temporal transitions between the various components of goal-directed behavior. III. SEARCH An important task underlying all "intelligent" goal directed activity is that of search. In artificial systems this involves application of several general problem-solving methods such as means-end analysis, generate and test methods and heuristic search methods. No attempt, as yet, has been made to fit any of these models to the food-tracking behavior of the catfish. However, behavioral observations suggest that the catfish uses a combinatorial approach resulting in a different yet optimal foraging strategy each time ~ What is interesting about biological models is that the intrinsic search strategy is expressed extrinsically by the behavior of the animal which, with a few precautions, can be observed ~uite easily. In addition, simple manipulations of either the animal or its environment can provide interesting data about the search 406 Fig. 3. ~ ..: . .•. ~I'· t -. , \ __ ",to • __ Fig. 4. SENSORY INPUT Ie xtra I-ora I r--:r a 5 t F ISH BRA IN ~ ac l al BEHAVIORAL OUT PUT (0 r a I VII food searc h e Jrp lobe ~~~~~ and -=-=IX DIe k uO b vagal ) u lobe selectIve d X s IngestIon ~~v~ ©(!l](L~~ [pl~®©~~~®(:l ~!pa~Cil[1. !p[3C!J(!J~~~C!J(3 I 407 strategy/ies being used by the animal, which in turn can highlight some of the computational (neuronal) search strategies adopted by the brain e.g. the catfish seems to minimize the probability of failure by continuously interacting with the environment so as to be able to correct any computational or knowledge-based errors. IV. KNOWLEDGE If an "intelligent" goal-directed system resets to zero knowledge before each search trial, its success would depend entirely upon the information obtained over the time period of a search. Such a system would also require a labile architecture to process the varying sets of information generated during each search. For such a system, the solution space can become very large and given the constraints of time (generally an important cri te:don in biological systems) this can lead to continuous failure. For these reasons, knowledge becomes an important ingredient of an "intelligent" agent since it can keep the search under control. For the gustatory system of the catfish too, randomly accessable knowledge, in combination with the immediately available information about the target, may playa critical role in the adoption of a successful search strategy. Although a significant portion of this knowledge is probably learned, it is not yet clear where and how this knowledge is stored in the catfish brain. The reduction in the solution space for a catfish which has gradually learned to find food in its environment may be attributed to the increase in the amount of knowledge, which to some extent may involve a restructuring of the neural networks during development. EXPERIMENTAL METHODS The methods employed for the present study are only briefly introduced here. Neuroanatomical tracing techniques exploit the phenomenon of axonal transport. Crystals of the enzyme, horseradish peroxidase (HRP) or some other substance, when injected at a small locus in the brain, are taken up by the damaged neurons and transported anterogradely and retrogradely from cell bodies and/or axons at the injection site. In the present study, small superficial injections of HRP (Sigma, Type VI) were made at various loci in the facial lobe (FL) in separate animals. After a survival period of 3 to 5 days, the animals were sacrificed and the brains sectioned and reacted for visualization of the neuronal tracer. In this manner, complex neural circuits can be gradually delineated. Electrophysiological recordings from neurons in the central nervous system were obtained using heat-pulled glass micropipettes. These glass electrodes had a tip diameter of approximately 1 um and an impedance of less than 1 megohm when filled with an electrolyte (3M KCl or 3M Nacl). Chemical stimulation of the receptive fields was accomplished by injection of stimuli (amino acids, amino acid mixtures and liver or bait-extract solutions) into a continuous flow of well-water over the receptive epithelium. Tactile stimulation was performed by gentle strokes of a sable hair brush or a glass probe. 408 EXPERIMENTAL OBSERVATIONS Injections of HRP into the spinal cord labelled two relevant populations of cells, (i) in the ipsilateral reticular formation at the level of the facial lobe (FL), and (ii) a few large scattered cells within the ipsilateral, rostral portion OI the lateral lobule of the FL (Fig. 5). Injection of HRP at several sites within the FL resulted in the identification of a small region in the FL from where anterogradely filled fibers project to the reticular formation (Fig. 5). Superimposition of these injection sites onto the anatomical map OI the extraoral surface of the catfish indicated that this small region, within the facial lobe, corresponds to the snout region of the extraoral surface. FACIO-RETICULAR PROJECTIONS FACIO- & RETICULO -SPINAL PROJECTIONS 1 injection site , 2 3 CB =cerebellum injection site ·r aID 1 SpC LL =lateral line lobe Fig. 5. Schematic chartings showing labelled-cell bodies(squares) and fibers transverse sections through the medulla. 4 (dots) in 409 FL = facial lobe Fl RF = reticular formation SpC= spinal cord VL = vagal lobe \. VL FLANK SNOUT RF Qjg LlP~ ~PC ~---------~ Fig. 6A. WATER SaUIRT -HEAD • GLIDING TOUCH -FLANK + !III1/1 III11I11 J11I11II IIHI~I J~ LIVER EXTRACT -SNOUT + ,. 'I I I I ! II 1-,'1 l UJi! -iJ,l" ILL !~~ I I I I I I , I· J.r '·n 1·1 (I ill'!Pll r~1"r if MIl?l "", i II I L I~ II '//1 I, 1/' "11' I, I I . II II AMINO ACID MIXTURE • (Receptive (Sample unit responses) fields) LI.J 11111,ltll!,llllliltUII.I~I'III~UII AU) Jlldll,IIIl~lIijkml!II,1. I J,[ I !11111,11II1".!L : CONTROL • til.llluI1Inlll.h!LIIIII,hll, 1,1/L . I LI ~ iLL.! L..IJ",1. , 11,1" I Lt J iJ,tt.i AMINO ACID MIXTURE • -SNOUT J ./, ,I lJ ,1,1 \ L [1 .. 1" 1,1,1./ d ,II "I..I1LLLlL/.i.LLU~LUt " ijl,ldqllljl,lJJ\lL,~Lld 1"ltH! I~d II~ 1,,1 LlLlII t LI J TOUCH -SNOUT • " 1 I d II I I ... • . ·.d.·' ,_J. L .. _,I ... ,:!...,J. ..... 1 ... · .... 1, 1 ...... 1 UJ"">1.-I",,JrlHk', ,Llk.-I " .+1.1.4. Jl;!d.t4.l,.lui, t Fig. 6B. 410 Multiunit electrophysiological recordings from various anteroposterior levels of the reticular formation indicated that the snout region (upper lip and proximal portion of the maxillary barbels) of the catfish project to a disproportionately large region of the reticular formation along with a mixed representation of the flank (Fig. 6A). Single unit recordings indicated that some neurons have receptive fields restricted to a bilateral portion of the snout region, while others had large receptive fields extending over the whole flank or over an anteroposterior half of the body (Fig. 6B). DISCUSSION The experimental results obtained here suggest that facial lobe projections to the reticular formation form a functional connection. The reticular neurons project to the spinal cord and, most likely, influence the general cycle of swimming-related activity of motoneurons within the spinal cord 16. The disproportionately large representation of the snout region within the medullary reticular formation, as determined electrophysiologically, is consistent with the anatomical data indicating that most of the fibers projecting to the reticular formatirnl originate from cells in that portion of the facial lobe where the snout region is mapped. The lateral lobule of the spinal cord has a second pathway which projects directly into the spinal cord upto the level of the anterior end of the caudal fin and may coordinate reflexive turning. The significance of the present results is best understood when considered together with previously known information about the anatomy and electrophysiology of the gustatory system. The information presented above is used to propose a model (Fig. 7) for a mechanism that may be involved during the homing phase of target tracking by the catfish. During homing, which refers to the last phase of target-tracking during food search, it may be assumed that the fish is rapidly approaching its target or moving through a steep signal intensity (stimulus concentration) gradient. The data presented above suggest that a neuronal mechanism exists which helps the catfish to lock on to the target during homing. This proposal is based upon the following considerations: 1. Owing to the rapidly adapting response of the peripheral filter, a tonic level of activity in the facial lobe input can occur only when the animal is moving through an increasing concentration gradient of the gustatory stimUlUS. 2. Facial lobe neurons, which receive inputs from the snout region, project to a group of cells in the reticular formation. Activity in the facio-reticular pathway causes a suppression in the spontaneous activity of the reticular neurons. 3. Direct and/or indirect spinal projections from the reticular neurons are involved in the modulation of activity of those spinal motoneurons which coordinate swimming. Thus, it may be hypothesized that during complete suppression of activity in a specific reticulospinal pathway, the fish swims straight ahead, but during excitation of certain reticulospinal neurons the fish dictated by the pattern of activation. Fig. 7. The snout region of the catfish has special significance because of its extensive representation in the reticular formation. In case the fish makes a random or computational error, while approaching its target, the snout is the first region to move out of the stimulus gradient. ~ . : .: 411 as .'. Thus, the spinal motoneurons, teleologically speaking, "seek" a gustatory stimulus in order to suppress activity of certain reticulospinal neurons, which in turn reduce variations in the pattern of activity of swimming-related spinal motoneurons. Accordingly, in a situation where the fish is rapidly approaching ~ target, ie. under the specific conditions of a continuously rising stimulus concentration at the snout region and an absence of a stimulus intensity difference across the barbels, there is a locking of the movement of the body (of the fish) towards the stationary or moving target (food or prey). It should be pointed out, however, that the empirical data available so far, only offers clues to the target-tracking mechanism proposed here. Clearly, more research is needed to validate this proposal and to identify other mechanisms of target-tracking utilized by this biological system. This research was supported in part by NIH Grant NS15258 to T.E. Finger. REFERENCES 1. P. B. Johnsen and J. H. Teeter, J. Compo Physiol. 140,95 (1981). 2. J. Atema, Brain Behav. and Evol. 4, 273-294, (1971). 3. J. E. Bardach, et a1., Science, 155,1276-1278, (1967). 4. A. Newell, Mc-Graw Hill Encyclopedia of Electronics and Computers, (1984), p.71-74. 5. C. J. Herrick, Bull. US. Fish. Comm. 22, 237-272, (1904). 6. J. Caprio, Compo Biochem. Physiol. 52A, 247-251, (1975). 7. C. J. Davenport and J. Caprio, J. Compo Physiol. 147, 217 (1982). 8. J. S. Kanwal and J. Caprio, Brain Res. 406, 105-112, (1987). 9. T. E. Finger, J. Compo Neurol. 165, 513-526 (1976). 10. T. Marui and J. Caprio', Brain Res. 231,185-190 (1982). 11. J. S. Kanwal and J. Caprio, J. Neurobiol. in press, (1988). 12. C. J. Herrick, J. Compo Neurol. 15, 375-456 (1905). 13. C. J. Herrick, J. Compo Neurol. 16, 403-440 (1906). 14. C. F. Lamb and J. Caprio, ISOT, #P70, (1986). 15. T. E. Finger and Y. Morita, Science, 227, 776-778 (1985). 16. P. S. G. Stein, Handbook of the Spinal Cord, (Marcel Dekker Inc., N.Y., 1984), p. 647.
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584 PHASOR NEURAL NETVORKS Andr~ J. Noest, N.I.B.R., NL-ll0S AZ Amsterdam, The Netherlands. ABSTRACT A novel network type is introduced which uses unit-length 2-vectors for local variables. As an example of its applications, associative memory nets are defined and their performance analyzed. Real systems corresponding to such 'phasor' models can be e.g. (neuro)biological networks of limit-cycle oscillators or optical resonators that have a hologram in their feedback path. INTRODUCTION Most neural network models use either binary local variables or scalars combined with sigmoidal nonlinearities. Rather awkward coding schemes have to be invoked if one wants to maintain linear relations between the local signals being processed in e.g. associative memory networks, since the nonlinearities necessary for any nontrivial computation act directly on the range of values assumed by the local variables. In addition, there is the problem of representing signals that take values from a space with a different topology, e.g. that of the circle, sphere, torus, etc. Practical examples of such a signal are the orientations of edges or the directions of local optic flow in images, or ~he phase of a set of (sound or EM) waves as they arrive on an array of detectors. Apart from the fact that 'circular' signals occur in technical as well as biological systems, there are indications that some parts of the brain (e.g. olfactory bulb, cf. Dr.B.Baird's contribution to these proceedings) can use limit-cycle oscillators formed by local feedback circuits as functional building blocks, even for signals without circular symmetry. Vith respect to technical implementations, I had speculated before the conference whether it could be useful to code information in the phase of the beams of optical neurocomputers, avoiding slow optical switching elements and using only (saturating) optical amplification and a © American Institute of Physics 1988 585 hologram encoding the (complex) 'synaptic' weight factors. At the conference, I learnt that Prof. Dana Anderson had independently developed an optical device (cf. these proceedings) that basically works this way, at least in the slow-evolution limit of the dynamic hologram. Hopefully, some of the theory that I present here can be applied to his experiment. In turn, such implementations call for interesting extensions of the present models. BASIC ELEMENTS OF GENERAL PHASOR NETVORKS Here I study the perhaps simplest non-scalar network by using unitlength 2-vectors (phasors) as continuous local variables. The signals processed by the network are represented in the relative phaseangles. Thus, the nonlinearities (unit-length 'clipping') act orthogonally to the range of the variables coding the information. The behavior of the network is invariant under any rigid rotation of the complete set of phasors, representing an arbitrary choice of a global reference I phase. Statistical physicists will recognize the phasor model as a generalization of 02-spin models to include vector-valued couplings. All 2-vectors are treated algebraically as complex numbers, writing Ixl for the length, Ixl for the phase-angle, and x for the complex conjugate of a 2-vector x. A phasor network then consists of N»l phasors s. , with Is.l=l, 1 1 interacting via couplings c .. , with C .. = O. The c .. are allowed 1J 11 1J to be complex-valued quantities. For optical implementations this is clearly a natural choice, but it may seem less so for biological systems. However, if the coupling between two limitcycle oscillators with frequency f is mediated via a path having propagationdelay d, then that coupling in fact acquires a phaseshift of f.d.2~ radians. Thus, complex couplings can represent such systems more faithfully than the usual models which neglect propagationdelays altogether. Only 2-point couplings are treated here, but multi-point couplings c. 'k' etc., can be treated similarly. 1) The dynamics of each phasor depends only on its local field h.= ! ~ c .. s. + n. 1 z:41J J 1 J where z is the number of inputs 586 c .. ~O per cell and n. is a local noise term (complex and Gaussian). 1J 1 Various dynamics are possible, and yield largely similar results: Continuous-time, parallel evolution: ("type A") d (/s./) = Ih. l.sin(/h.1 - Is./) (IT 1 1 1 1 Discrete-time updating: s.(t+dt)= h.1 Ih. I , either serially in 111 random i-sequence ("type B"), or in parallel for all i ("type C"). The natural time scale for type-B dynamics is obtained by scaling the discrete time-interval eft as ,.., liN ; type-C dynamics has cl't=l. LYAPUNOV FUNCTION (alias "ENERGY", or "HAMILTONIAN" ) If one limits the attention temporarily to purely deterministic (n.=O) models, then the question suggests itself whether a class of 1 couplings exists for which one can easily find a Lyapunov function i.e. a function of the network variables that is monotonic under the 1 dynamics. A well-known example is the 'energy' of the binary and scalar Hopfield models with symmetric interactions. It turns out that a very similar function exists for phasor networks with type-A or B dynamics and a Hermitian matrix of couplings. -H = L 5. h. = • 1 1 (lIz) L 5. c .. s. • . 1 1J J 1,J 1 Hermiticity (c .. =c .. ) makes H real-valued and non-increasing in time. 1J J 1 This can be shown as follows, e.g. for the serial dynamics (type B). Suppose, without loss of generality, that phasor i=l is updated. Then -z H = + Ls. c ' l sl 1>1 1 1 + I. I: i ,j>l -s. c .. s. 1 1J J z 51 h1 + Vith Hermitian couplings, sl' 2: c ' 1 5. + constant. i>l 1 1 H becomes real-valued, and one also has I:c' l 5. l:c1· z h1 s. = i>1 1 1 i>l 1 1 Thus, - H - constant = 51 h1 + sl h1 = 2 Re(sl h1) . Clearly, H is minimized with respect to sl by sl(t+1) = hll I h11 • Type-A dynamics has the same Lyapunovian, but type C is more complex. The existence of Hermitian interactions and the corresponding energy function simplifies greatly the understanding and design of phasor networks, although non-Hermitian networks can still have a Lyapunov587 function, and even networks for which such a function is not readily found can be useful, as will be illustrated later. AN APPLICATION: ASSOCIATIVE MEMORY. A large class of collective computations, such as optimisations and content-addressable memory, can be realised with networks having an energy function. The basic idea is to define the relevant penalty function over the solution-space in the form of the generic 'energy' of the net, and simply let the network relax to minima of this energy. As a simple example, consider an associative memory built within the framework of Hermitian phasor networks. In order to store a set of patterns in the network, i.e. to make a set of special states (at least approximatively) into attractive fixed points of the dynamics, one needs to choose an appropriate set of couplings. One particularly simple way of doing this is via the phasor-analog of "Hebb's rule" (note the Hermiticity) rp s(.k). -s(.k), h (k). h .. I d k c .. = were s. IS p asor 1 In earne pattern . IJ k 1 J 1 The rule is understood to apply only to the input-sets 'i of each i. Such couplings should be realisable as holograms in optical networks, but they may seem unrealistic in the context of biological networks of oscillators since the phase-shift (e.g. corresponding to a delay) of a connection may not be changeable at will. However, the required coupling can still be implemented naturally if e.g. a few paths with different fixed delays exist between pairs of cells. The synaps in each path then simply becomes the projection of the complex coupling on the direction given by the phase of its path, i.e. it is just a classical Hebb-synapse that computes the correlation of its pre- and post-synaptic (imposed) signals, which now are phase-shifted versions of the phasors s~~)The required complex c .. are then realised as the 1 IJ vector sum over at least two signals arriving via distinct paths with corresponding phase-shift and real-valued synaps. Two paths suffice if they have orthogonal phase-shifts, but random phases will do as well if there are a reasonable number of paths. Ve need to have a concise way of expressing how 'near' any state of the net is to one or more of the stored patterns. A natural way 588 of doing this is via a set of p order parameters called "overlaps" 1 N N -(k) 11: s .. s. I • 1 1 1 ; 1 < k < p • Note the constraint on the p overlaps P 2 I Mk ~ 1 if all the patterns k are orthogonal, or merely random in the limit N-.QO. This will be assumed from now on. Also, one sees at once that the whole behaviour of the network does not depend on any rigid rotation of all phasors over some angle since H, Mk, c .. and the dynamics are invariant under 1J multiplication of all s. by a fixed phasor : s~ = S.s. with ISI=1. I I I Let us find the performance at low loading: N,p,z .. oo, with p/z .. O and zero local noise. Also assume an initial overlap m)O with only one pattern, say with k=1. Then the local field is hi 1 f s~k~ s~k) h(1) h7 where = ~s .. i + , z j' i J k 1 J 1 hP~ 1 s~1~ I: sP~s. (1) + O(1//Z) with S~f(i);ISI=1, = m1 . si • S 1 Z 1 jl'i J J * ~ fs~k). L: s~k~s. O( ./( p-l) Iz') and h. = . 1 z k=2 1 j(~i J J Thus, perfect recall (M1=1) occurs in one 'pass' at loadings p/z ... O. EXACTLY SOLVABLE CASE: SPARSE and ASYMMETRIC couplings Although it would be interesting to develop the full thermodynamics of Hermitian phasor networks with p and z of order N (analogous to the analysis of the finite-T Hopfield model by the teams of Amit2 and van Hemmen3), I will analyse here instead a model with sparse, asymmetric connectivity, which has the great advantages of being exactly solvable with relative ease, and of being arguably more realistic biologically and more easily scalable technologically. In neurobiological networks a cell has up to z;104 asymmetric connections, whereas N;101~ This probably has the same reason as applies to most VLSI chips, namely to alleviate wiring problems. For my present purposes, the theoretical advantage of getting some exact results is of primary interest4 Suppose each cell has z incoming connections from randomly selected other cells. The state of each cell at time t depends on at most zt . t 112 cells at time t=O. Thus, If z «N and N large, then the respective 589 4 trees of 'ancestors' of any pair cells have no cells in common. In x particular, if z_ (logN) , for any finite x, then there are no common ancestors for any finite time t in the limit N-.OO. For fundamental information-theoretic reasons, one can hope to be able to store p patterns with p at most of order z for any sort of 2-point couplings. Important questions to be settled are: Yhat are the accuracy and speed of the recall process, and how large are the basins of the attractors representing recalled patterns? Take again initial conditions (t=O) with, say, m(t)= Hl > H>l = O. Allowing again local random Gaussian (complex) noise n., the local f · ld b . f '1' . h h(l) h* 1 Ie s ecome, In now amI Iar notatIon, .= . + . + n .• 1 1 1 1 As in the previous section, the h~l)term consists of the 'signal' 1 m(t).s. (modulo the rigid rotation S) and a random term of variance 1 * at most liz. For p _ z, the h. term becomes important. Being sums of 1 * z(p-1) phasors oriented randomly relative to the signal, the h. are 1 independent Gaussian zero-mean 2-vectors with variance (p-1)/z , as p,z and N .. oo . Finally, let the local noises n. have variance r2. 1 Then the distribution of the s.(t+l) phasors can be found in terms of 1 2 * the signal met) and the total variance a=(p/z)+r of the random h.+n .• 1 1 After somewhat tedious algebraic manipulations (to be reported in detail elsewhere) one obtains the dynamic behaviour of met) and m(t+1) = F(m(t),a) for discrete parallel (type-C) dynamics, d met) = F(m(t),a) - met) Tt for type-A or type-B dynamics , where the function F(m,a) = m +" 2 Idx.(1+cos2x).expl-(m.sinx) la].(l+erfl(m.cosx)/~) -1'C * * The attractive fixed points H (a)= F(H ,a) represent the retrieval accuracy when the loading-pIus-noise factor equals a. See figure 1. * 2 3 For a«l one obtains the expansion 1-H (a) = a/4 + 3a 132 + O(a ). * 112 The recall solutions vanish continuously as H _(a -a) at a =tc/4. c c One also obtains (at any t) the distribution of the phase scatter of the phasors around the ideal values occurring in the stored pattern. 590 P(/u./) = (1/2n).exp(-m2/a).(1+I1t.L.exp(L2).(1+erf(L» , 1 where L = (m/la).cos(/u./) , and 1 -(k) u.= s. s. (modulo S). 111 Useful approximations for the high, respectively low M regimes are: M »ra: PUu./) (MIl'a1l).exp[-(M./u./)2 /a ] ; I/u./1 «"XI2 1 1 1 M «fi : PUu./) = (1I21t).(1+L • ./;l) 1 Figure ~ RETRIEVAL-ERROR and BASIN OF ATTRACTION versus LOADING + NOISE. Q Q Q en Q Q .,; Q " .,; I: Q UI ..,) Q C -0 Q a. Q 1:) CD x Q c- .. Q Q '" Q 0 Q 0 Q 0 c: "'0.00 0.10 0.20 O. 30 0 • 40 0 • 50 0 • 60 O. 70 0 • 80 0 • 90 1. 00 a = p/z + r-r 591 DISCUSSION It has been shown that the usual binary or scalar neural networks can be generalized to phasor networks, and that the general structure of the theoretical analysis for their use as associative memories can be extended accordingly. This suggests that many of the other useful applications of neural nets (back-prop, etcJ can also be generalized to a phasor setting. This may be of interest both from the point of view of solving problems naturally posed in such a setting, as well as from that of enabling a wider range of physical implementations, such as networks of limit-cycle oscillators, phase-encoded optics, or maybe even Josephson-junctions. The performance of phasor networks turns out to be roughly similar to that of the scalar systems; the maximum capacity p/z=~/4 for phasor nets is slightly larger than its value 2/n for binary nets, but there is a seemingly faster growth of the recall error 1-M at small a (linear for phasors, against exp(-1/(2a» for binary nets). However, the latter measures cannot be compared directly since they stem from quite different order parameters. If one reduces recalled phasor patterns to binary information, performance is again similar. Finally, the present methods and results suggest several roads to further generalizations, some of which may be relevant with respect to natural or technical implementations. The first class of these involves local variables ranging over the k-sphere with k>l. The other generalizations involve breaking the O(n) (here n=2) symmetry of the system, either by forcing the variables to discrete positions on the circle (k-sphere), and/or by taking the interactions between two variables to be a more general function of the angular distance between them. Such models are now under development. REFERENCES 1. J.J.Hopfield, Proc.Nat.Acad.Sci.USA 79, 2554 (1982) and idem, Proc.Nat.Acad.Sci.USA 81, 3088 (1984). 2. D.J.Amit, H.Gutfreund and H.Sompolinski, Ann.Phys. 173, 30 (1987). 3. D.Grensing, R.Kuhn and J.L. van Hemmen, J.Phys.A 20, 2935 (1987). 4. B.Derrida, E.Gardner and A.Zippelius, Europhys.Lett. 4, 167 (1987)
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422 COMPUTING MOTION USING RESISTIVE NETWORKS Christof Koch, Jin Luo, Carver Mead California Institute of Technology, 216-76, Pasadena, Ca. 91125 James Hutchinson Jet Propulsion Laboratory, California Institute of Technology Pasadena, Ca. 91125 INTRODUCTION To us, and to other biological organisms, vision seems effortless. We open our eyes and we "see" the world in all its color, brightness, and movement. Yet, we have great difficulties when trying to endow our machines with similar abilities. In this paper we shall describe recent developments in the theory of early vision which lead from the formulation of the motion problem as an illposed one to its solution by minimizing certain "cost" functions. These cost or energy functions can be mapped onto simple analog and digital resistive networks. Thus, we shall see how the optical flow can be computed by injecting currents into resistive networks and recording the resulting stationary voltage distribution at each node. These networks can be implemented in cMOS VLSI circuits and represent plausible candidates for biological vision systems. APERTURE PROBLEM AND SMOOTHNESS ASSUMPTION In this study, we use intensity-based schemes for recovering motion. Let us derive an equation relating the change in image brightness to the motion of the image (seel ). Let us assume that the brightness of the image is constant over time: dI(~,y,t)/dt = o. On the basis of the chain rule of differentiation, this transforms into 81 d~ 81 dy 81 8~ dt + 8y dt + at = Izu + Iyv + It = 'V I· v + It = 0, (1) where we define the velocity v as (u,v) = (d:1)/dt,dy/dt). Because we assume that we can compute these spatial and temporal image gradients, we are now left with a single linear equation in two unknowns, u and v, the two components of the velocity vector (aperture problem). Any measuring system with a finite aperture, whether biological or artificial, can only sense the velocity component perpendicular to the edge or along the spatial gradient (-It! 1 'V I I). The component of motion perpendicular to the gradient cannot, in principle, be registered. The problem remains unchanged even if we measure these velocity components at many points throughout the image. How can this problem be made well-posed, that is, having a unique solution depending continuously on the data? One form of "regularizing" ill-posed @ American Institute of Physics 1988 423 problems is to restrict the class of admissible solutions by imposing appropriate constraints2 • Applying this method to motion, we shall argue that in general objects are smooth-except at isolated discontinuities-undergoing smooth movements. Thus, in general, neighboring points in the world will have similar velocities and the projected velocity field should reflect this fact. We therefore impose on the velocity field the constraint that it should be the smoothest as well as satisfying the data. As measure of smoothness we choose, the square of the velocity field gradient. The final velocity field (u, v) is the one that minimizes A J J [ (::)' + (::)' + (~:)' + (:~)'] dz dy (2) + + + + ., + II:, + _I~ + + + !al (b) Fig. 1. ( a) The location of the horizontal (lfj) and vertical (Iij) line processes relative to the motion field nngrid. (b) The hybrid resistive network, computing the optical flow in the presence of discontinuities. The conductances T c - ij connecting both grids depend on the brightness gradient, as do the conductances gij and gij connecting each node with the battery. For clarity, only two such elements are shown. The battery Eij depends on both the temporal and the spatial gradient and is zero if no brightness change occurs. The ~ (resp. y) component of the velocity is given by the voltage in the top (resp. bottom) network. Binary switches, which make or break the resistive connections between nodes, 424 implement motion discontinuities. These switches could be under the control of distributed digital processors. Analog cMOS implementations are also feasible3 • The first term implements the constraint that the final solution should follow as closely as possible the measured data whereas the second term imposes the smoothness constraint on the solution. The degree to which one or the other terms are minimized is governed by the parameter).. If the data is very accurate, it should be "expensive" to violate the first term and), will be small. If, conversely, the data is unreliable (low signal-to-noise), much more emphasis will be placed on the smoothness term. Horn and Schunck1 first formulated this variational approach to the motion problem. The energy E( u, v) is quadratic in the unknown u and v. It then follows from standard calculus of variation that the associated Euler-Lagrange equations will be linear in u and v: I~u + IzIyv ). \721.£ + IzIt = 0 I z I 1Iu + I:v ). \72 v + Iylt = O. (3) We now have two linear equations at every point and our problem is therefore completely determined. ANALOG RESISTIVE NETWORKS Let us assume that we are formulating eqs. (2) and (3) on a discrete 2-D grid, such as the one shown in fig. 1a. Equation (3) then transforms into I~ijuij + IzijI1Iijvij ). (UHlj + Uij+l 4Uij + Ui-lj + Uij-l) + IZijltij = 0 Izijlyijuij + I:ijvij ). (VHlj + Vij+l 4Vij + Vi-lj + Vij-l) + Iyijltij = 0 (4) where we replaced the Laplacian with its 5 point approximation on a rectangular grid. We shall now show that this set of linear equations can be solved naturally using a particular simple resistive network. Let us apply Kirchhoff's current law to the nodne i, j in the top layer of the resistive network shown in fig. lb. We then have the following update equation: du·· C d;' = T (Ui+lj + Uij+l 4Uij + Ui-lj + Uij-l) (5) + gij (Eij Uij) + Tc-ij( Vij - Uij). where Vij is the voltage at node i, j in the bottom network. Once dUij / dt = 0 and dVij/dt = 0, this equation is seen to be identical with eq. (4), if we identify (a) (c) (e) Tc-ij ~ -IzijIyij gij ~ Izij (Izij + IJlij) gij ~ Iyii (Izii + Iyij) -It Eij~---Izii + Iyij (b) (d) ~ (f) 425 (6) Fig. 2. Motion sequence using synthetic data. (a) and (b) Two images of three high contrast squares on a homogeneous background. (c) The initial velocity data. The inside of both squares contain no data. (d) The final state 426 of the network after 240 iterations, corresponding to the smooth optical flow field. (e) Optical flow in the presence of motion discontinuities (indicated by solid lines). (f) Discontinuities are strongly encouraged to form at the location of intensity edges4 • Both (e) and (f) show the state of the hybrid network after six analog-digital cycles. Once we set the batteries and the conductances to the values indicated in eq. (6), the network will settle-following Kirchhoff's laws-into the state of least power dissipation. The associated stationary voltages correspond to the sought solution: uii is equivalent to the :c component and Vii to the y component of the optical flow field. We simulated the behavior of these networks by solving the above circuit equations on parallel computers of the Hypercube family. As boundary conditions we copied the initial velocity data at the edge of the image into the nodes lying directly adjacent but outside the image. The sequences in figs. 2 and 3 illustrate the resulting optical flow for synthetic and natural images. As discussed by Horn and Schunck1 , the smoothness constraint leads to a qualitatively correct estimate of the velocity field. Thus, one undifferentiated blob appears to move to the lower right and one blob to the upper left. However, at the occluding edge where both squares overlap, the smoothness assumption results in a spatial average of the two opposing velocities, and the estimated velocity is very small or zero. In parts of the image where the brightness gradient is zero and thus no initial velocity data exists (for instance, the interiors of the two squares), the velocity estimates are simply the spatial average of the neighboring velocity estimates. These empty areas will eventually fill in from the boundary, similar to the How of heat for a uniform flat plate with "hot" boundaries. MOTION DISCONTINUITIES The smoothness assumption of Horn and Schunck1 regularizes the aperture problem and leads to the qualitatively correct velocity field inside moving objects. However, this approach fails to detect the locations at which the velocity changes abruptly or discontinuously. Thus, it smoothes over the figure-ground discontinuity or completely fails to detect the boundary between two objects with differing velocities because the algorithm combines velocity information across motion boundaries. A quite successful strategy for dealing with discontinuities was proposed by Geman and Geman5 • We shall not rigorously develop their approach, which is based on Bayesian estimation theory (for details see5,6). Suffice it to say that a priori knowledge, for instance, that the velocity field should in general be smooth, can be formulated in terms of a Markov Random Field model of the image. Given such an image model, and given noisy data, we then estimate the "best" flow field by some likelihood criterion. The one we will use here 427 is the maximum a posteriori estimate, although other criteria are possible and have certain advantages6 • This can be shown to be equivalent to minimizing an expression such as eq. (2). In order to reconstruct images consisting of piecewise constant segments, Geman and Geman5 further introduced the powerful idea of a line process 1. For our purposes, we will assume that a line process can be in either one of two states: "on" (1 = 1) or "off" (1 = 0). They are located on a regular lattice set between the original pixel lattice (see fig. 1a), such that each pixel i,j has a horizontallfi and a verticallij line process associated with it. If the appropriate line process is turned on, the smoothness term between the two adjacent pixels will be set to zero. In order to prevent line processes from forming everywhere and, furthermore, in order to incorporate additional knowledge regarding discontinuities into the line processes, we must include an additional term Vc(l) into the new energy function: E( 'IL, v, lh., IV) = L (Iz'ILii + IyVii + It )2 + i.i ). L (1 -It) [('lLi+1i - 'lLii)2 + (Vi+li - Vii)2] + i.i ). L (1 -Iii) [('lLij+l - 'lLii)2 + (vii+1 - Vij)2] + Vc(l). i.i (7) Vc contains a number of different terms, penalizing or encouraging specific configurations of line processes: i.; i.i plus the corresponding expression for the vertical line process Iii (obtained by interchanging i with j and Iii with Ifi). The first term penalizes each introduction of a line process, since the cost Cc has to be "payed" every time a line process is turned on. The second term prevents the formation of parallel lines: if either lfi+l or Ifi+2 is turned on, this term will tend to prevent It from turning on. The third term, CIVI , embodies the fact that in general, motion discontinuities occur along extended contours and rarely intersect (for more details see7 ). We obtain the optical flow by minimizing the cost function in eq. (7) with respect to both the velocity v and the line processes Ih. and IV. To find an optimal solution to this non-quadratic minimization problem, we follow Koch et a1. 7 and use a purely deterministic algorithm, based on solving Kirchhoff's equations for a mixed analogi digital network (see also 8). Our algorithm exploits the fact that for a fixed distribution of line processes, the energy function (7) is quadratic. Thus, we first initialize the analog resistive network (see fig. 2b) according to eq. (6) and with no line processes on. The network then converges to 428 the smoothest solution. Subsequently, we update the line processes by deciding at each site of the line process lattice whether the overall energy can be lowered by setting or breaking the line proceSSj that is, lfi will be turned on if E( u, v, lfi = 1, IV) < E( u, v, Ifi = 0, IV); otherwise, Ifj = o. Line processes are switched on by breaking the appropriate resistive connection between the two neighboring nodes. After the completion of one such analog-digital cycle, we reiterate and compute-for the newly updated distribution of line processes-the smoothest state of the analog network. Although there is no guarantee that the system will converge to the global minimum, since we are using a gradient descent rule, it seems to find next-to-optimal solutions in about 10 to 15 analog-digital cycles. (8) (c) (e) Figure 3. Optical flow of a moving person. (a) and (b) Two 128 by 128 pixel images captured by a video camera. The person in the foreground is moving toward the right while the person in the background is stationary. The noise in the lower part of the image is a camera artifact. (c) Zero-crossings superimposed on the initial velocity data. (d) The smooth optical flow after 1000 iterations. Note that the noise in the lower part of both images is completely smoothed away. (e) The final piecewise smooth optical flow. The velocity field is subsampled to improve visibility. The evolution of the hybrid network is shown after the 1. (a), 3. (b), 5. (c), 7. (d), 10. (e), and 13. (f) analog-digital cycle in the right part of the figure. The synthetic motion sequence in fig. 2 demonstrates the effect of the line 429 processes. The optical flow outside the discontinuities approximately delineating the boundaries of the moving squares is zero, as it should be (fig. 2e). However, where the two squares overlap the velocity gradient is high and multiple intersecting discontinuities exist. To restrict further the location of discontinuities, we adopt a technique used by Gamble and Poggio4 to locate depth discontinuities by requiring that depth discontinuities coincide with the location of intensity edges. Our rationale behind this additional constraint is that with very few exceptions, the physical processes and the geometry of the 3-dimensional scene giving rise to the motion discontinuity will also give rise to an intensity edge. As edges we use the zero-crossings of a Laplacian of a Gaussian convolved with the original image9 • We now add a new term VZ-Cii to our energy function E, such that Vz -Cii is zero if Iii is off or if Iii is on and a zero-crossing exists between locations i and j. If Iii = 1 in the absence of a zero-crossing, V Z - Cii is set to 1000. This strategy effectively prevents motion discontinuities from forming at locations where no zero-crossings exist, unless the data strongly suggest it. Conversely, however, zero-crossings by themselves will not induce the formation of discontinuities in the absence of motion gradients (figs. 2f and 3). ANALOG VLSI NETWORKS Even with the approximations and optimizations described above, the computations involved in this and similar early vision tasks require minutes to hours on computers. It is fortunate then that modern integrated circuit technology gives us a medium in which extremely complex, analog real-time implementations of these computational metaphors can be realized3 • We can achieve a very compact implementation of a resistive network using an ordinary cMOS process, provided the transistors are run in the sub-threshold range where their characterstics are ideal for implementing low-current analog functions. The effect of a resistor is achieved by a circuit configuration, such as the one shown in fig. 4, rather than by using the resistance of a special layer in the process. The value of the resulting resistance can be controlled over three orders of magnitude by setting the bias voltages on the upper and lower current source transistors. The current-voltage curve saturates above about 100 mVj a feature that can be used to advantage in many applications. When the voltage gradients are small, we can treat the circuit just as if it were a linear resistor. Resistances with an effective negative resistance value can easily be realized. In two dimensions, the ideal configuration for a network implementation is shown in fig. 4. Each point on the hexagonal grid is coupled to six equivalent neighbors. Each node includes the resistor apparatus, and a set of sample-andhold circuits for setting the confidence and signal the input and output voltages. Both the sample-and-hold circuits and the output buffer are addressed by a scanning mechanism, so the stored variables can be refreshed or updated, and the map of node voltages read out in real time. 430 ~ I, I VI v, (a) v (b) Figure 4. Circuit design for a resistive network for interpolating and smoothing noisy and sparsely sampled depth measurements. (a) Circuit-consisting of 8 transistors-implementing a variable nonlinear resistance. (b) If the voltage gradient is below 100 mV its approximates a linear resistance. The voltage VT controls the maximum current and thus the slope of the resistance, which can vary between 1 MO and 1 GO 3. This cMOS circuit contains 20 by 20 grid points on a hexagonal lattice. The individual resistive elements with a variable slope controlled by VT correspond to the term governing the smoothness, A. At those locations where a depth measurement dij is present, the battery is set to this value (Vin = dij ) and the value of the conductance G is set to some fixed value. If no depth data is present at that node, G is set to zero. The voltage at each node corresponds to the discrete values of the smoothed surface fitted through the noisy and sparse measurements7 • A 48 by 48 silicon retina has been constructed that uses the hexagonal network of fig. 4 as a model for the horizontal cell layer in the vertebrate retinal 0 • In this application, the input potentials were the outputs of logarithmic photoreceptors-implemented via phototransistors-and the potential difference across the conductance T formed an excellent approximation to the Laplacian operator. DISCUSSION We have demonstrated in this study that the introduction of binary motion 431 discontinuities into the algorithm of Horn and Schunck1 leads to a dramatically improved performance ~f their method, in particular for the optical flow in the presence of a number of moving non-rigid objects. Moreover, we have shown that the appropriate computations map onto simple resistive networks. We are now implementing these resistive networks into VLSI circuits, using subtheshold cMOS technology. This approach is of general interest, because a great number of problems in early vision can be formulated in terms of similar non-convex energy functions that need to be minimized, such as binocular stereo, edge detection, surface interpolation, structure from motion, etc.2 ,6,8. These networks share several features with biological neural networks. Specifically, they do not require a system-wide clock, they rely on many connections between simple computational nodes, they converge rapidly-within several time constants-and they are quite robust to hardware errors. Another interesting feature is that our networks only consume very moderate amounts of powerj the entire retina chip requires about 100 J.L W 10 Acknowledgments: An early version of this model was developed and implemented in collaboration with A. L. Yuille8 • M. Avalos and A. Hsu wrote the code for the Imaging Technology system and E. Staats for the NCUBE. C.K. is supported by an ONR Research Young Investigator Award and by the Sloan and the Powell Foundations. C.M. is supported by ONR and by the System Development Foundation. A portion of this research was carried out at the Jet Propulsion Laboratory and was sponsored by NSF grant No. EET-8714710, and by NASA. REFERENCES 1. Horn, B. K. P. and Schunck, B. G. Artif. Intell. 17,185-203 (1981). 2. Poggio, T., Torre, V. and Koch, C. Nature 317,314-319 (1985). 3. Mead, C. Analog VLSI and Neural Systems. Addison-Wesley: Reading, MA (1988). 4. Gamble, E. and Poggio, T. Artif. Intell. Lab. Memo. No. 970, MIT, Cambridge MA (1987). 5. Geman, S. and Geman, D. IEEE Trans. PAMI 6, 721-741 (1984). 6. Marroquin, J., Mitter, S. and Poggio, T. J. Am. Stat. Assoc. 82, 76-89 (1987). 7. Koch, C., Marroquin, J. and Yuille, A. Proc. Natl. Acad. Sci. USA 83, 4263-4267 (1986). 8. Yuille, A. L. Artif. Intell. Lab. Memo. No. 987, MIT, Cambridge, MA (1987). 9. Marr, D. and Hildreth, E. C. Proc. R. Soc. Lond. B 207, 187-217 (1980). 10. Sivilotti, M. A., Mahowald, M. A. and Mead, C. A. In: 1987 Stanford VLSI Conference, ed. P. Losleben, pp. 295-312 (1987).
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EXPERIMENTAL DEMONSTRATIONS OF OPTICAL NEURAL COMPUTERS Ken Hsu, David Brady, and Demetri Psaltis Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125 ABSTRACT 377 We describe two expriments in optical neural computing. In the first a closed optical feedback loop is used to implement auto-associative image recall. In the second a perceptron-Iike learning algorithm is implemented with photorefractive holography. INTRODUCTION The hardware needs of many neural computing systems are well matched with the capabilities of optical systemsl ,2,3. The high interconnectivity required by neural computers can be simply implemented in optics because channels for optical signals may be superimposed in three dimensions with little or no cross coupling. Since these channels may be formed holographically, optical neural systems can be designed to create and maintain interconnections very simply. Thus the optical system designer can to a large extent avoid the analytical and topological problems of determining individual interconnections for a given neural system and constructing physical paths for these interconnections. An archetypical design for a single layer of an optical neural computer is shown in Fig. 1. Nonlinear thresholding elements, neurons, are arranged on two dimensional planes which are interconnected via the third dimension by holographic elements. The key concerns in implementing this design involve the need for suitable nonlinearities for the neural planes and high capacity, easily modifiable holographic elements. While it is possible to implement the neural function using entirely optical nonlinearities, for example using etalon arrays\ optoelectronic two dimensional spatial light modulators (2D SLMs) suitable for this purpose are more readily available. and their properties, i.e. speed and resolution, are well matched with the requirements of neural computation and the limitations imposed on the system by the holographic interconnections5 ,6. Just as the main advantage of optics in connectionist machines is the fact that an optical system is generally linear and thus allows the superposition of connections, the main disadvantage of optics is that good optical nonlinearities are hard to obtain. Thus most SLMs are optoelectronic with a non-linearity mediated by electronic effects. The need for optical nonlinearities arises again when we consider the formation of modifiable optical interconnections, which must be an all optical process. In selecting @ American Institute of Physics 1988 378 a holographic material for a neural computing application we would like to have the capability of real-time recording and slow erasure. Materials such as photographic film can provide this only with an impractical fixing process. Photorefractive crystals are nonlinear optical materials that promise to have a relatively fast recording response and long term memory4,5,6,7,B. . " ..... '. '. ". '. Fourier lens '. . . .' .. - ~ :-w:-=7 -~---...... . '. Fourier hologro.phlc I"IealuI"I lens Figure 1. Optical neural computer architecture. In this paper we describe two experimental implementations of optical neural computers which demonstrate how currently available optical devices may be used in this application. The first experiment we describe involves an optical associative loop which uses feedback through a neural plane in the form of a pinhole array and a separate thresholding plane to implement associate regeneration of stored patterns from correlated inputs. This experiment demonstrates the input-output dynamics of an optical neural computer similar to that shown in Fig. 1, implemented using the Hughes Liquid Crystal Light Valve. The second experiment we describe is a single neuron optical perceptron implemented with a photorefractive crystal. This experiment demonstrates how the learning dynamics of long term memory may be controlled optically. By combining these two experiments we should eventually be able to construct high capacity adaptive optical neural computers. OPTICAL ASSOCIATIVE LOOP A schematic diagram of the optical associative memory loop is shown in Fig. 2. It is comprised of two cascaded Vander Lugt correlators9. The input section of the system from the threshold device P1 through the first hologram P2 to the pinhole array P3 forms the first correlator. The feedback section from P3 through the second hologram P4 back to the threshold device P1 forms the second correlator. An array of pinholes sits on the back focal plane of L2, which coincides with the front focal plane of L3. The purpose of the pinholes is to link the first and the second (reversed) correlator to form a closed optical feedback loop 10. There are two phases in operating this optical loop, the learning phase and the recal phase. In the learning phase, the images to be stored are spatially multiplexed and entered simultaneously on the threshold device. The 379 thresholded images are Fourier transformed by the lens Ll. The Fourier spectrum and a plane wave reference beam interfere at the plane P2 and record a Fourier transform hologram. This hologram is moved to plane P4 as our stored memory. We then reconstruct the images from the memory to form a new input to make a second Fourier transform hologram that will stay at plane P2. This completes the learning phase. In the recalling phase an input is imaged on the threshold Input ~~~*+++~~~~ device. This image is correlated with the reference images in the hologram at P2. If the correlation between the input and one of the stored images is high a bright peak appears at one of the pinholes. This peak is sampled by the pinhole to reconstruct the stored image from the hologram at P4. The reconstructed beam is then imaged back to the threshold device to form a closed loop. If the overall optical gain in the loop exceeds the loss the loop signal will grow until the threshold device is saturated. In this case, we can cutoff the external input image and the optical loop will be latched at the stable memory. ~ -,.....,.- Second Hologram Pinhole Array --.... L z I I Figure. 2. All-optical associative loop. The threshold device is a LCLV, and the holograms are thermoplastic plates. The key elements in this optical loop are the holograms, the pinhole array, and the threshold device. If we put a mirror10 or a phase conjugate mirror7 ,11 at the pinhole plane P3 to reflect the correlation signal back through the system then we only need one hologram to form a closed loop. The use of two holograms, however, improves system performance. We make the hologram at P2 with a high pass characteristic so that the input section of the loop has high spectral discrimination. On the other hand we want the images to be reconstructed with high fidelity to the original images. Thus the hologram at plane P4 must have broadband characteristics. We use a diffuser to achieve this when making this hologram. Fig. 3a shows the original images. Fig. 3b and Fig. 3c are the images reconstructed from first and second holograms, respectively. As desired, Fig. 3b is a high pass version of the stored image while Fig. 3c is broadband. Each of the pinholes at the correlation plane P3 has a diameter of 60 j.lm. The separations between the pinholes correspond to the separations of the input images at plane P 1. If one of the stored images appears at P 1 there will be a bright spot at the corresponding pinhole on plane P3. If the input image shifts to the position of another image the correlation peak will also 380 ~ '" • L ,. ' . . ~. ~ , . ( i .a:..J \~ .~ ~ • -y::' • • • . .. ·Il, .' .r ... I K~·';t .. .# a. b. c. Figure 3. (a) The original images. (b)The reconstructed images from the highpass hologram P2. (c) The reconstructed images from the band-pass hologram P4. shift to another pinhole. But if the shift is not an exact image spacing the correlation peak can not pass the pinhole and we lose the feedback signal. Therefore this is a loop with "discrete" shift invariance. Without the pinholes the cross-correlation noise and the auto-correlation peak will be fed back to the loop together and the reconstructed images won't be recognizable. There is a compromise between the pinhole size and the loop performance. Small pinholes allow good memory discrimination and sharp reconstructed images, but can cut the signal to below the level that can be detected by the threshold device and reduce the tolerance of the system to shifts in the input. The function of the pinhole array in this system might also be met by a nonlinear spatial light modulator, in which case we can achieve full shift invariance12• The threshold device at plane PI is a Hughes Liquid Crystal Light Valve. The device has a resolution of 16 Ip/mm and uniform aperture of 1 inch diameter. This gives us about 160,000 neurons at PI. In order to compensate for the optical loss in the loop, which is on the order of 10- 5 , we need the neurons to provide gain on the order of 105. In our system this is achieved by placing a Hamamatsu image intensifier at the write side of the LCLV. Since the microchannel plate of the image intensifier can give gains of 104 , the combination of the LCLV and the image intensifier can give gains of 106 with sensitivity down to n W /cm2 . The optical gain in the loop can be adjusted by changing the gain of the image intensifier. Since the activity of neurons and the dynamics of the memory loop is a continuously evolving phenomenon, we need to have a real time device to monitor and record this behavior. We do this by using a prism beam splitter to take part of the read out beam from the LCLV and image it onto a CCD camera. The output is displayed on a CRT monitor and also recorded on a video tape recorder. Unfortunately, in a paper we can only show static pictures taken from the screen. We put a window at the CCD plane so that each time we can pick up one of the stored images. Fig. 4a shows the read out image 381 a. b. c. Figure 4. (a) The external input to the optical loop. (b) The feedback image superimposed with the input image. (c) The latched loop image. from the LCLV which comes from the external input shifted away from its stored position. This shift moves its correlation peak so that it does not match the position of the pinhole. Thus there is no feedback signal going through the loop. If we cut off the input image the read out image will die out with a characteristic time on the order of 50 to 100 ms, corresponding to the response time of the LCLV. Now we shift the input image around trying to search for the correct position. Once the input image comes close enough to the correct position the correlation peak passes through the right pinhole, giving a strong feedback signal superimposed with the external input on the neurons. The total signal then goes through the feedback loop and is amplified continuously until the neurons are saturated. Depending on the optical gain of the neurons the time required for the loop to reach a stable state is between 100 ms and several seconds. Fig. 4b shows the superimposed images of the external input and the loop images. While the feedback signal is shifted somewhat with respect to the input, there is sufficient correlation to induce recall. If the neurons have enough gain then we can cut off the input and the loop stays in its stable state. Otherwise we have to increase the neuron gain until the loop can sustain itself. Fig. 4c shows the image in the loop with the input removed and the memory latched. If we enter another image into the system, again we have to shift the input within the window to search the memory until we are close enough to the correct position. Then the loop will evolve to another stable state and give a correct output. The input images do not need to match exactly with the memory. Since the neurons can sense and amplify the feedback signal produced by a partial match between the input and a stored image, the stored memory can grow in the loop. Thus the loop has the capability to recall the complete memory from a partial input. Fig. 5a shows the image of a half face input into the system. Fig. 5b shows the overlap of the input with the complete face from the memory. Fig. 5c shows the stable state of the loop after we cut off the external input. In order to have this associative behavior the input must have enough correlation with the stored memory to yield a strong feedback signal. For instance, the loop does not respond to the the presentation of a picture of 382 a. c. Figure 5. (a) Partial face used as the external input. (b) The superimposed images of the partial input with the complete face recalled by the loop. (c) The complete face latched in the loop. a. b. c. Figure 6. (a) Rotated image used as the external input. (b) The superimposed images of the input with the recalled image from the loop. (c) The image latched in the optical loop. a person not stored in memory. Another way to demonstrate the associative behavior of the loop is to use a rotated image as the input. Experiments show that for a small rotation the loop can recognize the image very quickly. As the input is rotated more, it takes longer for the loop to reach a stable state. If it is rotated too much, depending on the neuron gain, the input won't be recognizable. Fig. 6a shows the rotated input. Fig. 6b shows the overlap of loop image with input after we turn on the loop for several seconds. Fig. 6c shows the correct memory recalled from the loop after we cut the input. There is a trade-off between the degree of distortion at the input that the system can tolerate and its ability to discriminate against patterns it has not seen before. In this system the feedback gain (which can be adjusted through the image intensifier) controls this trade-off. PHOTOREFRACTIVE PERCEPTRON Holograms are recorded in photorefractive crystals via the electrooptic modulation of the index of refraction by space charge fields created by the migration of photogenerated charge13,14. Photorefractive crystals are attractive for optical neural applications because they may be used to store 383 long term interactions between a very large number of neurons. While photorefractive recording does not require a development step, the fact that the response is not instantaneous allows the crystal to store long term traces of the learning process. Since the photorefractive effect arises from the reversible redistribution of a fixed pool of charge among a fixed set of optically addressable trapping sites, the photorefractive response of a crystal does not deteriorate with exposure. Finally, the fact that photorefractive holograms may extend over the entire volume of the crystal has previously been shown to imply that as many as 1010 interconnections may be stored in a single crystal with the independence of each interconnection guaranteed by an appropriate spatial arrangement of the interconnected neurons6 ,5. In this section we consider a rudimentary optical neural system which uses the dynamics of photorefractive crystals to implement perceptron-like learning. The architecture of this system is shown schematically in Fig. 7. The input to the system, x, corresponds to a two dimensional pattern recorded from a video monitor onto a liquid crystal light valve. The light valve transfers this pattern on a laser beam. This beam is split into two paths which cross in a photorefractive crystal. The light propagating along each path is focused such that an image of the input pattern is formed on the crystal. The images along both paths are of the same size and are superposed on the crystal, which is assumed to be thinner than the depth of focus of the images. The intensity diffracted from one of the two paths onto the other by a hologram stored in the crystal is isolated by a polarizer and spatially integrated by a single output detector. The thresholded output of this detector corresponds to the output of a neuron in a perceptron. laser PB LCL V TV ~---,t+ - --f4HJ ucl BS$- PM COl"lputer Xtal Figure 7. Photorefractive perceptron. PB is a polarizing beam splitter. Ll and L2 are imaging lenses. WP is a quarter waveplate. PM is a piezoelectric mirror. P is a polarizer. D is a detector. Solid lines show electronic control. Dashed lines show the optical path. The ith component of the input to this system corresponds to the intensity in the ith pixel of the input pattern. The interconnection strength, Wi, between the ith input and the output neuron corresponds to the diffraction efficiency of the hologram taking one path into the other at the ith pixel of the image plane. While the dynamics of Wi can be quite complex in some geometries 384 and crystals, it is possible to show from the band transport model for the photorefractive effect that under certain circumstances the time development of Wi may be modeled by (1) where m(s) and 4>(s) are the modulation depth and phase, respectively, of the interference pattern formed in the crystal between the light in the two paths15• T is a characteristic time constant for crystal. T is inversely proportional to the intensity incident on the ith pixel of the crystal. Using Eqn. 1 it is possible to make Wi(t) take any value between 0 and W m l1Z by properly exposing the ith pixel of the crystal to an appropriate modulation depth and intensity. The modulation depth between two optical beams can be adjusted by a variety of simple mechanisms. In Fig. 7 we choose to control met) using a mirror mounted on a piezoelectric crystal. By varying the frequency and the amplitude of oscillations in the piezoelectric crystal we can electronically set both met) and 4>(t) over a continuous range without changing the intensity in the optical beams or interrupting readout of the system. With this control over met) it is possible via the dynamics described in Eqn. (1) to implement any learning algorithm for which Wi can be limited to the range (0, wmaz ). The architecture of Fig. 7 classifies input patterns into two classes according to the thresholded output of the detector. The goal of a learning algorithm for this system is to correctly classify a set of training patterns. The perceptron learning algorithm involves simply testing each training vector and adding training vectors which yield too Iowan output to the weight vector and subtracting training vectors which yield too high an output from the weight vector until all training vectors are correctly classified 16. This training algorithm is described by the equation L\wi = aXj where alpha is positive (negative) if the output for x is too low (high). An optical analog of this method is implemented by testing each training pattern and exposing the crystal with each incorrectly classified pattern. Training vectors that yield a high output when a low output is desired are exposed at zero modulation depth. Training vectors that yield a low output when high output is desired are exposed at a modulation depth of one. The weight vector for the k + 1 th iteration when erasure occurs in the kth iteration is given by (2) where we assume that the exposure time, L\t, is much less than T. Note that since T is inversely proportional to the intensity in the ith pixel, the change in 385 Wi is proportional to the ith input. The weight vector at the k + 1 th iteration when recording occurs in the kth iteration is given by -2~t _ / -~t -~t -~t 2 wi(k+ 1) = e-r-Wi(k) +2y Wi(k)Wmcue-r- (l-e-r-) +wmaz(l-e-r-) (3) To lowest order in 6.t and ~, Eqn. (3) yields .,. w m .... _ / ~t ~t 2 wi(k + 1) = wi(k) + 2y wi(k)Wmaz(-) + Wmaz(-) (4) T T Once again the change in Wi is proportional to the ith input. We have implemented the architecture of Fig. 7 using a SBN60:Ce crystal provided by the Rockwell International Science Center. We used the 488 nm line of an argon ion laser to record holograms in this crystal. Most of the patterns we considered were laid out on 10 x 10 grids of pixels, thus allowing 100 input channels. Ultimately, the number of channels which may be achieved using this architecture is limited by the number of pixels which may be imaged onto the crystal with a depth of focus sufficient to isolate each pixel along the length of the crystal. •• +.+ ... 1'1 I j 2 Y a. 8. 1 ! l t I ....... • • • ..... • • , 0 0 3 4 aCOftClS ~ W CIII) Figure 8. Training patterns. Figure 9. Output in the second training cycle. Using the variation on the perceptron learning algorithm described above with a fixed exposure times ~tr and ~te for recording and erasing, we have been able to correctly classify various sets of input patterns. One particular set which we used is shown in Fig. 8. In one training sequence, we grouped patterns 1 and 2 together with a high output and patterns 3 and 4 together with a low output. After all four patterns had been presented four times, the system gave the correct output for all patterns. The weights stored in the crystal were corrected seven times, four times by recording and three by erasing. Fig. 9a shows the output of the detector as pattern 1 is recorded in the second learning cycle. The dashed line in this figure corresponds to the threshold level. Fig. 9b shows the output of the detector as pattern 3 is erased in the second learning cycle. 386 CONCLUSION The experiments described in this paper demonstrate how neural network architectures can be implemented using currently available optical devices. By combining the recall dynamics of the first system with the learning capability of the second, we can construct sophisticated optical neural computers. ACKNOWLEDGEMENTS The authors thank Ratnakar Neurgaonkar and Rockwell International for supplying the SBN crystal used in our experiments and Hamamatsu Photonics K.K. for assistance with image intesifiers. We also thank Eung Gi Paek and Kelvin Wagner for their contributions to this research. This research is supported by the Defense Advanced Research Projects Agency, the Army Research Office, and the Air Force Office of Scientific Research. REFERENCES 1. Y. S. Abu-Mostafa and D. Psaltis, Scientific American, pp.88-95, March, 1987. 2. D. Psaltis and N. H. Farhat, Opt. Lett., 10,(2),98(1985). 3. A. D. Fisher, R. C. Fukuda, and J. N. Lee, Proc. SPIE 625, 196(1986). 4. K. Wagner and D. Psaltis, Appl. opt., 26(23), pp.5061-5076(1987). 5. D. Psaltis, D. Brady, and K. Wagner, Applied optics, March 1988. 6. D. Psaltis, J. Yu, X. G. Gu, and H. Lee, Second Topical Meeting on Optical Computing, Incline Village, Nevada, March 16-18,1987. 7. A. Yariv, S.-K. Kwong, and K. Kyuma, SPIE proc. 613-01,(1986). 8. D. Z. Anderson, Proceedings of the International Conference on Neural Networks, San Diego, June 1987. 9. A. B. Vander Lugt, IEEE Trans. Inform. Theory, IT-I0(2), pp.139145(1964). 10. E. G. Paek and D. Psaltis, Opt. Eng., 26(5), pp.428-433(1987). 11. Y. Owechko, G. J. Dunning, E. Marom, and B. H. Soffer, Appl. Opt. 26,(10) ,1900(1987). 12. D. Psaltis and J. Hong, Opt. Eng. 26,10(1987). 13. N. V. Kuktarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics, 22,949(1979). 14. J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51,1297(1980). 15. T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, Prog. Quan. Electr. 10,77(1985). 16. F. Rosenblatt, ' Principles of Neurodynamics: Perceptron and the Theory of Brain Mechanisms, Spartan Books, Washington,(1961).
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544 MURPHY: A Robot that Learns by Doing Bartlett W. Mel Center for Complex Systems Research University of Illinois 508 South Sixth Street Champaign, IL 61820 January 2, 1988 Abstract MURPHY consists of a camera looking at a robot arm, with a connectionist network architecture situated in between. By moving its arm through a small, representative sample of the 1 billion possible joint configurations, MURPHY learns the relationships, backwards and forwards, between the positions of its joints and the state of its visual field. MURPHY can use its internal model in the forward direction to "envision" sequences of actions for planning purposes, such as in grabbing a visually presented object, or in the reverse direction to "imitate", with its arm, autonomous activity in its visual field. Furthermore, by taking explicit advantage of continuity in the mappings between visual space and joint space, MURPHY is able to learn non-linear mappings with only a single layer of modifiable weights. Background Current Focus Of Learning Research Most connectionist learning algorithms may be grouped into three general catagories, commonly referred to as supenJised, unsupenJised, and reinforcement learning. Supervised learning requires the explicit participation of an intelligent teacher, usually to provide the learning system with task-relevant input-output pairs (for two recent examples, see [1,2]). Unsupervised learning, exemplified by "clustering" algorithms, are generally concerned with detecting structure in a stream of input patterns [3,4,5,6,7]. In its final state, an unsupervised learning system will typically represent the discovered structure as a set of categories representing regions of the input space, or, more generally, as a mapping from the input space into a space of lower dimension that is somehow better suited to the task at hand. In reinforcement learning, a "critic" rewards or penalizes the learning system, until the system ultimately produces the correct output in response to a given input pattern [8]. It has seemed an inevitable tradeoff that systems needing to rapidly learn specific, behaviorally useful input-output mappings must necessarily do so under the auspices of an intelligent teacher with a ready supply of task-relevant training examples. This state of affairs has seemed somewhat paradoxical, since the processes of Rerceptual and cognitive development in human infants, for example, do not depend on the moment by moment intervention of a teacher of any sort. Learning by Doing The current work has been focused on a fourth type of learning algorithm, i.e. learning-bydoing, an approach that has been very little studied from either a connectionist perspective © American Institute of Physics 1988 or in the context of more traditional work in machine learning. In its basic form, the learning agent • begins with a repertoire of actions and some form of perceptual input, • exercises its repertoire of actions, learning to predict i) the detailed sensory consequences of its actions, and, in the other direction, ii) its actions that are associated with incoming sensory patterns, and • runs its internal model (in one or both directions) in a variety of behaviorally-relevant tasks, e.g. to "envision" sequences of actions for planning purposes, or to internally "imitate" , via its internal action representation, an autonomously generated pattern of perceptual activity. In comparison to standard supenJised learning algorithms, the crucial property of learning-by-doing is that no intelligent teacher is needed to provide input-output pairs for learning. Laws of physics simply translate actions into their resulting percepts, both of which are represented internally. The learning agent need only notice and record this relationship for later use. In contrast to traditional unsupervised learning approaches, learning-by-doing allows the acquisition of specific, task-relevant mappings, such as the relationship between a simultaneously represented visual and joint state. Learning-bydoing differs as well from reinforcement paradigms in that it can operate in the absence of a critic, i.e. in situations where reward or penalty for a particular training instance may be inappropriate. Learning by doing may therefore by described as an unsupeMJised associative algorithm, capable of acquiring rich, task-relevant associations, but without an intelligent teacher or critic. Abridged History of the Approach The general concept of leaning by doing may be attributed at least to Piaget from the 1940's (see [9] for review). Piaget, the founder of the "constructivist" school of cognitive development, argued that knowledge is not given to a child as a passive observer, but is rather discovered and constructed by the child, through active manipulation of the environment. A handful of workers in artificial intelligence have addressed the issue of learning-by-doing, though only in highly schematized, simulated domains, where actions and sensory states are represented as logical predicates [10,11,12,13]. Barto & Sutton [14] discuss learning-by-doing in the context of system identification and motor control. They demonstrated how a simple simulated automaton with two actions and three sensory states can build a model of its environment through exploration, and subsequently use it to choose among behavioral alternatives. In a similar vein, Rumelhart [15] has suggested this same approach could be used to learn the behavior of a robot arm or a set of speech articulators. Furthermore, the forward-going "mental model" , once learned, could be used internally to train an inverse model using back-propagation. In previous work, this author [16] described a connectionist model (VIPS) that learned to perform 3-D visual transformations on simulated wire-frame objects. Since in complex sensory-motor environments it is not possible, in general, to learn a direct relationship between an outgoing command state and an incoming sensory state, VIPS was designed to predict changes in the state of its visual field as a function of its outgoing motor command. VIPS could then use its generic knowledge of motor-driven visual transformations to "mentally rotate" objects through a series of steps. 545 546 Recinolopic Visual RepresemaDOII --I Value-Coded loint RepresenlatiOll - -Figure 1: MURPHY's Connectionist Architecture. 4096 coarsely-tuned visual units are organized in a square, retinotopic grid. These units are bi-directionally interconnected with a population of 273 joint units. The joint population is subdivided into 3 sUbpopulations, each one a value-coded representation of joint angle for one of the three joints. During training, activity in the joint unit population determines the physical arm configuration. Inside MURPY The current work has sought to further explore the process of learning-by-doing in a complex sensory-motor domain, extending previous work in three ways. First, the learning of mappings between sensory and command (e.g. motor) representations should be allowed to proceed in both directions simultaneously during exploratory behavior, where each mapping may ultimately subserve a very different behavioral goal. Secondly, MURPHY has been implemented with a real camera and robot arm in order to insure representational realism to the greatest degree possible. Third, while the specifics of MURPHY's internal structures are not intended as a model of a specific neural system, a serious attempt has been made to adhere to architectural components and operations that have either been directly suggested by nervous system structures, or are at least compatible with what is currently known. Detailed biological justification on this point awaits further work. MURPHY's Body MURPHY consists of a 512 x 512 JVC color video camera pointed at a Rhino XR-3 robotic arm. Only the shoulder, elbow, and wrist joints are used, such that the arm can move only in the image plane of the camera. (A fourth, waist joint is not used). White spots are stuck to the arm in convenient places; when the image is thresholded, only the white spots appear in the image. This arrangement allows continuous control over the complexity of the visual image of the arm, which in turn affects time spent both in computing visual features and processing weights during learning. A Datacube image processing system is used for the thresholding operation and to "blur" the image in real time with a gaussian mask. The degree of blur is variable and can be used to control the degree of coarse-coding (i.e. receptive field overlap) in the camera-topic array of visual units. The arm is software controllable, with a stepper motor for each joint. Arm dynamics are not considered in this work. MURPHY's Mind MURPHY is currently based on two interconnected populations of neuron-like units. The first is organized as a rectangular, visuotopically-mapped 64 x 64 grid of coarsely-tuned visual units that each responds when a visual feature (such as a white spot on the arm) falls into its receptive field (fig. 1). Coarse coding insures that a single visual feature will activate a small population of units whose receptive fields overlap the center of stimulation. The upper trace in figure 2 shows the unprocessed camera view, and the center trace depicts the resulting pattern of activation over the grid of visual units. The second population of 273 units consists of three subpopulations, representing the angles of each of the three joints. The angle of each joint is value-coded in a line of units dedicated to that joint (fig. 1). Each unit in the population is "centered" at a some joint angle, and is maximally activated when the joint is to be sent to that angle. Neighboring joint units within a joint sub population have overlapping "projective fields" and progressively increasing joint-angle centers. It may be noticed that both populations of units are coarsely tuned, that is, the units have overlapping receptive fields whose centers vary in an orderly fashion from unit to neighboring unit. This style of representation is extremely common in biological sensory systems [17,18,19]' and has been attributed a number ofrepresentational advantages (e.g. fewer units needed to encode range of stimuli, increased immunity to noise and unit malfunction, and finer stimulus discriminations). A number of additional advantages of this type of encoding scheme are discussed in section, in relation to ease of learning, speed of learning, and efficacy of generalization. MURPHY's Education By moving its arm through a small, representative sample (approximately 4000) of the 1 billion possible joint configurations, MURPHY learns the relationships, backwards and forwards, between the positions of its joints and the state of its visual field. During training, the physical environment to which MURPHY's visual and joint representations are wired enforces a particular mapping between the states of these two representations. The mapping comprises both the kinematics of the arm as well as the optical parameters and global geometry of the camera/imaging system. It is incrementally learned as each unit in population B comes to "recognize" , through a process of weight modification, the states of population A in which it has been strongly activated. After sufficient training experience therefore, the state of popUlation A is sufficient to generate a "mental image" on population B, that is, to predictively activate the units in B via the weighted interconnections developed during training. In its current configuration, MURPHY steps through its entire joint space in around 1 hour, developing a total of approximately 500,000 weights between the two popUlations. The Learning Rule Tradeoffs in Learning and Representation It is well known in the folklore of connectionist network design that a tradeoff exists between the choice of representation (i.e. the "semantics") at the single unit level and the consequent ease or difficulty of learning within that representational scheme. At one extreme, the single-unit representation might be completely decoded, calling for a separate unit for each possible input pattern. While this scheme requires a combinatorially explosive number of units, and the system must "see" every possible input pattern during training, the actual weight modification rule is rendered very simple. At another extreme, the single unit representation might be chosen in a highly encoded fashion with complex interactions among input units. In this case, the activation of an output unit 547 548 may be a highly non-linear or discontinuous function of the input pattern, and must be learned and represented in mUltiple layers of weights. Research in connectionism has often focused on Boolean functions [20,21,1,22,23], typified by the encoder problem [20], the shifter problem [21] and n-bit parity [22]. Since Boolean functions are in general discontinuous, such that two input patterns that are close in the sense of Hamming distance do not in general result in similar outputs, much effort has been directed toward the development of sophisticated, multilayer weight-modification rules (e.g. back-propagation) capable of learning arbitrary discontinuous functions. The complexity of such learning procedures has raised troubling questions of scaling behavior and biological plausibility. The assumption of continuity in the mappings to be learned, however, can act to significationly simplify the learning problem while still allowing for full generalization to novel input patterns. Thus, by relying on the continuity assumption, MURPHY's is able to learn continuous non-linear functions using a weight modification procedure that is simple, locally computable, and confined to a single layer of modifiable weights. How MURPHY learns For sake of concrete illustration, MURPHY's representation and learning scheme will be described in terms of the mapping learned from joint units to visual units during training. The output activity of a given visual unit may be described as a function over the 3dimensional joint space, whose shape is determined by the kinematics of the arm, the location of visual features (i.e. white spots) on the arm, the global properties of the camera/imaging system, and the location of the visual unit's receptive field. In order for the function to be learned, a visual unit must learn to "recognize" the regions of joint space in which it has been visually activated during training. In effect, each visual unit learns to recognize the global arm configurations that happen to put a white spot in its receptive field. It may be recalled that MURPHY's joint unit population is value-coded by dimension, such that each unit is centered on a range of joint angles (overlapping with neighboring units) for one of the 3 joints. In this representation, a global arm configuration can be represented as the conjunctive activity of the k (where k = 3) most active joint units. MURPHY's visual units can therefore learn to recognize the regions of joint space in which they are strongly activated by simply ''memorizing'' the relevant global joint configurations as conjunctive clusters of input connections from the value-coded joint unit population. To realize this conjunctive learning scheme, MURPHY's uses sigma-pi units (see [24]), as described below. At training step S, the set of k most active joint units are first identified. Some subset of visual units is also strongly activated in state S, each one signalling the presence of a visual feature (such as a white spot) in its receptive fields. At the input to each active visual unit, connections from the k most highly active joint units are formed as a multiplicative k-tuple of synaptic weights. The weights Wi on these connections are initially chosen to be of unit strength. The output Cj of a given synaptic conjunction is computed by multiplying the k joint unit activation values Xi together with their weights: The output y of the entire unit is computed as a weighted sum of the outputs of each conjunction and then applying a sigmoidal nonlinearity: y = u(L WjCj). i Sigma-pi units of this kind may be thought of as a generalization of a logical disjunction of conjunctions (OR of AND's). The multiplicative nature of the conjunctive clusters insures that every input to the conjunct is active in order for the conjunct to have an effect on the unit as a whole. If only a single input to a conjunct is inactive, the effect of the conjunction is nullified. Specific-Instance Learning in Continuous Domains MURPHY's learning scheme is directly reminiscent of specific-instance learning as discussed by Hampson &: Vol per [23] in their excellent review of Boolean learning and representational schemes. Specific-instance learning requires that each unit simply ''memorize" all relevant input states, i.e. those states in which the unit is intended to fire. Unfortunately, simple specific-instance learning allows for no generalization to novel inputs, implying that each desired system responses will have to have been explicitly seen during training. Such a state of affairs is clearly impractical in natural learning contexts. Hampson &: Volper [23] have further shown that random Boolean functions will require an exponential number of weights in this scheme. For continous functions on the other hand, two kinds of generalization are possible within this type of specific-instance learning scheme. We consider each in turn, once again from the perspective of MURPHY's visual units learning to recognize the regions in joint space in which they are activated. Generalization by Coarse-Coding When a visual unit is activated in a given joint configuration, and acquires an appropriate conjunct of weights from the set of highly active units in the joint popUlation, by continuity the unit may assume that it should be at least partially activated in nearby joint configurations as well. Since MURPHY's joint units are coarse-coded in joint angle, this will happen automatically: as the joints are moved a small distance away from the specific training configuration, the output of the conjunct encoding that training configuration will decay smoothly from its maximum. Thus, a visual unit can "fire" predictively in joint configurations that it has never specifically seen during training, by interpolating among conjuncts that encode nearby joint configurations. This scheme suggests that training must be sufficiently dense in joint space to have seen configurations ''nearby'' to all points in the space by some criterion. In practice, the training step size is related to the degree of coarse-coding in the joint population, which is chosen in tUrn such that a joint pertubation equal to the radius of a joint unit's projective field (i.e. the range of joint angles over which the unit is active) should on average push a feature in the visual field a distance of about one visual receptive field radius. As a rule of thumb, the visual receptive field radius is chosen small enough so as to contain only a single feature on average. Generalization by Extrapolation The second type of generalization is based on a heuristic principle, again illustrated in terms of learning in the visual population. If a visual unit has during training been very often activated over a large, easy-to-specify region of joint space, such as a hyperrectangular region, then it may be assumed that the unit is activated over the entire region of joint space, i.e. even at points not yet seen. At the synaptic level, "large regions" can be represented as conjuncts with fewer terms. In its simplest form, this kind of generalization amounts to simply throwing out one or more joints as irrelevant to the activation of a given visual unit. What synaptic mechanism can achieve this effect? Competition among joint unit afferents can be used to drive irrelevant variables from the sigma-pi conjuncts. Thus, if a visual unit is activated repeatedly during training, and the elbow and shoulder angle units are constantly active while the most active wrist unit varies from step to step, then the weighted connections from the repeatedly activated elbow and shoulder units 549 550 -- . • • • • • • • • • • .. • • • • -, • .. I .; , til ,., , I Figure 2: Three Visual Traces. The top trace shows the unprocessed camera view of MURPHY's arm. White spots have been stuck to the arm at various places, such that a thresholded image contains only the white spots. This allows continuous control over the visual complexity of the image. The center trace represents the resulting pattern of activation over the 64 x 64 grid of coarsely-tuned visual units. The bottom trace depicts an internally-produced "mental" image of the arm in the same configuration as driven by weighted connections from the joint population. Note that the "mental" trace is a sloppy, but highly recognizable approximation to the camera-driven trace. will become progressively and mutually reinforced at the expense of the set of wrist unit connections, each of which was only activated a single time. This form of generalization is similar in function to a number of "covering" algorithms designed to discover optimal hyper-rectangular decompositions (with possible overlap) of a set of points in a multi-dimensional space (e.g. [25,26]). The competitive feature has not yet been implemented explicitly at the synaptic level, rather, the full set of conjuncts acquired during training are currently collapsed en masse into a more compact set of conjuncts, according to the above heuristics. In a typical run, MURPHY is able to eliminate between 30% and 70% of its conjuncts in this way. What MURPHY Does Grabbing A Visually Presented Target Once MURPHY has learned to image its arm in an arbitrary joint configuration, it can use heuristic search to guide its arm "mentally" to a visually specified goal. Figure 3(a) depicts a hand trajectory from an initial position to the location of a visually presented target. Each step in the trajectory represents the position of the hand (large blob) at an intermediate joint configuration. MURPHY can visually evaluate the remaining distance to the goal at each position and use best-first search to reduce the distance. Once a complete trajectory has been found, MURPHY can move its arm to the goal in a single physical step, dispensing with all backtracking dead-ends, and other wasted operations (fig. 3(b» . It would also be possible to use the inverse model, i.e. the map from a desired visual into an internal joint image, to send the arm directly to its final position. Unfortunately, MURPHY has no means in its current early state of development to generate a full-blown _ ... M1RPI!Y'. HInt&! Tra jeetory Figure 3: Grabbing a.n Object. (a) Irregular trajectory represents sequence of "mental" steps taken by MURPHY in attempting to "grab" a visually-presented target (shown in (b) as white cross). Mental image depicts MURPHY's arm in its final goal configuration, i.e. with hand on top of object. Coarse-coded joint activity is shown at right. (b) Having mentally searched a.nd found the target through a series of steps, MURPHY moves its arm phY6ically in a single step to the target, discarding the intermediate states of the trajectory that are not relevant in this simple problem. visual image of its arm in one of the final goal positions, of which there are many possible. Sending the tip of a robot arm to a given point in space is a classic task in robotics. The traditional approach involves first writing explicit kinematic equations for the arm based on the specific geometric details of the given arm. These equations take joint angles as inputs and produce manipUlator coordinates as outputs. In general, however, it is most often useful to specify the coordinates of the manipulator tip (i.e. its desired final position), and compute the joint angles necessary to achieve this goal. This involves the solution of the kinematic equations to generate an inverse kinematic model. Deriving such expressions has been called "the most difficult problem we will encounter" in vision-based robotics [27]. For this reason, it is highly desirable for a mobile agent to learn a model of its sensory-motor environment from scratch, in a way that depends little or not at all on the specific parameters of the motor apparatus, the sensory apparatus, or their mutual interactions. It is interesting to note that in this reaching task, MURPHY appears from the outside to be driven by an inverse kinematic model of its arm, since its first official act after training is to reach directly for a visually-presented object. While it is clear that best-first search is a weak method whose utility is limited in complex problem solving domains, it may be speculated that given the ability to rapidly image arm configurations, combined with a set of simple visual heuristics and various mechanism for escaping local minima (e.g. send the arm home), a number of more interesting visual planning problems may be within MURPHY's grasp, such as grabbing an object in the presence of obstacles. Indeed, for problems that are either difficult to invert, or for which the goal state is not fully known a priori, the technique of iterating a forward-going model has a long history (e.g. Newton's Method). 551 552 Imitating Autonomous Arm Activity A particularly interesting feature of "learning-by-doing" is that for every pair of unit populations present in the learning system, a mapping can be learned between them both backwards and forwards. Each such mapping may enable a unique and interesting kind of behavior. In MURPHY's case, we have seen that the mapping from a joint state to a visual image is useful for planning arm trajectories. The reverse mapping from a visual state to ajoint image has an altogether different use, i.e. that of "imitation". Thus, if MURPHY's arm is moved passively, the model can be used to ''follow'' the motion with an internal command (i.e. joint) trace. Or, if a substitute arm is positioned in MURPHY's visual field, MURPHY can "assume the position", i.e. imitate the model arm configuration by mapping the afferent visual state into a joint image, and using the joint image to move the arm. As of this writing, the implementation of this behavior is still somewhat unreliable. Discussion and Future Work In short, this work has been concerned with learning-by-doing in the domain of visionbased robotics. A number of features differentiate MURPHY from most other learning systems, and from other approaches to vision-based robotics: • No intelligent teacher is needed to provide input-ouput pairs. MURPHY learns by exercising its repertoire of actions and learning the relationship between these actions and the sensory images that result. • Mappings between populations of units, regardless of modality, can be learned in both directions simultaneously during exploratory behavior. Each mapping learned can support a distinct class of behaviorally useful functions. • MURPHY uses its internal models to first solve problems "mentally". Plans can therefore be developed and refined before they are actually executed. • By taking explicit advantage of continuity in the mappings between visual and joint spaces, and by using a variant of specific-instance learning in such a way as to allow generalization to novel inputs, MURHPY can learn "difficult" non-linear mappings with only a single layer of modifiable weights. Two future steps in this endeavor are as follows: • Provide MURPHY with direction-selective visual and joint units both, so that it may learn to predict relationships between rates of change in the visual and joint domains. In this way, MURPHY can learn how to perturb its joints in order to send its hand in a particular direction, greatly reducing the current need to search for hand trajectories. • Challenge MURPHY to grab actual objects, possibly in the presence of obstacles, where path of approach is crucial. The ability to readily envision intermediate arm configurations will become critical for such tasks. . Acknowledgements Particular thanks are due to Stephen Omohundro for his unrelenting scientific and moral support, and for suggesting vision and robotic kinematics as ideal domains for experimentation. References [1] T.J. Sejnowski & C.R. Roaenberg, Complex Systems, 1, 145, (1969). [2] G.J. Tesauro & T.J. Sejnowski. A parallel network that leMIUI to play backgammon. Submitted for publication. [3] S. Grossberg, BioI. Cybern., f3, 187, (1976). [4] T. Kohonen, Self organization and auociati"e memory., (Springer-Verlag, Berlin 1984). [5] D.E. Rumelhart & D. Zipser. In Parallel diatri6uted proceuing: e:rplorationa in tA.e microatructure oj cognition, "01. 1, D.E. Rumelhart, J.L. McClelland, Eds., (Bradford: Cambridge, MA, 1986), p. 151. [6] R. Linsker, Proc. Natl. Acad. Sci., 83, 8779, (1986). [7] G.E. Hinton & J.L. McClelland. Learning representations by recirculation. Oral presentation, IEEE conference on Neural Information Processing Systems, Denver, 1987. [8] A.G. Barto, R.S. Sutton, & C.W. Anderson, IEEE Trans. on Sy •. , Man, Cybern., amc-13, 834, (1983). [9] H. Ginsburg & S. Opper, Piaget'a tA.eor, oj intellectual de"elopment., (Prentice Hall, New Jersey, 1969). [10] J.D. Becker. In Computer modela Jor tA.ougA.t and language., R. Schank & K.M. Colby, Eds., (FreelllAIl, San Francisco, 1973). [11] R.L. Rivest & R.E. Schapire. In Proc. of the 4th into workshop on ma.ch.ine learning, 364-375, (1987). [12] J .G. Carbonell & Y. Gil. In Proc. of the 4th into workshop on machlne learning, 256-266, (1987). [13] K. Chen, Tech Report, Dept. of Computer Science, University of illinois, 1987. [14] A.G. Barto & R.S. Sutton, AFWAL-TR-81-1070, Avionics Laboratory, Air Force Wright Aeronautical Laboratories, Wright-Patterson AFB, Ohio 45433, 1981. [15] D.E. Rumelhart, "On learning to do what you want". Talk given at CMU Connectionist Summer School,1986a. [16] B.W. Melin Proc. of 8th Ann. Con!. of the Cognitive Science Soc., 562-571, (1986). [17] D.H. Ballard, G.E. Hinton, & T.J Sejnowski, Nature, 306, 21, (1983). [18] R.P. Erikson, American Scientist, May-June 1984, p. 233. [19] G.E. Hinton, J.L. McClelland, & D.E. Rumelhart. In Parallel diatri6uted proceuing: e:rplorationa in tA.e microatructure oj cognition, "01. 1, D.E. Rumelhart, J .L. McClelland, Eds., (Bradford, Cambridge, 1986), p. 77. [20] D.H. Ackley, G.E. Hinton, & T.J. Sejnowski, Cognitive Science, 9, 147, (1985). [21] G.E. Hinton & T .J. Sejnowski. In Parallel diatri6uted proceuing: e:rplorationa in tA.e microatructure oj cognition, "01. 1, D.E. Rumelhart, J.L. McClelland, Eds., (Bradford, Cambridge, 1986), p. 282. [22] G.J. Tes&llro, Complex Systems, 1,367, (1987). [23] S.E. Hampson & D.J. Volper, Biological Cybernetics, 56, 121, (1987). [24] D.E. Rumelhart, G.E. Hinton, & J.L. McClelland. In Parallel diatri6uted proceuing: e:rplorationa in tA.e microatructure oj cognition, "01. 1, D.E. Rumelhart, J.L. McClelland, Eds., (Bradford, Cambridge, 1986), p. 3. [25] R.S. Michalski, J.G. Carbonell, & T.M. Mitchell, Eds., MacA.ine learning: an artificial intelligence approacA., Vois. I and II, (Morgan KauflllAll, Los Altos, 1986). [26] S. Omohundro, Complex Systems, 1, 273, (1987). [27] Paul, R. R060t manipulatora: matA.ematica, programming, and control., (MIT Press, Cambridge, 1981). 553
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SPONTANEOUS AND INFORMATION-TRIGGERED SEGMENTS OF SERIES OF HUMAN BRAIN ELECTRIC FIELD MAPS 467 D. lehmann, D. Brandeis*, A. Horst, H. Ozaki* and I. Pal* Neurol09Y Department, University Hospital, 8091 Zurich, Switzerland ABSTRACT The brain works in a state-dependent manner: processin9 strate9ies and access to stored information depends on the momentary functional state which is continuously re-adjusted. The state is manifest as spatial confi9uration of the brain electric field. Spontaneous and information-tri9gered brain electric activity is a series of momentary field maps. Adaptive segmentation of spontaneous series into spatially stable epochs (states) exhibited 210 msec mean segments, discontinuous changes. Different maps imply different active neural populations, hence expectedly different effects on information processing: Reaction time differred between map classes at stimulus arrival. Segments might be units of brain information processin9 (content/mode/step), possibly operationalizin9 consciousness time. Related units (e.9. tri9gered by stimuli durin9 fi9ure perception and voluntary attention) mi9ht specify brain submechanisms of information treatment. BRAIN FUNCTIONAL STATES AND THEIR CHANGES The momentary functional state of the brain is reflected by the confi9uration of the brain's electro-ma9netic field. The state manifests the strate9Y, mode, step and content of brain information processing, and the state constrains the choice of strate9ies and modes and the access to memory material available for processin9 of incoming information (1). The constraints include the available range of changes of state in PAVLOV's classical ·orienting reaction" as response to new or important informations. Different states mi9ht be viewed as different functional connectivities between the neural elements. The orienting reaction (see 1,2) is the result of the first (Mpre-attentiveM) stage of information processing. This stage operates automatically (no involvement of consciousness) and in a parallel mode, and quickly determines whether (a) the information is important or unknown and hence requires increased attention and alertness, i.e. an orienting reaction which means a re-adjustment of functional state in order to deal adequately with the information invokin9 consciousness for further processing, or whether (b) the information is known or unimportant and hence requires no readjustment of state, i.e. that it can be treated further with well* Present addresses: D.B. at Psychiat. Dept., V.A. Med. Center, San Francisco CA 94121; H.O. at lab. Physiol. for the Developmentally Handicapped, Ibaraki Univ., Mito, Japan 310; I.P. at Biol09ic Systems Corp., Mundelein Il 60060. © American Institute of Physics 1988 468 established (·automatic·) strategies. Conscious strategies are slow but flexible (offer wide choice), automatic strategies are fast but rigid. Examples for functional states on a gross scale are wakefulness, drowsin.ss and sleep in adults, or developmental stages as infancy, childhood and adolesc.nce, or drug states induced by alcohol or other psychoactive agent •• The different states are associated with distinctly different ways of information processing. For example, in normal adults, reality-close, abstracting strategies based on causal relationships predominate during wakefulness, whereas in drowsiness and sleep (dreams), reality-remote, visualizing, associative concatenations of contents are used. Other well-known examples are drug states. HUMAN BRAIN ELECTRIC FIELD DATA AND STATES While alive, the brain produces an ever-changing el.ctromagnetic fi.ld, which very sensitively reflects global and local states as effected by spontaneous activity, incoming information, metabolism, drugs, and diseases. The .lectric component of the brain~s electromagnetic field as non-invasively measured from the intact human scalp shows voltages between 0.1 and 250 microVolts, temporal fr.quencies between 0.1 and 30, 100 or 3000 Hz depending on the examined function, and spatial frequencies up to 0.2 cycles/em. Brain electric field data are traditionally viewed as time series of potential differences betwe.n two scalp locations (the electroencephalogram or EE6). Time series analysis has offered an effective way to class different gross brain functional states, typically using EE6 power spectral values. Differences between power spectra during different gross states typically are greater than between different locations. States of lesser functional complexity such as childhood vs adult states, sleep vs wakefulness, and many drug-state. vs non-drug states tend to increased power in slower frequencies (e.g. 1,4). Time series analyses of epochs of intermediate durations between 30 and 10 seconds have demonstrated (e.g. 1,5,6) that there are significant and reliable relations between spectral power or coh.rency values of EE6 and characteristics of human mentation (reality-close thoughts vs free associations, visual vs non-visual thoughts, po.itive vs negative ~otions). Viewing brain electric field data as series of momentary field maps (7,8) opens the possibility to investigate the temporal microstructure of brain functional states in the sub-second range. The rationale is that the momentary configuration of activated neural elements represents a given brain functional state, and that the spatial pattern of activation is reflected by the momentary brain electric field which is recordable on the scalp as a momentary field map. Different configurations of activation (different field maps) are expected to be associated with different modes, strategies, steps and contents of information processing. 469 SE(J1ENTATI~ OF BRAIN ELECTRIC HAP SERIES INTO STABLE SE(J1ENTS When Viewing brain electric activity as series of maps of momentary potential distributions, changes of functional state are recognizable as changes of the ·electric landscapes· of these maps. Typically, several successive maps show similar landscapes, then quickly change to a new configuration which again tends to persist for a number of successive maps, suggestive of stable states concatenated by non-linear transitions (9,10). Stable map landscapes might be hypothesized to indicate the basic building blocks of information processing in the brain, the -atoms of thoughts·. Thus, the task at hand is the recognition of the landscape configurations; this leads to the adaptive segmentation of time series of momentary maps into segments of stable landscapes during varying durations. We have proposed and used a method which describes the configuration of a momentary map by the locations of its maximal and minimal potential values, thus invoking a dipole model. The goal here is the phenomenological recognition of different momentary functional states using a very limited number of major map features as classifiers, and we suggest conservative interpretion of the data as to real brain locations of the generating processes which always involve millions of neural elements. We have studied (11) map series recorded from 16 scalp locations over posterior skull areas from normal subjects during relaxation with closed eyes. For adaptive segmentation, the maps at the times of maximal map relief were selected for optimal signal/nOise conditions. The locations of the maximal and minimal (extrema) potentials were extracted in each map as descriptors of the landscape; taking into account the basically periodic nature of spontaneous brain electric activity (Fig. 1), extrema locations were treated disregarding polarity information. If over time an extreme left its pre-set spatial window (say, one electrode distance), the segment was terminated. The map series showed stable map configurations for varying durations (Fig. 2), and discontinuous, step-wise changes. Over 6 subjects, resting alpha-type EEG showed 210 msec mean segment duration; segments longer than 323 msec covered 50% of total time; the most prominent segment class (1.5% of all classes) covered 20% of total time (prominence varied strongly over classes; not all possible classes occurred). Spectral power and phase of averages of adaptive and pre-determined segments demonstrated the adequacy of the strategy and the homogeneity of adaptive segment classes by their reduced within-class variance. Segmentation using global map dissimilarity (sum of Euklidian difference vs average reference at all measured points) emulates the results of the extracted-characteristics-strategy. FUNCTIONAL SIGNIFICANCE OF MOMENTARY MICRO STATES Since different maps of momentary EEG fields imply activity of different neural populations, different segment classes must manifest different brain functional states with expectedly different 470 189 to 189 117 to 117 125 to 125 132 to 132 WItS 148 to 148 148 to 148 156 to 156 164 to 164 WItS 171 to 171 179 to 179 187 to 187 195 to 195 WItS RECORD=1 FILE=A:VP3EC2A NORMAL SUBJECT, EYES CLOSED Fig. 1. Series of momentary potential distribution maps of the brain field recorded from the scalp of a normal human during relaxation with closed eyes. Recording with 21 electrodes (one 5-electrode row added to the 16-electrode array in Fig. 2) using 128 samples/sec/ channel. Head seen from above, left ear left; white positive, dark negative, 8 levels from +32 to -32 microVolts. Note the periodic reversal of field polarity within the about 100 msec (one cycle of the 8-12Hz so-called ·EEG alpha- activity) while the field configuration remains largely constant. - This recording and display was done with a BRAIN ATLAS system (Biologic Systems, Mundelein, Il). effects on ongoing information processing. This was supported by measurements of selective reaction time to acoustic stimuli which were randomly presented to eight subjects during different classes of EEG segments (323 responses for each subject). We found significant reaction time differences over segment classes (ANOVA p smaller than .02), but similar characteristics over subjects. This indicates that the momentary sub-second state as manifest in the potential distribution map significantly influences the behavioral consequence of information reaching the brain. Presentation of information is followed by a sequence of potential distribution maps (Nevent-related potentials· or ERP's, averaged over say, 100 presentations of the same stimulus, see 12). The different spatial configurations of these maps (12) are thought to reflect the sequential stages of information processing associated with Mcomponents· of event-related brain activity (see e.g. 13) which are traditionally defined as times of maximal voltages after information input (maximal response strength). 45 '-X _--L...-7 2 ~~' : : , 5 ~,~.4 \' .~ ,.,. 55 ' 56 ' , " 3 l ~' / 3 471 4 5 59 'X _ .l..LU..L/ Fig. 2. Sequence of spatially stable segments durin9 a spontaneous series of momentary EEG maps of 3.1 sec duration in a normal volunteer. Each map shows the occurrence of the extreme potential values durin9 one adaptively determined segment: the momentary maps were searched for the locations of the two extreme potentials; these locations were accumulated, and linearly interpolated between electrodes to construct the present maps. (The number of isofrequency-of-occurrence lines therefore is related to the number of searched maps). - Head seen from above, left ear left, electrode locations indicated by crosses, most forward electrode at vertex. Data FIR filtered to 8-12Hz (alpha EEG). The fi9ure to the left below each map is a running segment number. The figure to the ri9ht above each map multiplied by 50 indicates the segment duration in msec. Application of the adaptive segmentation procedure described above for identification of functional components of event-related brain electric map sequences requires the inclusion of polarity information (14); such adaptive segmentation permits to separate different brain functional states without resortin9 to the strength concept of processing stages. An example (12) might illustrate the type of results obtained with this analysis: Given segments of brain activity which were triggered by visual information showed different map configurations when subjects paid attention vs when they paid no attention to the stimulus, and when they viewed figures vs meanin9less shapes as 472 LVF RVF Fig. 3. Four difference maps, computed as differences between maps obtained during (upper row) perception of a visual -illusionarytriangle figure (left picture) minus a visual non-figure (right) shown to the left and right visual hemi-fields (LVF, RVF) , and obtained during (lower row) attending minus during ignoring the presented display. The analysed segment covered the time from 168 to 200 msec after stimulus presentations. - Mean of 12 subjects. Head seen from above, left ear left, 16 electrodes as in Fig. 2, isopotential contour lines at 0.1 microVolt steps, dotted negative referred to mean of all values. The -illusionary- figure stimulus wa. studied by Kanisza (16); see also (12). - Note that the mirror symmetric configuration of the difference maps for LVF and RVF is found for the -figure- effect only, not for the -attention- effect, but that the anterior-posterior difference is similar for both cases. stimuli. Fig. 3 illustrates such differences in map configuration. The -attention--induced and -figureR-induced changes in map configuration showed certain similarities e.g. in the illustrated segment 168-200 msec after information arrival, supporting the hypothesis that brain mechanisms for figure perception draw on brain resources which in other circumstances are utilized in volontary attention. The spatially homogeneous temporal segments might be basic building blocks of brain information processing, possibly operationalizing consciousness time (15), and offering a common concept for analysis of brain spontaneous activity and event related brain potentials. The functional significance of the segments might be types/ modes/ steps of brain information processing or performance. Identification of related building blocks during different brain functions accordingly could specify brain submechanisms of information treatment. 473 Acknowledgement: Financial support by the Swiss National Science Foundation (including Fellowships to H.O. and I.P.) and by the 8HDO, the Hartmann Muller and the SANDOZ Foundation is gratefully acknowledged. REFERENCES 1. M. Koukkou and D. Lehmann, Brit. J. Psychiat. 142, 221-231 (1983). 2. A. Ohman, In: H.D. Kimmel, E.H. von Olst and J.F. Orlebeke (Eds.), Drug-Discrimination and State Dependent Learning (Academic Press, New York, 1979), pp. 283-318. 3. A. Katada, H. Ozaki, H. Suzuki and K. Suhara, Electroenceph. Clin. Neurophysiol. 52, 192-201 (1981). 4. M. Koukkou and D. Lehmann, BioI. Psychiat. 11, 663-677 (1976). 5. J. Berkhout, D.O. Walter and W.R. Adey, Electroenceph. clin. Neurophysiol. 27, 457-469 (1969). 6. P. Grass, D. Lehmann, B. Meier, C.A. Meier and I. Pal, Sleep Res. 16, 231 (1987). 7. D. Lehmann, Electroenceph. Clin. Neurophysiol. 31, 439-449 {1971). 8. D. Lehmann, In: H.H. Petsche and M.A.B. Brazier (eds.), Synchronization of EEG Activity in Epilepsies (Springer, Wien, 1972), pp. 307-326. 9. H. Haken, Advanced Synergetics (Springer, Heidelberg, 1983). 10. J.J. Wright, R.R. Kydd and G.L. Lees, BioI. Cybern., 1985, 53, 11-17. 11. D. Lehmann, H. Ozaki and I. Pal, Electroenceph. Clin. Neurophysiol. 67, 271-288 (1987). 12. D. Brandeis and D. Lehmann, Neuropsychologia 24, 151-168 (1986). 13. A.S. Gevins, N.H. Morgan, S.L. Bressler, B.A. Cutillo, R.M. White, J. Illes, D.S. Greer, J.C.Doyle and M. Zeitlin, Science 235,580-585 (1987). 14. D. Lehmann and W. Skrandies, Progr. Neurobiol. 23, 227-250 (1984). 15. B. Libet, Human Neurobiol. 1, 235-242 (1982). 16. G. Kanisza, Organization of Vision (Praeger, New York, 1979).
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82 SIMULATIONS SUGGEST INFORMATION PROCESSING ROLES FOR THE DIVERSE CURRENTS IN HIPPOCAMPAL NEURONS Lyle J. Borg-Graham Harvard-MIT Division of Health Sciences and Technology and Center for Biological Information Processing, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ABSTRACT A computer model of the hippocampal pyramidal cell (HPC) is described which integrates data from a variety of sources in order to develop a consistent description for this cell type. The model presently includes descriptions of eleven non-linear somatic currents of the HPC, and the electrotonic structure of the neuron is modelled with a soma/short-cable approximation. Model simulations qualitatively or quantitatively reproduce a wide range of somatic electrical behavior i~ HPCs, and demonstrate possible roles for the various currents in information processing. 1 The Computational Properties of Neurons There are several substrates for neuronal computation, including connectivity, synapses, morphometries of dendritic trees, linear parameters of cell membrane, as well as non-linear, time-varying membrane conductances, also referred to as currents or channels. In the classical description of neuronal function, the contribution of membrane channels is constrained to that of generating the action potential, setting firing threshold, and establishing the relationship between (steady-state) stimulus intensity and firing frequency. However, it is becoming clear that the role of these channels may be much more complex, resulting in a variety of novel "computational operators" that reflect the information processing occurring in the biological neural net. © American Institute of Physics 1988 2 Modelling Hippocampal Neurons Over the past decade a wide variety of non-linear ion channels, have been described for many excitable cells, in particular several kinds of neurons. One such neuron is the hippocampal pyramidal cell (HPC). HPC channels are marked by their wide range of temporal, voltage-dependent, and chemical-dependent characteristics, which results in very complex behavior or responses of these stereotypical cortical integrating cells. For example, some HPC channels are activated (opened) transiently and quickly, thus primarily affecting the action potential shape. Other channels have longer kinetics, modulating the response of HPCs over hundreds of milliseconds. The measurement these channels is hampered by various technical constraints, including the small size and extended electrotonic structure of HPCs and the diverse preparations used in experiments. Modelling the electrical behavior of HPCs with computer simulations is one method of integrating data from a variety of sources in order to develop a consistent description for this cell type. In the model referred to here putative mechanisms for voltage-dependent and calcium-dependent channel gating have been used to generate simulations of the somatic electrical behavior of HPCs, and to suggest mechanisms for information processing at the single cell level. The model has also been used to suggest experimental protocols designed to test the validity of simulation results. Model simulations qualitatively or quantitatively reproduce a wide range of somatic electrical behavior in HPCs, and explicitly demonstrate possible functional roles for the various currents [1]. The model presently includes descriptions of eleven non-linear somatic currents, including three putative N a+ currents - INa-trig, INa-rep, and INa-tail; six K+ currents that have been reported in the literature - IDR (Delayed Rectifier), lA, Ie, IAHP (After-hyperpolarization), 1M, and IQ; and two Ca2+ currents, also reported previously - lea and leas. The electrotonic structure of the HPC is modelled with a soma/shortcable approximation, and the dendrites are assumed to be linear. While the conditions for reducing the dendritic tree to a single cable are not met for HPC (the so-called Rall conditions [3]), the Zin of the cable is close to that of the tree. In addition, although HPC dendrites have non-linear membrane, it assumed that as a first approximation the contribution of currents from this membrane may be ignored in the somatic response to somatic stimulus. Likewise, the model structure assumes that axon-soma current under these conditions can be lumped into the soma circuit. 83 84 In part this paper will address the following question: if neural nets are realizable using elements that have simple integrative all-or-nothing responses, connected to each other with regenerative conductors, then what is the function for all the channels observed experimentally in real neurons? The results of this HPC model study suggest some purpose for these complexities, and in this paper we shall investigate some of the possible roles of non-linear channels in neuronal information processing. However, given the speculative nature of many of the currents that we have presented in the model, it is important to view results based on the interaction of the many model elements as preliminary. 3 Defining Neural Information Coding is the First Step in Describing Biological Computations Determination of computational properties of neurons requires a priori assumptions as to how information is encoded in neuronal output. The classical description assumes that information is encoded as spike frequency. However, a single output variable, proportional to firing frequency, ignores other potentially information-rich degrees of freedom, including: • Relative phase of concurrent inputs. • Frequency modulation during single bursts. • Cessation of firing due to intrinsic mechanisms. • Spike shape. Note that these variables apply to patterns of repetitive firingl. The relative phase of different inputs to a single cell is very important at low firing rates, but becomes less so as firing frequency approaches the time constant of the postsynaptic membrane or some other rate-limiting process in the synaptic transduction (e.g. neurotransmitter release or post synaptic channel activation/deactivation kinetics). Frequency modulation during bursts/spike trains may be important in the interaction of a given axon's output with other inputs at the target neuron. Cessation of firing due to mechanisms intrinsic to the cell (as opposed to the end of input) may be lSingle spikes may be considered as degenerate cases of repetitive firing responses. important, for example, in that cell's transmission function. Finally, modulation of spike shape may have several consequences, which will be discussed later. 4 Physiological Modulation of HPC Currents In order for modulation of HPC currents to be considered as potential information processing mechanisms in vivo, it is necessary to identify physiological modulators. For several of the currents described here such factors have been identified. For example, there is evidence that 1M is inhibited by muscarinic (physiologically, cholinergic) agonists [2], that 1A is inhibited by acetylcholine [6], and that 1AHP is inhibited by noradrenaline [5]. In fact, the list of neurotransmitters which are active non-synaptically is growing rapidly. It remains to be seen whether there are as yet undiscovered mechanisms for modulating other HPC currents, for example the three N a+ currents proposed in the present model. Some possible consequences of such mechanisms will be discussed later. 5 HPC Currents and Information Processing The role of a given channel on the HPC electrical response depends on its temporal characteristics as a function of voltage, intracellular messengers, and other variables. This is complicated by the fact that the opening and closing of channels is equivalent to varying conductances, allowing both linear and non-linear operations (e.g. [4] and [7]). In particular, a current which is activated/deactivated over a period of hundreds of milliseconds will, to a first approximation, act by slowly changing the time constant of the membrane. At the other extreme, currents which activate/deactivate with sub-millisecond time constants act by changing the trajectory of the membrane voltage in complicated ways. The classic example of this is the role of N a+ currents underlying the action potential. To investigate how the different HPC currents may contribute to the information processing of this neuron, we have looked at how each current shapes the HPC response to a simple repertoire of inputs. At this stage in our research the inputs have been very basic - short somatic current steps that evoke single spikes, long lasting somatic current steps that evoke spike trains, and current steps at the distal end of the dendritic cable. By examining the response to these inputs the functional roles of the HPC 85 86 I Current" Spike Shape I Spike Threshold I Tm/Frequency-Intensity I INa-trig + +++ INa-rep + ++ +++ ICa - (++) -(+) + (+++) IDR ++ + ++ IA + ++ ++ Ic + +++ IAHP ++ +++ 1M + + Table 1: Putative functional roles of HPC somatic currents. Entries in parentheses indicate secondary role, e.g. Ca2+ activation of J(+ current. currents can be tentatively grouped into three (non-exclusive) categories: • Modulation of spike shape. • Modulation of firing threshold, both for single and repetitive spikes. • Modulation of semi-steady-state membrane time constant. • Modulation of repetitive firing, specifically the relationship between strength of tonic input and frequency of initial burst and later "steady state" spike train. Table 1 summarizes speculative roles for some of the HPC currents as suggested by the simulations. Note that while all four of the listed characteristics are interrelated, the last two are particularly so and are lumped together in Table 1. 5.1 Possible Roles for Modulation of FI Characteristic Again, it has been traditionally assumed that neural information is encoded by (steady-state) frequency modulation, e.g. the number of spikes per second over some time period encodes the output information of a neuron. For example, muscle fiber contraction is approximately proportional to the spike frequency of its motor neuron 2. If the physiological inhibition of a specific 2In fact, where action potential propagation is a stereotyped phenomena, such as in long axons, then the timing of spikes is the only parameter that may be modulated. . ~ .. --'--;......... , , , , , , , , , , , Stimulus Intensity (Constant Current) \ \ \ \ \ Figure 1: Classical relation between total neuronal input (typically tonic current stimulus) and spike firing frequency [solid line] and (qualitative) biological relationships [dashed and dotted lines]. The dotted line applies when INa-rep is blocked. current changes the FI characteristic, this allows one way to modulate that neuron's information processing by various agents. Figure 1 contrasts the classical input-output relation of a neuron and more biological input-output relations. The relationships have several features which can be potentially modulated either physiologically or pathologically, including saturation, threshold, and shape of the curves. Note in particular the cessation of output with increased stimulation, as the depolarizing stimulus prevents the resetting of the transient inward currents. For the HPC, simulations show (Figure 2 and Figure 3) that blocking the putative INa-rep has the effect of causing the cell to "latch-up" in response to tonic stimulus that would otherwise elicit stable spike trains. Both depolarizing currents and repolarizing currents playa role here. First, spike upstroke is mediated by both INa-rep and the lower threshold INa-trig; at high stimuli repolarization between spikes does not get low enough to reset INa-trig' Second, spikes due to only one of these N a+ currents are weaker and as a result do not activate the repolarizing [(+ currents as much as normal because a) reduced time at depolarized levels activates the voltagedependent [(+ currents less and b) less Ca2+ influx with smaller spikes reduces the Ca2+ -dependent activation of some [(+ currents. The net result is that repolarization between spikes is weaker and, again, does not reset INa-trig. Although the current being modulated here (INa-rep) is theoretical, the 87 88 Voltage (nV) Tine (sec) (x 1.ge-3) ,~ ;299.9 499.9 699.9 ,899.9 ~~VL--2 nA Stinulus, Nornal '--Vo leage (nV) Tine (sec) (x 1.ge-3) b~ ,299.9 499.9 699.9 ,899.9 I!J(~VVVVVVL.--'L.--V--V--~~N~ Voltage (P'lV) Tine (se~ (x 1.ge-3) h~ ,299.9 499.9 699.9 99.9 I I ~VVl/VvI/\/VV\/\/VVVVV1.,/VVVVV-~~ 6 nA StiP'lulus, Nornal Figure 2: Simulation of repetitive firing in response to constant current injection into the soma. In this series, with the "normal" cell, a stimulus of about 8 nA (not shown) will cause to cell to fire a short burst and then cease firing. possibility of selective blocking of INa-rep allows a mechanism for shifting the saturation of the neuron's response to the left and, as can be seen by comparing Figures 2 and 3, making the FI curve steeper over the response range. 5.2 Possible Roles for Modulation of Spike Threshold The somatic firing threshold determines the minimal input for eliciting a spike, and in effect change the sensitivity of a cell. As a simple example, blocking INa-trig in the HPe model raises threshold by about 10 millivolts. This could cause the cell to ignore input patterns that would otherwise generate action potentials. There are two aspects of the firing "threshold" for a cell - static and dynamic. Thus, the rate at which the soma membrane approaches threshold is important along with the magnitude of that threshold. In general the threshold level rises with a slower depolarization for several reasons, including partial inactivation of inward currents (e.g. INa-trig) and partial activation of outward currents (e.g. IA [8]) at subthreshold levels. Tine (sec) (x 1.ge-3) 499.9 99.9 899.9 2 nA Stinulus, u~o I-Na-Rep 4 nA Stinulus, u~o I-Na-Rep Tine (sec) ex 1.ge-3} 499 . 9 99.9 899.9 -89.9 6 nA Stinulus, u~o I-Na-Rep Figure 3: Blocking one of the putative N a+ currents (INa-rep) causes the HPC repetitive firing response to fail at lower stimulus than "normal". This corresponds to the leftward shift in the saturation of the response curve shown in Figure 1. Thus it is possible, for example, that IA helps to distinguish tonic dendritic distal synaptic input from proximal input. For input that eventually will supply the same depolarizing current at the soma, dendritic input will have a slower onset due to the cable properties of the dendrites. This slow onset could allow IA to delay the onset of the spike or spikes. A similar depolarizing current applied more proximally would have a faster onset. Sub-threshold activation of IA on the depolarizing phase would then be insufficient to delay the spike. 5.3 Possible Roles for Modulation of Somatic Spike Shape How important is the shape of an individual spike generated at the soma? First, we can assume that spike shape, in particular spike width, is unimportant at the soma spike-generating membrane - once the soma fires, it fires. However, the effect of the spike beyond the soma mayor may not depend on the spike shape, and this is dependent on both the degree which spike propagation is linear and on the properties of the pre-synaptic membrane. Axon transmission is both a linear and non-linear phenomena, and the shorter the axon's electrotonic length, the more the shape of the somatic 89 90 action potential will be preserved at the distal pre-synaptic terminal. At one extreme, an axon could transmit the spike a purely non-linear fashion - once threshold was reached, the classic "all-or-nothing" response would transmit a stereotyped action potential whose shape would be independent of the post-threshold soma response. At the other extreme, i.e. if the axonal membrane were purely linear, the propagation of the somatic event at any point down the axon would be a linear convolution of the somatic signal and the axon cable properties. It is likely that the situation in the brain lies somewhere between these limits, and will depend on the wavelength of the spike, the axon non-linearities and the axon length. What role could be served by the somatic action potential shape modulating the pre-synaptic terminal signal? There are at least three possibilities. First, it has been demonstrated that the release of transmitter at some presynaptic terminals is not an "all-or-nothing" event, and in fact is a function of the pre-synaptic membrane voltage waveform. Thus, modulation of the somatic spike width may determine how much transmitter is released down the line, providing a mechanism for changing the effective strength of the spike as seen by the target neuron. Modulation of somatic spike width could be equivalent to a modulation ofthe "loudness" of a given neuron's message. Second, pyramidal cell axons often project collateral branches back to the originating soma, forming axo-somatic synapses which result in a feedback loop. In this case, modulation of the somatic spike could affect this feedback in complicated ways, particularly since the collaterals are typically short. Finally, somatic spike shape may also playa role in the transmission of spikes at axonal branch points. For example, consider a axonal branch point with an impedance mismatch and two daughter branches, one thin and one thick. Here a spike that is too narrow may not be able to depolarize the thick branch sufficiently for transmission of the spike down that branch, with the spike propagating only down the thin branch. Conversely, a wider spike may be passed by both branches. Modulation of the somatic spike shape could then be used to direct how a cell's output is broadcast, some times allowing transmission to all the destinations of an HPC , and at other times inhibiting transmission to a limited set of the target neurons. For HPCs much evidence has been obtained which implicate the roles of various HPC currents on modulating somatic spike shape, for example the Ca2+ -dependent K+ current Ie [9]. Simulations which demonstrate the effect of Ie on the shape of individual action potentials are shown in Figure 4. Volt"9" (!'IV) -81l.1l Tin" (~"c) (x 1.9,,-3) Il 3.1l 4. 9 S.1l " . ,:" . 1'\ .... , : \ ... ... I" .. Ti~,, : (~ec) (x.1.1l,,-3) .Il • .9 3.'8- .'\. • .il"_".5.1l I-Na-Tris -11l.1l -_. I-DR " .. " I-C Volts" (nU) Tin" (~"c) (x 1.9,,-3) .9 Il 3.1l 4.1l 5.9 -81l.9 Curr"nt (nA)" 1l.1l " " I Figure 4: Role of Ie during repolarization of spike. In the simulation on the left, Ie is the largest repolarizing current. In the simulation on the right, blocking Ie results in an wider spike. 6 The Assumption of Somatic Vs. Non-Somatic Currents In this research the somatic response of the HPC has been modelled under the assumption that the data on HPC currents reflect activity of channels localized at the soma. However, it must be considered that all channel proteins, regardless of their final functional destination, are manufactured at the soma. Some of the so-called somatic channels may therefore be vestiges of channels intended for dendritic, axonal, or pre-synaptic membrane. For example, if the spike-shaping channels are intended to be expressed for pre-synaptic membrane, then modulation of these channels by endogenous factors (e.g. ACh) takes place at target neuron. This may seem disadvantageous if a factor is to act selectively on some afferent tract. On the other hand, in the dendritic field of a given neuron it is possible only some afferents have certain channels, thus allowing selective response to modulating agents. These possibilities further expand the potential roles of membrane channels for computation. 91 92 7 Other Possible Roles of Currents for Modulating HPC Response There are many other potential ways that HPC currents may modulate the HPC response. For example, the relationship between intracellular Ca2+ and the Ca2+-dependent K+ currents, Ic and IAHP, may indicate possible information processing mechanisms. Intracellular Ca2+ is an important second messenger for several intracellular processes, for example muscular contraction, but excessive [Ca2+]in is noxious. There are at least three negative feedback mechanisms for limiting the flow of Ca2+ : voltage-dependent inactivation of Ca2+ currents; reduction of ECa (and thus the Ca2+ driving force) with Ca2+ influx; and the just mentioned Ca2+ -mediation of repolarizing currents. A possible information processing mechanism could be by modulation of IAHP, which plays an important role in limiting repetitive firing;. Simulations suggest that blocking this current causes Ic to step in and eventually limit further repetitive firing, though after many more spikes in a train. Blocking both these currents may allow other mechanisms to control repetitive firing, perhaps ones that operate independently of [Ca2+]in. Conceivably, this could put the neuron into quite a differen t operating region. 8 Populations of Neurons V s. Single Cells: Implications for Graded Modulation of HPC Currents In this paper we have considered the all-or-nothing contribution of the various channels, Le. the entire population of a given channel type is either activated normally or all the channels are disabled/blocked. This description may be oversimplified in two ways. First, it is possible that a blocking mechanism for a given channel may have a graded effect. For example, it is possible that cholinergic input is not homogeneous over the soma membrane, or that at a given time only a portion of these afferents are activated. In either case it is possible that only a portion of the cholinergic receptors are bound, thus inhibiting a portion of channels. Second, the result of channel inhibition by neuromodulatory projections must consider both single cell 3The slowing down of the spike trains in Figure 2 and Figure 3 is mainly due to the buildup of [Ca 2+];n, which progressively activates more IAHP. response and population response, the size of the population depending on the neuro-architecture of a cortical region and the afferents. For example, activation of a cholinergic tract which terminates in a localized hippocampal region may effect thousands of HPCs. Assuming that the 1M of individual HPCs in the region may be either turned on or off completely with some probability, the behavior of the population will be that of a graded response of 1M inhibition. This graded response will in turn depend on the strength of the cholinergic tract activity. The key point is that the information processing properties of isolated neurons may be reflected in the behavior of a population, and vica-versa. While it is likely that removal of a single pyramidal cell from the hippocampus will have zero functional effect, no neuron is an island. Understanding the central nervous system begins with the spectrum of behavior in its functional units, which may range from single channels, to specific areas of a dendritic tree, to the single cell, to cortical or nuclear subfields, on up through the main subsystems of CNS. References [1] L. Borg-Graham. Modelling the Somatic Electrical Behavior of Hippocampal Pyramidal Neurons. Master's thesis, Massachusetts Institute of Technology, 1987. [2] J. Halliwell and P. Adams. Voltage clamp analysis of muscarinic excitation in hippocampal neurons. Brain Research, 250:71-92, 1982. [3] J. J. B. Jack, D. Noble, and R. W. Tsien. Electric Current Flow In Excitable Cells. Clarendon Press, Oxford, 1983. [4] C. Koch and T. Poggio. Biophysics of computation: neurons, synapses and membranes. G. B.!. P. Paper, (008), 1984. Center for Biological Information Processing, MIT. [5] D. Madison and R. Nicoll. Noradrenaline blocks accommodation ofpyramidal cell discharge in the hippocampus. Nature, 299:, Oct 1982. [6] Y. Nakajuma, S. Nakajima, R. Leonard, and K. Yamaguchi. Actetylcholine inhibits a-current in dissociated cultured hippocampal neurons. Biophysical Journal, 49:575a, 1986. 93 94 [7] T. Poggio and V. Torre. Theoretical Approaches to Complex Systems, Lecture Notes in Biomathematics, pages 28- 38. Volume 21, Springer Verlag, Berlin, 1978. A New Approach to Synaptic Interaction. [8] J. Storm. A-current and ca-dependent transient outward current control the initial repetitive firing in hippocampal neurons. Biophysical Journal, 49:369a, 1986. [9] J. Storm. Mechanisms of action potential repolarization and a fast afterhyperpolarization in rat hippocampal pyramidal cells. Journal of Physiology, 1986.
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AN ARTIFICIAL NEURAL NETWORK FOR SPATIOTEMPORAL BIPOLAR PATTERNS: APPLICATION TO PHONEME CLASSIFICATION Toshiteru Homma Les E. Atlas Robert J. Marks II Interactive Systems Design Laboratory Department of Electrical Engineering, Ff-l0 University of Washington Seattle, Washington 98195 ABSTRACT An artificial neural network is developed to recognize spatio-temporal bipolar patterns associatively. The function of a formal neuron is generalized by replacing multiplication with convolution, weights with transfer functions, and thresholding with nonlinear transform following adaptation. The Hebbian learning rule and the delta learning rule are generalized accordingly, resulting in the learning of weights and delays. The neural network which was first developed for spatial patterns was thus generalized for spatio-temporal patterns. It was tested using a set of bipolar input patterns derived from speech signals, showing robust classification of 30 model phonemes. 1. INTRODUCTION 31 Learning spatio-temporal (or dynamic) patterns is of prominent importance in biological systems and in artificial neural network systems as well. In biological systems, it relates to such issues as classical and operant conditioning, temporal coordination of sensorimotor systems and temporal reasoning. In artificial systems, it addresses such real-world tasks as robot control, speech recognition, dynamic image processing, moving target detection by sonars or radars, EEG diagnosis, and seismic signal processing. Most of the processing elements used in neural network models for practical applications have been the formal neuronl or" its variations. These elements lack a memory flexible to temporal patterns, thus limiting most of the neural network models previously proposed to problems of spatial (or static) patterns. Some past solutions have been to convert the dynamic problems to static ones using buffer (or storage) neurons, or using a layered network with/without feedback. We propose in this paper to use a "dynamic formal neuron" as a processing element for learning dynamic patterns. The operation of the dynamic neuron is a temporal generalization of the formal neuron. As shown in the paper, the generalization is straightforward when the activation part of neuron operation is expressed in the frequency domain. Many of the existing learning rules for static patterns can be easily generalized for dynamic patterns accordingly. We show some examples of applying these neural networks to classifying 30 model phonemes. © American Institute of Physics 1988 32 2. FORMAL NEURON AND DYNAMIC FORMAL NEURON The formal neuron is schematically drawn in Fig. l(a), where Input Activation Output Transmittance Node operator Neuron operation r = [Xl Xz ... xd1 Yi' i = 1,2 •... • N Zi, i = 1,2. . . . • N W = [Wil WiZ ... wiLf 11 where 11(') is a nonlinear memory less transform Zi = 11(wTr> (2.1) Note that a threshold can be implicitly included as a transmittance from a constant input. In its original form of formal neuron, Xi E {O,I} and 110 is a unit step function u ('). A variation of it is a bipolar formal neuron where Xi E {-I, I} and 110 is the sign function sgn O. When the inputs and output are converted to frequency of spikes, it may be expressed as Xi E Rand 110 is a rectifying function rO. Other node operators such as a sigmoidal function may be used. We generalize the notion of formal neuron so that the input and output are functions of time. In doing so, weights are replaced with transfer functions, multiplication with convolution, and the node operator with a nonlinear transform following adaptation as often observed in biological systems. Fig. 1 (b) shows a schematic diagram of a dynamic formal neuron where Input Activation Output Transfer function Adaptation Node operator Neuron operation r(l) = [Xl(t) xz(t) ... xdt)f Yi(t), i == 1,2 •... . N Zi(t), i = 1,2 •... • N w(t) = [Wjl(t) wiZ(t) ... WiL(t)]T ai (t) 1l where 110 is a nonlinear memoryless transform Zj(t) = ll(ai (-t). W;(t)T .x(t» (2.2) For convenience, we denote • as correlation instead of convolution. Note that convolving a(t) with b(t) is equivalent to correlating a( -t) with b(t). If the Fourier transforms r(f)=F{r(t)}, w;(f)=F{W;(t)}, Yj(f)=F{Yi(t)}, and aj(f) = F {ai(t)} exist, then Yi (f) = ai (f) [Wi (f fT r(f)] (2.3) where Wi (f fT is the conjugate transpose of Wi (t). x,(1) I----zt • 1----zt(I) (b) Fig. 1. Formal Neuron and Dynamic Formal Neuron. 33 3. LEARNING FOR FORMAL NEURON AND DYNAMIC FORMAL NEURON A number of learning rules for formal neurons has been proposed in the past. In the following paragraphs, we formulate a learning problem and describe two of the existing learning rules, namely, Hebbian learning and delta learning, as examples. Present to the neural network M pairs of input and desired output samples {X<k), (lk)}, k ::;: 1,2, ... ,M , in order. Let W(k)::;: [w/k) w!k) '" wJk~T where wr) is the transmittance vector at the k-th step of learning. Likewise, let K (k) = [X<I) x'-2) ... X<k)], r(k) = rfl) t 2) ... t k)], ~(k) = [z<I) z<2) ... ~k)], and D(k) = [(ll) (l2) '" (lk)] , where 'Ik) = W(k)x'-k), z<k) = n<tk», and n<Y> = [T1(Y I) T1(Y2) .. . T1(yN)]T. The Hebbian learning rule 2 is described as follows *: W(k) ::;: W(k-I) + a;JC.k)X<k)T (3.1) The delta learning (or LMS learning) rule3,4 is described as follows: W(k) = W(k-I) _ o.{W(k-l)t:k) _ (lk)}X<k)T (3.2) The learning rules described in the previous section are generalized for the dynamic formal neuron by replacing multiplication with correlation. First, the problem is reformulated and then the generalized rules are described as follows. Present to the neural network M pairs of time-varing input and output samples {X<k)(t), (lk)(t)), k = 1,2, .. . ,M , in order. Let W(k)(t) = [WI(t)(k)(t) w~k)(t)·· . wJk)(t)f where w/k)(t) is the vector whose elements W;)t)(t) are transfer functions connecting the input j to the neuron i at the k-th step of learning. The Hebbian learning rule for the dynamic neuron is then W(kl(t) = W(k-I)(t) + 0.(-1}. (lk)(t). X<k)(t)T . (3.3) The delta learning rule for dynamic neuron is then W(kl(t) ::;: W(k-I)(t) - o.(-t). {W(k-Il(t). X<k)(t) - (It)(t)} .X<k)(t)T . (3.4) This generalization procedure can be applied to other learning rules in some linear discriminant systems5 , the self-organizing mapping system by Kohonen6 , the perceptron 7 , the backpropagation model3 , etc. When a system includes a nonlinear operation, more careful analysis is necesssay as pointed out in the Discussion section. 4. DELTA LEARNING,PSEUDO INVERSE AND REGULARIZATION This section reviews the relation of the delta learning rule to the pseudo-inverse and the technique known as regularization.4, 6, 8, 9,10 Consider a minimization problem as described below: Find W which minimizes R = LII'Ik) - (lk)U i = <f-k) - (lky <tk) - (lk» (4.1) subject to t k) = WX<k) • A solution by the delta rule is, using a gradient descent method, W(k) = W(k-I) _ o.-1...R(k) aw (4.2) • This interpretation assumes a strong supervising signal at the output while learning. 34 where R (k) = II y<k) ... ~A:)1I1. The minimum norm solution to the problem, W*, is unique and can tie expressed as W* == D xt (4.3) where !. t is the Moore-Penrose pseudo-inverse of!. , i.e., X t = lim(XTX + dl/)-lXT = limXT (X XT + dl/)-l. a-.o a-+O(4.4) On the condition that 0 < a < ~ where An- is the max.imum eigenvalue of !.T!., J'.k) and (jC.k) are independent, and WCl) is uncorrelated with ~l), E {W*} = E (~c .. )} (4.5) where E {x} denotes the expected value of x. One way to make use of this relation is to calculate W* for known standard data and refine it by (4.2), thereby saving time in the early stage of learning. However, this solution results in an ill-conditioned W often in practice. When the problem is ill-posed as such, the technique known as regularization can alleviate the ill-conditioning of W . The problem is reformulated by finding W which minimizes R(a) = Dly<t) - (jC.k)IIl + dlLII wkll 1 (4.6) 1 k subject to t k ) = ~k) where W = [Wlw2 ... WN]T . This reformulation regularizes (4.3) to W (a) = D!.T (!.!.T + a2n-1 (4.7) which is statistically equivalent to Wc .. ) when the input has an additive noise of variance dl utlcorrelated with ~l). Interestingly, the leaky LMS algorithmll leads to a statistically equivalent solution W(l) = ~WCk-l) _ tx~(k-l)~l) - {jC.l)};f<l)T (4.8) 2 whete 0 < ~ < 1 and 0 < a < Amax • E {W(a)} = E {Wc .. )} These solutions are related as if dl = ..!::J! when WCl) is uncorrelated with ;f<l) .11 a (4.9) Equation (4.8) can be generalized for a network using dynamic formal neurons, resulting in a equation similar to (3.4). Making use of (4.9), (4.7) can be generalized for a dynamic neuron network as W (t ; a) = F-1 {Q if )!. if fT (!. if )!. if)CT + a2n-1} where F-1 denotes the inverse Fourier transform. s. SYNTHESIS OF BIPOLAR PHONEME PATTERNS (4.10) This section illustrates the scheme used to synthesize bipolar phoneme patterns and to form prototype and test patterns. The fundamental and first three formant frequencies, along with their bandwidths, of phonemes provided by Klattl2 were taken as parameters to synthesize 30 prototype phoneme patterns. The phonemes were labeled as shown in Table 1. An array of L (=100) input neurons OOVered the range of 100 to 4000 Hz. Each neuron had a bipolar state which was + 1 only when one of the frequency bands in the phoneme presented to the network was within the critical band 35 of the neuron and -1 otherwise. The center frequencies if e) of critical bands were obtained by dividing the 100 to 4000 Hz range into a log scale by L. The critical bandwidth was a constant 100 Hz up to the center frequency Ie = 500 Hz and 0.2/e Hz when Ie >500 Hz.13 The parameters shown in Table 1 were used to construct Table 1. Labels of Phonemes 30 prototype phoneme patterns. For 9. it was constructed as a combination of t and 9. Fl. F 2 .F 3 were the first. second. and third formants. and B I' B 2. and B 3. were corresponding bandwidths. The fundamental frequency F 0 = 130 Hz with B 0 = 10 Hz was added when the phoneme was voiced. For plosives. there was a stop before formant traces start. The resulting bipolar patterns are shown in Fig.2. Each pattern had length of 5 time units, composed by linearly interpolating the frequencies when the formant frequency was gliding. Label Phoneme A sequence of phonemes converted from a continuous pronunciation of digits, {o, zero, one, two, three. four, five, six. seven, eight, nine }, was translated into a bipolar pattern, adding two time units of transition between two consequtive phonemes by interpolating the frequency and bandwidth parameters linearly. A flip noise was added to the test pattern and created a noisy test pattern. The sign at every point in the original clean test pattern was flipped with the probability 0.2. These test patterns are shown in Fig. 3. I'IlDM_ Label I 1 5 7 , II Il 15 .7 ., JI 21 Z5 17 It 2 4 , I II 11 14 16 II II U 14 I' II JO II. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Fig. 2. Prototype Phoneme Patterns. (Thirty phoneme patterns are shown in sequence with intervals of two time units.) [iY] [Ia] leY] [Ea] [3e'] [el] [~] [It ] [ow] [\I~] [uw] [a;J [a ] [aWl loY] [w] [y] [r] [I] [f] [v] [9] [\] [s] [z] [p] [t] [d] [k] [n] 6. SIMULATION OF SPATIO-TEMPORAL FILTERS FOR PHONEME CLASSIFICATION The network system described below was simulated and used to classify the prototype phoneme patterns in the test patterns shown in the previoius section. It is an example of generalizing a scheme developed for static patterns13 to that for dynamic patterns. Its operation is in two stages. The first stage operation is a spatio-temporal filter bank: 36 • e = z .. 4 • (a) !! ~ :! ! • I ' , I "I ' , I ' 'I • if ~.. lU U' (b) Fig. 3. Test Patterns. (a) Clean Test Pattern. (b) Noisy Test Pattern. 1(t) = W(t).r(t) , and r(t) = !:(a(-t)y(t» . The second stage operation is the "winner-take-all" lateral inhibition: (/(t) = zt(t) , and (/(t+A) = !:(~(-t).(/(t) - Ii), and 114 A(t) = (1 + -)/O(t) - -S "fiI' 2,O(t-nA). SN N 11=0 (6.1) (6.2) (6.3) where Ii is a constant threshold vector with elements hi = h and 0(.) is the Kronecker delta function. This operation is repeated a sufficient number of times, No .13,14 The output is (/(t + No ·A). Two models based on different leaming rules were simulated with parameters shown below. Model 1 (Spatio-temporal Matched Filter Bank) Let a(t) = O(t) , (/tk) = et in (3.3) where ek is a unit vector with its elements eki W(t)=!(t)T. 4 1 h=200, and a(t) = 2,-O(t-nA). 11=0 S Model 2 (Spatio-temporal Pseudo-inverse Filter) Let D = L in (4.10). Using the alternative expression in (4.4), W (t) = F-1{(! (j fT! (j) + cr2n-lXCT}. h = O.OS ,cr2 = 1000.0, and a(t) = O(t). This minimizes R (cr,!) = DI1k)(j) - (/t")(j )lIi + cr22,11 w" if )lIi for all ! . " k = O(k-i) . (6.4) (6.5) (6.6) 37 Because the time and frequency were finite and discrete in simulation, the result of the inverse discrete Fourier transform in (6.5) may be aliased. To alleviate the aliasing, the transfer functions in the prototype matrix:! (t) were padded with zeros, thereby doubling the lengths. Further zero-padding the transfer functions did not seem to change teh result significantly. The results are shown in Fig. 4(a)-(d). The arrows indicate the ideal response positions at the end of a phoneme. When the program was run with different thresholds and adaptation function a (t), the result was not very sensitive to the threshold value, but was, nevertheless affected by the choice of the adaptation function. The maximum number of iterations for the lateral inhibition network to converge was observed: for the experiments shown in Fig. 4(a) - (d), the numbers were 44, 69, 29, and 47, respectively. Model 1 missed one phoneme and falsely responded once in the clean test pattern. It missed three and had one false response in the noisy test pattern. Model 2 correctly recognized all phonemes in the clean test pattern, and falsealarmed once in the noisy test pattern. 7. DISCUSSION The notion of convolution or correlation used in the models presented is popular in engineering disciplines and has been applied extensively to designing filters, control systems, etc. Such operations also occur in biological systems and have been applied to modeling neural networks.IS,16 Thus the concept of dynamic formal neuron may be helpful for the improvement of artificial neural network models as well as the understanding of biological systems. A portion of the system described by Tank and Hopfield 11 is similar to the matched filter bank model simulated in this paper. The matched filter bank model (Modell) performs well when all phonemes (as above) are of the same duration. Otherwise, it would perform poorly unless the lengths were forced to a maximum length by padding the input and transfer functions with -1' s during calculation. The pseudo-inverse filter model, on the other hand, should not suffer from this problem. However, this aspect of the 11KXlel (Model 2) has not yet been explicitly simulated. Given a spatio-temporal pattern of size L x K, i.e., L spatial elements and K temporal elements, the number of calculations required to process the first stage of filtering by both models is the same as that by a static formal neuron network in which each neuron is connected to the L x K input elements. In both cases, L x K multiplications and additions are necessary to calculate one output value. In the case of bipolar patterns, the rnutiplication used for calculation of activation can be replaced by sign-bit check and addition. A future investigation is to use recursive filters or analog filters as transfer functions for faster and more efficient calculation. There are various schemes to obtain optimal recursive or analog filters.t8,19 Besides the lateral inhibition scheme used in the models, there are a number of alternative procedures to realize a "winnertake-all" network in analog or digital fashion. IS, 20, 21 As pointed out in the previous section, the Fourier transform in (6.5) requires a precaution concerning the resulting length of transfer functions. Calculating the recursive correlation equation (3.4) also needs such preprocessing as windowing or truncation.22 The generalization of static neural networks to dynamic ones along with their learning rules is strainghtforward as shown if the neuron operation and the learning rule are linear. Generalizing a system whose neuron operation and/or learning rule are nonlinear requires more careful analysis and remains for future work. The system described by Watrous and Shastril6 is an example of generalizing a backpropagation model. Their result showed a good potential of the model and a need for more rigorous analysis of the model. Generalizing a system with recurrent connections is another task to be pursued. In a system with a certain analytical nonlinearity, the signals are expressed by Volterra functionals, for example. A practical learning system can then be constructed if higher kernels are neglected. For example, a cubic function can be used instead of a sigmoidal function. 38 3. 1'1 1\ 0-{'-r. ~ j"-~ ;~. 1\ U ! --{. e ~ (a) z ~ '\ 0 .t ·f-t 7\ -. • I I , I I I I I I I • 51 t .. IS. en TIme "t ~ l~ ~ ~ !. ~7 1 1 ! ~ e Ii (b) z "; .:~ • ~ 1. l j r-• I I I • u t .. lSI tu TIme Fig. 4. Performance of Models. (a) Modell with Clean Test Pattern. (b) Model 2 with Clean Test Pattern. (c) Modell with Noisy Test Pattern. (d) Model 2 with Noisy Test Pattern. Arrows indicate the ideal response positions at the end of phoneme. 8. CONCLUSION The formal neuron was generalized to the dynamic formal neuron to recognize spatiotemporal patterns. It is shown that existing learning rules can be generalized for dynamic formal neurons. An artificial neural network using dynamic formal neurons was applied to classifying 30 model phonemes with bipolar patterns created by using parameters of formant frequencies and their bandwidths. The model operates in two stages: in the first stage, it calculates the correlation between the input and prototype patterns stored in the transfer function matrix, and, in the second stage, a lateral inhibition network selects the output of the phoneme pattern close to the input pattern. i!! e ii (C) z ~ C i!! e ii (d) z ~ C 3. at It • I I • 3. " I u It • I I , • 39 ---{'.-\ 1"'• j ,--;' !" P, X I I I 51 t .. t51 u. nme .~ ~0 .'--~ '1 " • ,.. • Fig. 4 (continued.) Two models with different transfer functions were tested. Model 1 was a matched filter bank model and Model 2 was a pseudo-inverse filter model. A sequence of phoneme patterns corresponding to continuous pronunciation of digits was used as a test pattern. For the test pattern, Modell missed to recognize one phoneme and responded falsely once while Model 2 correctly recognized all the 32 phonemes in the test pattern. When the flip noise which flips the sign of the pattern with the probability 0.2, Model 1 missed three phonemes and falsely responded once while Model 2 recognized all the phonemes and false-alarmed once. Both models detected the phonerns at the correct position within the continuous stream. References 1. W. S. McCulloch and W. Pitts, "A logical calculus of the ideas imminent in nervous activity," Bulletin of Mathematical Biophysics, vol. 5, pp. 115-133, 1943. 2. D. O. Hebb, The Organization of Behavior, Wiley, New York, 1949. 40 3. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, "Learning internal representations by error propagation," in Parallel Distributed Processing. Vol. 1, MIT, Cambridge, 1986. 4. B. Widrow and M. E. Hoff, "Adaptive switching circuits," Institute of Radio Engineers. Western Electronics Show and Convention, vol. Convention Record Part 4, pp. 96-104, 1960. 5. R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis. Chapter 5, Wiley, New York, 1973. 6. T. Kohonen, Self-organization and Associative Memory, Springer-Verlag, Berlin, 1984. 7. F. Rosenblatt, Principles of Neurodynamics, Spartan Books, Washington, 1962. 8. 1. M. Varah, "A practical examination of some numerical methods for linear discrete illposed problems," SIAM Review, vol. 21, no. 1, pp. 100-111, 1979. 9. C. Koch, J. Marroquin, and A. Y uiIle, "Analog neural networks in early vision," Proceedings of the National Academy of Sciences. USA, vol. 83, pp. 4263-4267, 1986. 10. G. O. Stone, "An analysis of the delta rule and the learning of statistical associations," in Parallel Distributed Processing .• Vol. 1, MIT, Cambridge, 1986. 11. B. Widrow and S. D. Stearns, Adaptive Signal Processing, Prentice-Hall, Englewood Cliffs, 1985. 12. D. H. Klatt, "Software for a cascade/parallel formant synthesizer," Journal of Acoustical Society of America, vol. 67, no. 3, pp. 971-995, 1980. 13. L. E. Atlas, T. Homma, and R. J. Marks II, "A neural network for vowel classification," Proceedings International Conference on Acoustics. Speech. and Signal Processing, 1987. 14. R. P. Lippman, "An introduction to computing with neural nets," IEEE ASSP Magazine, April, 1987. 15. S. Amari and M. A. Arbib, "Competition and cooperation in neural nets," in Systems Neuroscience, ed. J. Metzler, pp. 119-165, Academic Press, New York, 1977. 16. R. L. Watrous and L. Shastri, "Learning acoustic features from speech data using connectionist networks," Proceedings of The Ninth Annual Conference of The Cognitive Science Society, pp. 518-530, 1987. 17. D. Tank and J. J. Hopfield, "Concentrating information in time: analog neural networks with applications to speech recognition problems," Proceedings of International Conference on Neural Netoworks, San Diego, 1987. 18. J. R. Treichler, C. R. Johnson,Jr., and M. G. Larimore, Theory and Design of Adaptive Filters. Chapter 5, Wiley, New York, 1987. 19. M Schetzen, The Volterra and Wiener Theories of Nonlinear Systems. Chapter 16, Wiley, New York, 1980. 20. S. Grossberg, "Associative and competitive principles of learning," in Competition and Cooperation in Neural Nets, ed. M. A. Arbib, pp. 295-341, Springer-Verlag, New York, 1982. 21. R. J. Marks II, L. E. Atlas, J. J. Choi, S. Oh, K. F. Cheung, and D. C. Park, "A performance analysis of associative memories with nonlinearities in the correlation domain," (submitted to Applied Optics), 1987. 22. D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing, pp. 230-234, Prentice-Hall, Englewood Cliffs, 1984.
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740 SPATIAL ORGANIZATION OF NEURAL NEn~ORKS: A PROBABILISTIC MODELING APPROACH A. Stafylopatis M. Dikaiakos D. Kontoravdis National Technical University of Athens, Department of Electrical Engineering, Computer Science Division, 15773 Zographos, Athens, Greece. ABSTRACT The aim of this paper is to explore the spatial organization of neural networks under Markovian assumptions, in what concerns the behaviour of individual cells and the interconnection mechanism. Spaceorganizational properties of neural nets are very relevant in image modeling and pattern analysis, where spatial computations on stochastic two-dimensional image fields are involved. As a first approach we develop a random neural network model, based upon simple probabilistic assumptions, whose organization is studied by means of discrete-event simulation. We then investigate the possibility of approXimating the random network's behaviour by using an analytical approach originating from the theory of general product-form queueing networks. The neural network is described by an open network of nodes, in which customers moving from node to node represent stimulations and connections between nodes are expressed in terms of suitably selected routing probabilities. We obtain the solution of the model under different disciplines affecting the time spent by a stimulation at each node visited. Results concerning the distribution of excitation in the network as a function of network topology and external stimulation arrival pattern are compared with measures obtained from the simulation and validate the approach followed. INTRODUCTION Neural net models have been studied for many years in an attempt to achieve brain-like performance in computing systems. These models are composed of a large number of interconnected computational elements and their structure reflects our present understanding of the organizing principles of biological nervous systems. In the begining, neural nets, or other equivalent models, were rather intended to represent the logic arising in certain situations than to provide an accurate description in a realistic context. However, in the last decade or so the knowledge of what goes on in the brain has increased tremendously. New discoveries in natural systems, make it now reasonable to examine the possibilities of using modern technology in order to synthesige systems that have some of the properties of real neural systems 8,9,10,11. In the original neural net model developed in 1943 by McCulloch and Pitts 1,2 the network is made of many interacting components, known as the "McCulloch-Pitts cells" or "formal neurons II , which are simple logical units with two possible states changing state accord® American Institute of Physics 1988 741 ing to a threshold function of their inputs. Related automata models have been used later for gene control systems (genetic networks) 3, in which genes are represented as binary automata changing state according to boolean functions of their inputs. Boolean networks constitute a more general model, whose dynamical behaviour has been studied extensively. Due to the large number of elements, the exact structure of the connections and the functions of individual components are generally unknown and assumed to be distributed at random. Several studies on these random boolean networks 5,6 have shown that they exhibit a surprisingly stable behaviour in what concerns their temporal and spatial organization. However, very few formal analytical results are available, since most studies concern statistical descriptions and computer simulations. The temporal and spatial organization of random boolean networks is of particular interest in the attempt of understanding the properties of such systems, and models originating from the theory of stochastic processes 13 seem to be very useful. Spatial properties of neural nets are most important in the field of image recognition 12. In the biological eye, a level-normalization computation is performed by the layer of horizontal cells, which are fed by the immediately preceding layer of photoreceptors. The horizontal cells take the outputs of the receptors and average them spatially, this average being weighted on a nearest-neighbor basis. This procedure corresponds to a mechanism for determining the brightness level of pixels in an image field by using an array of processing elements. The principle of local computation is usually adopted in models used for representing and generating textured images. Among the stochastic models applied to analyzing the parameters of image fields, the random Markov field model 7,14 seems to give a suitably structured representation, which is mainly due to the application of the markovian property in space. This type of modeling constitutes a promising alternative in the study of spatial organization phenomena in neura 1 nets. The approach taken in this paper aims to investigate some aspects of spatial organization under simple stochastic assumptions. In the next section we develop a model for random neural networks assuming boolean operation of individual cells. The behaviour of this model, obtained through simulation experiments, is then approximated by using techniques from the theory of queueing networks. The approximation yields quite interesting results as illustrated by various examples. THE RANDOM NETWORK MODEL We define a random neural network as a set of elements or cells, each one of which can be in one of two different states: firing or quiet. Cells are interconnected to form an NxN grid, where each grid point is occupied by a cell. We shall consider only connections between neighbors, so that each cell is connected to 4 among the other cells: two input and two output cells (the output of a cell is equal 'to its internal state and it is sent to its output cells which use ;it as one of their inputs). The network topology is thus specified 742 by its incidence matrix A of dimension MxM, where M=N2. This matrix takes a simple form in the case of neighbor-connection considered here. We further assume a periodic structure of connections in what concerns inputs and outputs; we will be interested in the following two types of networks depending upon the period of reproduction for elementary square modules 5, as shown in Fig.l: - Propagative nets (Period 1) - Looping nets (Period 2) \ .... "' "';> I , (a) (b) -- '"' Fig.1. (a) Propagative connections, (b) Looping connections At the edges of the grid, circular connections are established (modulo N), so that the network can be viewed as supported by a torus. The operation of tile network is non-autonomous: changes of state are determined by both the interaction among cells and the influence of external stimulations. We assume that stimulations arrive from the outside world according to a Poisson process with parameter A. Each arriving stimulation is associated with exactly one cell of the network; the cell concerned is determined by means of a given discrete probability distribution qi (l~i~M), considering an one-dimensional labeling of the M cells. The operation of each individual cell is asynchronous and can be described in terms of the following rules: - A quiet cell moves to the firing state if it receives an arriving stimulation or if a boolean function of its inputs becomes true. - A firing cell moves to the quiet state if a boolean function of its inputs becomes false. - Changes of state imply a reaction delay of the cell concerned; these delays are independent identically distributed random variables following a negative exponential distribution with parameter y. According to these rules, the operation of a cell can be viewed as illustrated by Fig.2, where the horizontal axis represents time and the numbers 0,1,2 and 3 represent phases of an operation cycle. Phases 1 and 3, as indicated in Fig.2, correspond to reaction delays. In this sense, the qui et and fi ri ng s ta tes, as defi ned in the begi ni ng of thi s section, represent the aggregates of phases 0,1 and 2,3 respectively. External stimulations affect the receiving cell only when it is in phase 0; otherwise we consider that the stimulation is lost. In the same way, we assume tha t changes of the value of the input boo 1 ean function do not affect the operation of the cell during phases land 3. The conditions are checked only at the end of the respective reaction delay. 743 quiet I firing state state ~ /r 2 ~ / 0 0 Fig.2. Changes of state for individual cells The above defi ned model i ncl udes some fea tures of the ori gi na 1 McCulloch-Pitts cells 1,2. In fact, it represents an asynchronous counterpart of the latter, in which boolean functions are considered instead of threshold functions. However, it can be shown that any McCulloch and Pitts' neural network can be implemented by a boolean network designed in an appropriate fashion 5. In what follows, we will consider that the firing condition for each individual cell is determined by an "or" function of its inputs. Under the assumptions adopted, the evolution of the network in time can be described by a conti nuous-parameter Markov process. However, the size of the state-space and the complexity of the system are such that no analytical solution is tractable. The spatial organization of the network could be expressed in terms of the steadystate probability distribution for the Markov process. A more useful representation is provided by the marginal probability distributions for all cells in the network, or equivalently by the probability of being in the firing state for each cell. This measure expresses the level of excitation for each point in the grid. We have studied the behaviour of the above model by means of simulation experiments for various cases depending upon the network size, the connection type, the distribution of external stimulation arrivals on the grid and the parameters A and V. Some examples are illustrated in the last section, in comparison with results obtained using the approach discussed in the next section. The estimations obta i ned concern the probabil i ty of bei ng in the fi ri ng s ta te for all cells in the network. The simulation was implemented according to the "batched means" method; each run was carried out unti 1 the width of the 95% confidence interval was less that 10% of the estimated mean value for each cell, or until a maximum number of events had been simulated depending upon the size of the network. THE ANALYTICAL APPROACH The neural network model considered in the previous section exhibited the markovian property in both time and space. Markovianity in space, expressed by the principle of "neighbor-connections", is the basic feature of Markov random fields 7,14, as already discussed. Our idea is to attempt an approximation of the random neural network model by usi ng a well-known model, wlli ch is markovi an in time, and applying the constraint of markovianity in space. The model considered is an open queueing network, which belongs to the general class of queueing networks admitting a product-form solution 4. In fact, one could distinguish several common features in the two network models. 744 A neural network, in general, receives information in the form of external stimulation signals and performs some computation on this information, which is represented by changes of its state. The operation of the network can be viewed as a flow of excitement among the cells and the spatial distribution of this excitement represents the response of the network to the information received. This kind of operation is particularly relevant in the processing of image fields. On the other hand, in queueing networks, composed of a number of service station nodes, customers arrive from the outside world and spend some time in the network, during which they more from node to node, waiting and receiving service at each node visited. Following the external arrival pattern, the interconnection of nodes and the other network parameters, the operation of the network is characterized by a distribution of activity among the nodes. Let us now consider a queueing network, where nodes represent cells and customers represent stimulations moving from cell to cell following the topology of the network. Our aim is to define the network's characteristics in a way to match those of the neural net model as much as possible. Our queueing network model is completely specified by the following assumptions: - The network is composed of M=N2 nodes arranged on an NxN rectangular grid, as in the previous case. Interconnections are expressed by means of a matrix R of routing probabilities: rij (l~i,j~) represents the probability that a stimulation (customer) leaving node i will next visit node j. Since it is an open network, after visiting an arbitrary number of cells, stimulations may eventually leave the network. Let riO denote the probability of leaving the network upon leaving node i. In what follows, we will assume that riO=s for all node's. In what concerns the routing probauilities rij, they are determined by the two interconnection schemata considered in the previous section (propagative and looping connections): each node i has two output nodes j, for which the routing probabilities are equally distributed. Thus, rij=(1-s)/2 for the two output nodes of i and equal to zero for a 11 0 ther nodes in the network. - External stimulation arrivals follow a Poisson process with parameter A and are routed to the nodes according to the probability distribution qi (l~i<M) as in the previous section. - Stimulations receive a "service time" at each node visited. Service times are independent identically distributed random variables, which are exponentially distributed with parameter V. The time spent by a stimulation at a node depends also upon the "service discipline" adopted. We shall consider two types of service disciplines according to the general queuei n9 network model 4: the fi rs t-come-fi rs t-served (FCFS) discipline, where customers are served in the order of their arrival to the node, and the infinite-server (IS) discipline, where a customer's service is immediately assumed upon arrival to the node, as if there were always a server available for each arriving customer (the second type includes no waiting delay). We will refer to the above two types of nodes as type 1 and type 2 respectively. In either case, all nodes of the network will be of the same type. The steady-state solution of the above network is a straightforwa rd app 1 i ca ti on of the general BCMP theorem 4 according to the 745 Isimple assumptions considered. The state of the system is described ,by the vector (kl,k2, ... ,kM), where ki is the number of customers present at node i. We first define the traffic intensity Pi for each node i as Pi = Aei/V i = 1,2, ... ,M (1) where the quantities {ei} are the solution of the following set of linear equations: M ei = qi + I e·r .. j=1 J Jl i = 1,2, ... ,M (2) It can be easily seen that, in fact, ei represents the average number of visits a customer makes to node, i during his sojourn in the network. The existence of a steady-state distribution for the system depends on the sol uti on of the above set. Fo 11 owi ng the general theorem 4, the joint steady-state distribution takes the form of a product of independent distributions for the nodes: p(k1,k2, ... ,kM} = ~1(k1)P2(k2} •.• Pr~(kM) (3) where p.(k.) = 1 1 1 ki (I-P.}P. 1 1 k· _p. p. 1 ell kiT (Type 1) (Type 2) provided that the stabtlity condition Pi<1 is satisfied for type 1 nodes. (4) The product form solution of this type of network expresses the idea of global and local balance which is characteristic of ergodic Ivlarkov processes. We can then proceed to deri vi ng the des i red measure for each node in the network; we are interested in the probability of being active for each node, which can be interpreted as the probability that at least one customer is present at the node: (Type 1) 1 Pi P(k .>O}=1-p.(O) = _po 1 1 1-e 1 (5 ) (Type 2) The variation in space of the above quantity will be studied with respect to the corresponding measure obtained from simulation experiments for the neural network model. . NUMERICAL AND SIMULATION EXAMPLES Simulations and numerical solutions of the queueing network motiel were run for different values of the parameters. The network sizes considered are relatively small but can provide useful information on the spatial organization of the networks. For both types of service discipline discussed in the previous section, the approach followed yields a very good approximation of the network's organization in most regions of the rectangular grid. The choice of the probability s of leaving the network plays a critical role in the beha746 (a) (b) Fig.3. A 10xiO network with A=l, V=l and propagative connections. External stimulations are uniformly distributed over a 3x3 square on the upper left corner of the grid. (a) simulation (b) Queueing network approach with s=0.05 and type 2 nodes. (a) (b) Fig.4. The network of Fig.3 with A=2 (a) Simulation (b) Queueing network approach with s=0.08 and type 2 nodes. viour of the queueing model,and must have a non-zero value in order for the network to be stable. Good results are obtained for very small values of s; in fact, this parameter represents the phenomenon of excitation being "lost" somewhere in the network. Graphical representations for various cases are shown in Figures 3-7. We have used a coloring of five "grey levels", defined by dividing into five segments the interval between the smallest and the largest value of the probability on the grid; the normalization is performed with respect to simulation results. This type of representation is less accurate than directly providing numerical values, but is more clear for describing the organization of the system. In each case, the results shown for the queueing model concern only one type of nodes, the one that best fits the simulation results, which is type 2 in the majority of cases examined. The graphical representation illustrates the structuring of the distribution of excitation on the grid in terms of functionally connected regions of high and low (a) (b) Fig.5. A 10xl0 network with A=l, V=l and looping connections. External stimulations are uniformly distributed over a 4x4 square on the center of the grid. (a) Simulation (b) Queueing network approach wi th s= 0.07 and type 2 nodes. (a) (b) Fig.6. The network of Fig.5 with A=0.5 (a) Simulation (b) Queuei ng network approach wi th s= 0.03 and type 2 nodes. excitation. We notice that clustering of nodes mainly follows the spatial distribution of external stimulations and is more sharply structured in the case of looping connections. CONCLUSION 747 We have developed in this paper a simple continuous-time probabilistic model of neural nets in an attempt to investigate their spatial organization. The model incorporates some of the main features of the McCulloch-Pitts "formal neurons" model and assumes boolean operation of the elementary cells. The steady-state behaviour of the model was approximated by means of a queueing network model with suitably chosen parameters. Results obtained from the solution of the above approximation were compared with simulation results of the initial model, which validate the approximation. This simplified approach is a first step in an attempt to study the organiza748 (a) (b) Fig.7. A 16x16 network with A=1, V=1 and looping connections. External stimulations are uniformly distributed over two 4x4 squares on the upper left and lower right corners of the grid. (a) Simulation (b) Queueing network approach with s=0.05 and type 1 nodes. tional properties of neural nets by means of markovian modeling techn; ques. REFERENCES 1. W. S. McCulloch, W. Pitts, "A Logical Calculus of the Ideas Immanent in Nervous Activity", Bull. of Math. Biophysics 5, 115133 (1943). 2. M. L. Minsky, Computation: Finite and Infinite Machines (Prentice Hall, 1967). 3. S. Kauffman, "Behaviour of Randomly Constructed Genetic Nets", in Towards a Theoretical Biology, Ed. C. H. Waddington (Edinburgh University Press, 1970). 4. F. Baskett, K. M. Chandy, R. R. Muntz, F. G. Palacios, "Open, Closed and Mixed Networks of Queues with Different Classes of Customers", J. ACM, 22 (1975). 5. H. Atlan, F. Fogelman-Soulie, J. Salomon, G. Weisbuch, "Random Boolean Networks", Cyb. and Syst. 12 (1981). 6. F. Folgeman-Soulie, E. Goles-Chacc, G. Weisbuch, "Specific Roles of the Different Boolean Mappings in Random Networks", Bull. of Math. Biology, Vol.44, No 5 (1982). 7. G. R. Cross, A. K. Jain, "Markov Random Field Texture Models", IEEE Trans. on PAMI, Vol. PAMI-5, No 1 (1983). 8. E. R. Kandel, J. H. Schwartz, Principles of Neural Science, (Elsevier, N.Y., 1985). 9. J. J. Hopfield, D. W. Tank, "Computing with Neural Circuits: A Model", Sc i ence, Vol. 233, 625-633 (1986). 10. Y. S. Abu-Mostafa, D. Psaltis, "Optical Neural Computers", Scient. Amer., 256, 88-95 (1987). 11. R. P. Lippmann, "An Introduction to Computing with Neural Nets", IEEE ASSP Mag. (Apr. 1987). 12. C. A. Mead, "Neural Hardware for Vision", Eng. and Scie. (June 1987) . 13. E. Gelenbe, A. Stafylopatis, "Temporal Behaviour of Neural Networks", IEEE First Intern. Conf. on Neural Networks, San Diego, CA (June 1987). 14. L. Onural, "A Systematic Procedure to Generate Connected Binary Fractal Patterns with Resolution-varying Texture", Sec. Intern. Sympt. on Compo and Inform. Sciences, Istanbul, Turkey (Oct. 1987) .
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Teaching Artificial Neural Systems to Drive: Manual Training Techniques for Autonomous Systems J. F. Shepanski and S. A. Macy TRW, Inc. One Space Park, 02/1779 Redondo Beach, CA 90278 Abetract 693 We have developed a methodology for manually training autononlous control systems based on artificial neural systems (ANS). In applications where the rule set governing an expert's decisions is difficult to formulate, ANS can be used to ext.ra.c:t rules by associating the information an expert receives with the actions h~ takes. Properly constructed networks imitate rules of behavior that permits them to function autonomously when they are trained on the spanning set of possible situations. This training can be provided manually, either under the direct. supervision or a system trainer, or indirectly using a background mode where the network assimilates training data as the expert perrorms his day-to-day tasks. To demonstrate these methods we have trained an ANS network to drive a vehicle through simulated rreeway traffic. I ntJooducticn Computational systems employing fine grained parallelism are revolutionizing the way we approach a number or long standing problems involving pattern recognition and cognitive processing. The field spans a wide variety or computational networks, rrom constructs emulating neural runctions, to more crystalline configurations that resemble systolic arrays. Several titles are used to describe this broad area or research, we use the term artificial neural systems (ANS). Our concern in this work is the use or ANS ror manually training certain types or autonomous systems where the desired rules of behavior are difficult to rormulate. Artificial neural systems consist of a number or processing elements interconnected in a weighted, user-specified fashion, the interconnection weights acting as memory ror the system. Each processing element calculatE',> an output value based on the weighted sum or its inputs. In addition, the input data is correlated with the output or desired output (specified by an instructive agent) in a training rule that is used to adjust the interconnection weights. In this way the ne~ work learns patterns or imitates rules of behavior and decision making. The partiCUlar ANS architecture we use is a variation of Rummelhart et. al. [lJ multi-layer perceptron employing the generalized delta rule (GD R). Instead of a single, multi-layer ,structure, our final network has a a multiple component or "block" configuration where one blOt'k'~ output reeds into another (see Figure 3). The training methodology we have developed is not tied to a particular training rule or architecture and should work well with alternative networks like Grossberg's adaptive resonance model[2J. © American Institute of Physics 1988 694 The equations describing the network are derived and described in detail by Rumelhart et. al.[l]. In summary, they are: Transfer function: Weight adaptation rule: Error calculation: • Sj = E WjiOi; i-O Aw ·· =( 1- a .. )n ··0 ·0· + a ··Awp.revious. l' l' ., l' J • l' l' ' '" OJ =0j{1- OJ) E0.tW.ti, .t=1 (1) ( 2) ( 3) where OJ is the output or processing element j or a sensor input, wi is the interconnection weight leading from element ito i, n is the number of inputs to j, Aw is the adjustment of w, '1 is the training constant, a is the training "momentum," OJ is the calculated error for element i, and m is the Canout oC a given element. Element zero is a constant input, equal to one, so that. WjO is equivalent to the bias threshold of element j. The (1- a) factor in equation (2) differs from standard GDR formulation, but. it is useful for keeping track of the relative magnitudes of the two terms. For the network's output layer the summation in equation (3) is replaced with the difference between the desired and actual output value of element j. These networks are usually trained by presenting the system with sets of input/output data vectors in cyclic fashion, the entire cycle of database presentation repeated dozens of times. This method is effective when the training agent is a computer operating in batch mode, but would be intolerable for a human instructor. There are two developments that will help real-time human training. The first is a more efficient incorporation of data/response patterns into a network. The second, which we are addressing in this paper, is a suitable environment wherein a man and ANS network can interact in training situation with minimum inconvenience or boredom on the human's part. The ability to systematically train networks in this fashion is extremely useful for developing certain types of expert systems including automatic signal processors, autopilots, robots and other autonomous machines. We report a number of techniques aimed at facilitating this type of training, and we propose a general method for teaching these networks. System. Development Our work focuses on the utility of ANS for system control. It began as an application of Barto and Sutton's associative search network[3]. Although their approach was useful in a number of ways, it fell short when we tried to use it for capturing the subtleties of human decision-making. In response we shifted our emphasis rrom constructing goal runctions for automatic learning, to methods for training networks using direct human instruction. An integral part or this is the development or suitable interraces between humans, networks and the outside world or simulator. In this section we will report various approaches to these ends, and describe a general methodology for manually teaching ANS networks. To demonstrate these techniques we taught a network to drive a robot vehicle down a simulated highway in traffic. This application combines binary decision making and control of continuous parameters. Initially we investigated the use or automatic learning based on goal functions[3] for training control systems. We trained a network-controlled vehicle to maintain acceptable following distances from cars ahead or it. On a graphics workstation, a one lane circular track was 695 constructed and occupied by two vehicles: a network-controlled robot car and a pace car that varied its speed at random .. Input data to the network consisted of the separation distance and the speed of the robot vehicle . The values of a goal function were translated into desired output for GDR training. Output controls consisted of three binary decision elements: 1) accelerate one increment of speed, 2) maintain speed, and 3) decelerate one increment of speed. At all times the desired output vector had exactly one of these three elements active . The goal runction was quadratic with a minimum corresponding to the optimal following distance. Although it had no direct control over the simulation, the goal function positively or negatively reinforced the system's behavior. The network was given complete control of the robot vehicle, and the human trainer had no influence except the ability to start and terminate training. This proved unsatisractory because the initial system behavior--governed by random interconnection weights--was very unstable. The robot tended to run over the car in rront of it before significant training occurred. By carerully halting and restarting training we achieved stable system behavior. At first the rollowing distance maintained by the robot car oscillated as ir the vehicle was attached by a sj)ring to the pace car. This activity gradually damped. Arter about one thousand training steps the vehicle maintained the optimal following distance and responded quickly to changes in the pace car's speed. Constructing composite goal functions to promote more sophisticated abilities proved difficult, even ill-defined, because there were many unspecified parameters. To generate goal runctions ror these abilities would be similar to conventional programming--the type or labor we want to circumvent using ANS. On the other hand, humans are adept at assessing complex situations and making decisions based on qualitative data, but their "goal runctions" are difficult ir not impossible to capture analytically. One attraction of ANS is that it can imitate behavior based on these elusive rules without rormally specifying them. At this point we turned our efforts to manual training techniques. The initially trained network was grafted into a larger system and augmented with additional inputs: distance and speed inrormation on nearby pace cars in a second traffic lane, and an output control signal governing lane changes. The original network's ability to maintain a safe following distance was retained intact. Thts grafting procedure is one of two methods we studied for adding ne .... abilities to an existin, system. (The second, which employs a block structure, is described below.) The network remained in direct control of the robot vehicle, but a human trainer instructed it when and when not to change lanes. His commands were interpreted as the desired output and used in the GDR training algorithm. This technique, which we call coaching, proved userul and the network quickly correlated its environmental inputs with the teacher's instructions. The network became adept at changing lanes and weaving through traffic. We found that the network took on the behavior pattern or its trainer. A conservative teacher produced a timid network, while an aggressive tzainer produced a network that tended to cut off other automobiles and squeeze through tight openings. Despite its success, the coaching method of training did not solve the problem or initial network instability. The stability problem was solved by giving the trainer direct control over the simulation. The system configuration (Figure 1), allows the expert to exert control or release it to the n~t work. During initial tzaining the expert is in the driver's seat while the network acts the role of 696 apprentice. It receives sensor information, predicts system commands, and compares its predictions. against the desired output (ie. the trainer's commands) . Figure 2 shows the data and command flow in detail. Input data is processed through different channels and presented to the trainer and network. Where visual and audio formats are effective for humans, the network uses information in vector form. This differentiation of data presentation is a limitation of the system; removing it is a cask for future ~search. The trainer issues control commands in accordance with his assigned ~k while the network takes the trainer's actions as desired system responses and correlates these with the input. We refer to this procedure as master/apprentice training, network training proceeds invisibly in the background as the expert proceeds with his day to day work. It avoids the instability problem because the network is free to make errors without the adverse consequence of throwing the operating environment into disarray. I World (--> sensors) or Simulation Input Expert Commands l+ I Ne',WOrk I ~ Actuation ~ J + ~------------------~ ~------~---------------------------~ Figure 1. A scheme for manually training ANS networks. Input data is received by both the network and trainer. The trainer issues commands that are actuated (solid command line). or he coaches the network in how it ought to respond (broken command line). Input data Preprocessing tortunan Preprocessing for network --+ Commands ~ 9'l. Actuation .1-r" N twork --+ Predicted e t commands Training '-------------. rule Coaching/emphasis Fegure 2. Data and convnand flow In the training system. Input data is processed and presented to the trainer and network. In master/appre~ice training (solid command Hne). the trainer's orders are actuated and the network treats his commands as the system's desired output. In coaching. the network's predicted oonvnands are actuated (broken command line). and the trainer influences weight adaptation by specifying the desired system output and controlHng the values of trailing constants-his -suggestions- are not cirec:tty actuated. Once initial. bacqround wainmg is complete, the expert proceeds in a more formal manner to teach the network. He releases control of the command system to the network in order to evaluate ita behavior and weaknesses. He then resumes control and works through a 697 series of scenarios designed to train t.he network out of its bad behavior. By switching back and forth. between human and network control, the expert assesses the network's reliability and teaches correct responses as needed. We find master/apprentice training works well for behavior involving continuous functions, like steering. On the other hand, coaching is appropriate for decision Cunctions, like when Ule car ought to pass. Our methodology employs both techniques. The Driving Network The fully developed freeway simulation consists of a two lane highway that is made of joined straight and curved segments which vary at. random in length (and curvature). Several pace cars move at random speeds near the robot vehicle. The network is given the tasks of tracking the road, negotiating curves. returning to the road if placed far afield, maintaining safe distances from the pace cars, and changing lanes when appropriate. Instead of a single multi-layer structure, the network is composed of two blocks; one controls the steering and the other regulates speed and decides when the vehicle should change lanes (Figure 3). The first block receives information about the position and speed of the robot vehicle relative to other ears in its vicinity. Its output is used to determine the automobile's speed and whet.her the robot should change lanes. The passing signal is converted to a lane assignment based on the car's current lane position. The second block receives the lane assignment and data pertinent to the position and orientation of the vehicle with respect to the road. The output is used to determine the steering angle of the robot car. Block 1 Inputs Constant. Speed. Disl. Ahead, Pl • Disl. Ahead, Ol • Dist. Behind, Ol • ReI. Speed Ahead, Pl • ReI. Speed Ahead, Ol • ReI. Speed Behind, Ol • Constant • Rei. Orientation • -..--t~ lane Nurmer • lateral Dist. • Curvature • Convert lane change to lane number • • • Outputs I Speed Change lanes • Steering Angle Figure 3. The two blocks of the driving ANS network. Heavy arrows Indicate total interconnectivity between layers. PL designates the traffic lane presently oca.apied by the robot vehicle, Ol refers to the other lane, QJrvature refers to the road, lane nurrber is either 0 or 1, relative orientation and lateral distance refers to the robot car's direction and podion relative to the road'l direction and center line. respectively. . 698 The input data is displayed in pictorial and textual form to the driving instructor. He views the road and nearby vehicles from the perspective of the driver's seat or overhead. The network receives information in the form of a vector whose elements have been scaled to unitary order, O( 1) . Wide ranging input parameters, like distance, are compressed using the hyperbolic tangent or logarithmic functions. In each block, the input layer is totally interconnected to both the ou~ put and a hidden layer. Our scheme trains in real time, and as we discuss later, it trains more smoothly with a small modification of the training algorithm . Output is interpreted in two ways: as a binary decision or as a continuously varying parameter. The first simply compares the sigmoid output against a threshold. The second scales the output to an appropriate range for its application. For example, on the steering output element, a 0.5 value is interpreted as a zero steering angle. Left and right turns of varying degrees are initiated when this output is above or below 0.5, respectively. The network is divided into two blocks that can be trained separately. Beside being conceptually easier to understand, we find this component approach is easy to train systematically. Because each block has a restricted, well-defined set of tasks, the trainer can concentrate specifically on those functions without being concerned that other aspects of the network behavior are deteriorating. "'e trained the system from bottom up, first teaching the network to stay on the road, negotiate curves, chan~e lanes, and how to return if the vehicle strayed off the highway. Block 2, responsible for steering, learned these skills in a few minutes using the master/apprentice mode. It tended to steer more slowly than a human but further training progressively improved its responsiveness. We experimented with different trammg constants and "momentum" values. Large " values, about 1, caused weights to change too coarsely. " values an order of magnitude smaller worked well. We found DO advantage in using momentum for this method of training, in fact, the system responded about three times more slowly when 0 =0.9 than when the momentt:m term was dropped. Our standard training parameters were" =0.2, and Cl' =00 a) ~ Db)~~ =D-=-~=~~--=~--= ~ Figure 4. Typical behavior of a network-controlled vehicle (dam rectangle) when trained by a) a conservative miYer, ItI:I b}. reckless driver. Speed Is indicated by the length of the arrows. After Block 2 "Was trained, we gave steering control to the network and concentrated on teaching the network to change lanes and adjust speed. Speed control in this ('"asP. was a continuous variable and was best taught using master/apprentice training. On the other hand, the binary decision to change lanes was best taught by coaching. About ten minutes of training were needed to teach the network to weave through traffic. We found that the network readily adapts the 699 behavioral pattern of its trainer. A conservative trainer generated a network that hardly ever passed, while an aggressive trainer produced a network that drove recklessly and tended to cut off other-cars (Figure 4). Discussion One of the strengths of el:pert 5ystf'mS based on ANS is that the use of input data in the decision making and control proc~ss does not have to be specified. The network adapts its internal weights to conform to input/output correlat.ions it discovers. It is important, however, that data used by the human expert is also available to the network. The different processing of sensor data for man and network may have important consequences, key information may be presented to the man but not. the machine. This difference in data processing is particularly worrisome for image data where human ability to extract detail is vastly superior to our au tomatic image processing capabilities. Though we would not require an image processing system to understand images, it would have to extract relevant information from cluttered backgrounds. Until we have sufficiently sophisticated algorithms or networks to do this, our efforts at constructing expert systems which halldle image data are handicapped. Scaling input data to the unitary order of magnitude is important for training stability. 111is is evident from equations (1) and (2) . The sigmoid transfer function ranges from 0.1 to 0.9 in approximat.eiy four units, that is, over an 0(1) domain. If system response must change in reaction to a large, O( n) swing of a given input parameter, the weight associated with that input will be trained toward an O( n- 1) magnitude. On the other hand, if the same system responds to an input whose range is O( 1), its associated weight will also be 0(1). The weight adjustment equation does not recognize differences in weight magnitude, therefore relatively small weights will undergo wild magnitude adjustments and converge weakly. On the other hand, if all input parameters are of the same magnitude their associated weights will reflect this and the training constant can be adjusted for gentle weight convergence . Because the output of hidden units are constrained between zero and one, O( 1) is a good target range for input parameters. Both the hyperbolic tangent and logarithmic functions are useful for scaling wide ranging inputs. A useful form of the latter is .8[I+ln(x/o)] if o<x, .8x/o if-o::;x::;o, -.8[I+ln(-%/o)] ifx<-o, ( 4) where 0>0 and defines the limits of the intermediate linear section, and .8 is a scaling factor. This symmetric logarithmic function is continuous in its first derivative, and useful when network behavior should change slowly as a parameter increases without bound. On the othl'r hand, if the system should approach a limiting behavior, the tanh function is appropriate. Weight adaptation is also complicated by relaxing the common practice of restricting interconnections to adjacent layers. Equation (3) shows that the calculated error for a hidden layergiven comparable weights, fanouts and output errors-will be one quarter or less than that of the 700 output layer. This is caused by the slope ractor, 0 .. ( 1- oil. The difference in error magnitudes is not noticeable in networks restricted to adjacent layer interconnectivity. But when this constraint is released the effect of errors originating directly from an output unit has 4" times the magnitude and effect of an error originating from a hidden unit removed d layers from the output layer. Compared to the corrections arising from the output units, those from the hidden units have little influence on weight adjustment, and the power of a multilayer structure is weakened. The system will train if we restrict connections to adjacent layers, but it trains slowly. To compensate for this effect we attenuate the error magnitudes originating from the output layer by the above factor. This heuristic procedure works well and racilitates smooth learning. Though we have made progress in real-time learning systems using GDR, compared to humans-who can learn from a single data presentation-they remain relatively sluggish in learning and response rates. We are interested in improvements of the GDR algorithm or alternative architectures that facilitate one-shot or rapid learning. In the latter case we are considering least squares restoration techniquesl4] and Grossberg and Carpenter's adaptive resonance modelsI3,5]. The construction of automated expert systems by observation of human personnel is attractive because of its efficient use of the expert's time and effort. Though the classic AI approach of rule base inference is applicable when such rules are clear cut and well organized, too often a human expert can not put his decision making process in words or specify the values of parameters that influence him. The attraction or ANS based systems is that imitations of expert behavior emerge as a natural consequence of their training. Referenees 1) D. E. Rumelhart, G . E. Hinton, and R. J. Williams, "Learning Internal Representations by Error Propagation," in Parallel D~tributed Proceuing: Ezploration~ in the Micro~trvcture 0/ Cognition, Vol. I, D. E. Rumelhart and J. L. McClelland (Eds.)' chap. 8, (1986), Bradford BooksjMIT Press, Cambridge 2) S. Grossberg, Studie~ 0/ Mind and Brain, (1982), Reidel, Boston 3) A. Barto and R. Sutton, "Landmark Learning: An Illustration of Associative Search," BiologicaIC,6emetiu,42, (1981), p.l 4) A. Rosenfeld and A . Kak, Digital Pieture Proeming, Vol. 1, chap. 7, (1982), Academic Press, New York 5) G. A. Carpenter and S. Grossberg, "A Massively Parallel Architecture for a Self-organizing Neural Pattern Recognition Machine," Computer Vision, Graphiu and Image Procu,ing, 37, ( 1987), p.54
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270 Correlational Strength and Computational Algebra of Synaptic Connections Between Neurons Eberhard E. Fetz Department of Physiology & Biophysics, University of Washington, Seattle, WA 98195 ABSTRACT Intracellular recordings in spinal cord motoneurons and cerebral cortex neurons have provided new evidence on the correlational strength of monosynaptic connections, and the relation between the shapes of postsynaptic potentials and the associated increased firing probability. In these cells, excitatory postsynaptic potentials (EPSPs) produce crosscorrelogram peaks which resemble in large part the derivative of the EPSP. Additional synaptic noise broadens the peak, but the peak area -- i.e., the number of above-chance firings triggered per EPSP -- remains proportional to the EPSP amplitude. A typical EPSP of 100 ~v triggers about .01 firings per EPSP. The consequences of these data for information processing by polysynaptic connections is discussed. The effects of sequential polysynaptic links can be calculated by convolving the effects of the underlying monosynaptic connections. The net effect of parallel pathways is the sum of the individual contributions. INTRODUCTION Interactions between neurons are determined by the strength and distribution of their synaptic connections. The strength of synaptic interactions has been measured directly in the central nervous system by two techniques. Intracellular recording reveals the magnitude and time course of postsynaptic potentials (PSPs) produced by synaptic connections, and crosscorrelation of extracellular spike trains measures the effect of the PSP's on the firing probability of the connected cells. The relation between the shape of excitatory postsynaptic potentials (EPSPs) and the shape of the crosscorrelogram peak they produce has been empirically investigated in cat motoneurons 2,4,5 and in neocortical cells 10. RELATION BETWEEN EPSP'S AND CORRELOGRAM PEAKS Synaptic interactions have been studied most thoroughly in spinal cord motoneurons. Figure 1 illustrates the membrane potential of a rhythmically firing motoneuron, and the effect of EPSPs on its firing. An EPSP occurring sufficiently close to threshold (8) will cause the motoneuron to fire and will advance an action potential to its rising edge (top). Mathematical analysis of this threshold-crossing process predicts that an EPSP with shape e(t) will produce a firing probability f(t), which resembles © American Institute of Phy~ics 1988 r f; 'I :: I 'I 'I i : I' 8 --.----r \ .. ) : ...",/ \ .",.""..,.", EPSP e(t) CROSSCORRELOGRAM f(t) ,; .,...,."" /..-::-.... /' ----.... ~-""" I .... , .,.,,,, I .... , .. , .... .. TIME t t 271 Fig. 1. The relation between EPSP's and motoneuron firing. Top: membrane trajectory of rhythmically firing motoneuron, showing EPSP crossing threshold (8) and shortening the normal interspike interval by advancing a spike. V(t) is difference between membrane potential and threshold. Middle: same threshold-crossing process aligned with EPSP, with v(t) plotted as falling trajectory. Intercept (at upward arrow) indicates time of the advanced action potential. Bottom: Cross-correlation histogram predicted by threshold crossings. The peak in the firing rate f(t) above baseline (fo) is produced by spikes advanced from baseline, as indicated by the changed counts for the illustrated trajectory. Consequently, the area in the peak equals the area of the subsequent trough. 272 the derivative of the EPSP 4,8. Specifically, for smooth membrane potential trajectories approaching threshold (the case of no additional synaptic noise): f(t) = fo + (fo/v) del dt (1) where fo is the baseline firing rate of the motoneuron and v is the rate of closure between motoneuron membrane potential and threshold. This relation can be derived analytically by tranforming the process to a coordinate system aligned with the EPSP (Fig. 1, middle) and calculating the relative timing of spikes advanced by intercepts of the threshold trajectories with the EPSP 4. The above relation (1) is also valid for the correlogram trough during the falling phase of the EPSP, as long as del dt > -v; if the EPSP falls more rapidly than -v, the trough is limited at zero firing rate (as illustrated for the correlogram at bottom). The fact that the shape of the correlogram peak above baseline matches the EPSP derivative has been empirically confirmed for large EPSPs in cat motoneurons 4. This relation implies that the height of the correlogram peak above baseline is proportional to the EPSP rate of rise. The integral of this relationship predicts that the area between the correlogram peak and baseline is proportional to the EPSP amplitude. This linear relation further implies that the effects of simultaneously arriving EPSPs will add linearly. The presence of additional background synaptic "noise", which is normally produced by randomly occurring synaptic inputs, tends to make the correlogram peak broader than the duration of the EPSP risetime. This broadening is produced by membrane potential fluctuations which cause additional threshold crossings during the decay of the EPSP by trajectories that would have missed the EPSP (e.g., the dashed trajectory in Fig. 1, middle). On the basis of indirect empirical comparisons it has been proposed 6,7 that the broader correlogram peaks can be described by the sum of two linear functions of e(t): f(t) = fo + a e(t) + b deldt (2) This relation provides a reasonable match when the coefficients (a and b) can be optimized for each case 5,7, but direct empirical comparisons 2,4 indicate that the difference between the correlogram peak and the derivative is typically briefer than the EPSP. The effect of synaptic noise on the transform -between EPSP and correlogram peak has not yet been analytically derived (except for the case of Gaussian noise1). However the threshold-crossing process has been simulated by a computer model which adds synaptic noise to the trajectories intercepting the EPSP 1. The correlograms generated by the simulation match the correlograms measured empirically for small EPSP's in motoneurons 2, confirming the validity of the model. Although synaptic noise distributes the triggered firings over a wider peak, the area of the correlogram peak, i.e., the number of motoneuron firings produced by an EPSP, is essentially preserved and remains proportional to EPSP amplitude for moderate noise levels. For unitary EPSP's (produced by 273 a single afferent fiber) in cat motoneurons, the number of firings triggered per EPSP (Np) was linearly related to the amplitude (h) of the EPSP 2: Np = (O.l/mv)· h (mv) + .003 (3) The fact that the number of triggered spikes increases in proportion to EPSP amplitude has also been confirmed for neocortical neurons 10; for cells recorded in sensorimotor cortex slices (probably pyramidal cells) the coefficient of h was very similar: 0.07/mv. This means that a typical unitary EPSP with amplitude of 100 Ilv, raises the probability that the postsynaptic cell fires by less than .01. Moreover, this increase occurs during a specific time interval corresponding to the rise time of the EPSP - on the order of 1 - 2 msec. The net increase in firing rate of the postsynaptic cell is calculated by the proportional decrease in interspike intervals produced by the triggered spikes 4. (While the above values are typical, unitary EPSP's range in size from several hundred IlV down to undetectable levels of severalllv., and have risetimes of.2 - 4 msec.) Inhibitory connections between cells, mediated by inhibitory postsynaptic potentials (IPSPs), produce a trough in the cross-correlogram. This reduction of firing probability below baseline is followed by a subsequent broad, shallow peak, representing the spikes that have been delayed during the IPSP. Although the effects of inhibitory connections remain to be analyzed more quantitatively, preliminary results indicate that small IPSP's in synaptic noise produce decreases in firing probability that are similar to the increases produced by EPSP's 4,5. DISYNAPTIC LINKS The effects of polysynaptic links between neurons can be understood as combinations of the underlying monosynaptic connections. A monosynaptic connection from cell A to cell B would produce a first-order cross-correlation peak P1(BIA,t), representing the conditional probability that neuron B fires above chance at time t, given a spike in cell A at time t = O. As noted above, the shape of this first-order correlogram peak is largely proportional to the EPSP derivative (for cells whose interspike interval exceeds the duration of the EPSP). The latency of the peak is the conduction time from A to B (Fig. 2 top left). In contrast, several types of disynaptic linkages betw.een A and B, mediated by a third neuron C, will produce a second-order correlation peak between A and B. A disynaptic link may be produced by two serial monosynaptic connections, from A to C and from C to B (Fig. 2, bottom left), or by a common synaptic input from C ending on both A and B (Fig. 2, bottom right). In both cases, the second-order correlation between A and B produced by the disynaptic link would be the convolution of the two firstorder correlations between the monosynaptically connected cells: (4) 274 As indicated by the diagram, the cross-correlogram peak P2(BIA,t) would be smaller and more dispersed than the peaks of the underlying first-order correlation peaks. For serial connections the peak would appear to the right of the origin, at a latency that is the sum of the two monosynaptic latencies. The peak produced by a common input typically straddles the origin, since its timing reflects the difference between the underlying latencies. Monosynaptic connection => First-order correlation @ I \ t \ LJA,,-_~_(_B_I_A_' t_)_ -----..'t~(AIB,t) = ~(~IA,-t) 1 Disynaptic connection ~ Second-order correlation Serial connection t \ I \ : \. A ~ (C I A) H\ t \ ~ t \ t \ t \ I \ @ : \~(BIC) P(BIA) _________ ~,_2 __ ---Common input @ r--A---t : t j I :" V I'" t \ ll(AIC) t \. P(BIC) 1 \/\ ' J t"-_-----___ /\.~(BIA) -L Fig. 2. Correlational effects of monosynaptic and disynaptic links between two neurons. Top: monosynaptic excitatory link from A to B produces an increase in firing probability of B after A (left). As with all correlograms this is the time-inverted probability of increased firing in A relative to B (right). Bottom: Two common disynaptic links between A and B are a serial connection via C (left) and a common input from C. In both cases the effect of the disynaptic link is the convolution of the underlying monosynaptic links. 275 This relation means that the probability that a spike in cell A will produce a correlated spike in cell B would be the product of the two probabilities for the intervening monosynaptic connections. Given a typical Np of .Ol/EPSP, this would reduce the effectiveness of a given disynaptic linkage by two orders of magnitude relative to a monosynaptic connection. However, the net strength of all the disynaptic linkages between two given cells is proportional to the number of mediating intemeurons (C}, since the effects of parallel pathways add. Thus, the net potency of all the disynaptic linkages between two cells could approach that of a monosynaptic linkage if the number of mediating interneurons were sufficiently large. It should also be noted that some intemeurons may fire more than once per EPSP and have a higher probability of being triggered to fire than motoneurons 11. For completeness, two other possible disynaptic links between A and B involving a third cell C may be considered. One is a serial connection from B to C to A, which is the reverse of the serial connection from A to B. This would produce a P2(BIA) with peak to the left of the origin. The fourth circuit involves convergent connections from both A and B to C; this is the only combination that would not produce any causal link between A and B. The effects of still higher-order polysynaptic linkages can be computed similarly, by convolving the effects produced by the sequential connections. For example, trisynaptic linkages between four neurons are equivalent to combinations of disynaptic and monosynaptic connections. The cross-correlograms between two cells have a certain symmetry, depending on which is the reference cell. The cross-correlation histogram of cell B referenced to A is identical to the time-inverted correlogram of A referenced to B. This is illustrated for the monosynaptic connection in Fig.2, top right, but is true for all correlograms. This symmetry represents the fact that the above-chance probability of B firing after A is the same as the probability of A firing before B: P(BIA, t) = P(AIB, -t) (5) As a consequence, polysynaptic correlational links can be computed as the same convolution integral (Eq. 4), independent of the direction of impulse propagation. P ARALLEL PATHS AND FEEDBACK LOOPS In addition to the simple combinations of pair-wise connections between neurons illustrated above, additional connections between the same cells may form circuits with various kinds of loops. Recurrent connections can produce feedback loops, whose correlational effects are also calculated by convolving effects of the underlying synaptic links. Parallel feed-forward paths can form multiple pathways between the same cells. These produce correlational effects that are the sum of the effects of the individual underlying connections. The simplest feedback loop is formed by reciprocal connections between a pair of cells. The effects of excitatory feedback can be computed by 276 successive cO?1volutions of the underlying monosynaptic connections (Fig. 3 top). Note that such a positive feedback loop would be capable of sustaining activity only if the connections were sufficiently potent to ensure postsynaptic firing. Since the probabilities of triggered firings at a single synapse are considerably less than one, reverberating activity can be sustained only if the number of interacting cells is correspondingly increased. Thus, if the probability for a single link is on the order of .01, reverberating activity can be sustained if A and B are similarly interconnected with at least a hundred cells in parallel. Connections between three neurons may produce various kinds of loops. Feedforward parallel pathways are formed when cell A is monosynaptically connected to B and in addition has a serial disynaptic connection through C, as illustrated in Fig. 3 (bottom left); the correlational effects of the two linkages from A to B would sum linearly, as shown for excitatory connections. Again, the effect of a larger set of cells {C} would be additive. Feedback loops could be formed with three cells by recurrent connections between any pair; the correlational consequences of the loop again are the convolution of the underlying links. Three cells can form another type loop if both A and B are monosynaptically connected, and simultaneously influenced by a common interneuron C (Fig. 3 bottom right). In this case the expected correlogram between A and B would be the sum of the individual components -- a common input peak around the origin plus a delayed peak produced by the serial connection. Feedback loop Parallel 1-----' .. l;---~ ... . ... / \.... . ...... -', " ..•. :.... . ......... . .... .... ;'" .... .... . ...... . .. ' -', .... '" I: ••••• •••. jeedfOrward path I I : PI (BIA) +P 2 (BIA) t t t Common input loop I t PI (BIA)+P 2 (BIA) :/\ ___ .;.J l~ ___ _ Fig. 3. Correlational effects of parallel connections between two neurons. Top: feedback loop between two neurons A and B produces higher-order effects equivalent to convolution of mono~aptic effects. Bottom: Loops formed by parallel feed forward paths (left) and by a common mput concurrent with a monosynaptic link (right) produce additive effects. 277 CONCLUSIONS Thus, a simple computational algebra can be used to derive the correlational effects of a given network structure. Effects of sequential connections can be computed by convolution and effects of parallel paths by summation. The inverse problem, of deducing the circuitry from the correlational data is more difficult, since similar correlogram features may be produced by different circuits 9. The fact that monosynaptic links produce small correlational effects on the order of .01 represents a significant constraint in the mechanisms of information processing in real neural nets. For example, secure propagation of activity through serial polysynaptic linkages requires that the small probability of triggered firing via a given link is compensated by a proportional increase in the number of parallel links. Thus, reliable serial conduction would require hundreds of neurons at each level, with appropriate divergent and convergent connections. It should also be noted that the effect of intemeurons can be modulated by changing their activity. The intervening cells need to be active to mediate the correlational effects. As indicated by eq. I, the size of the correlogram peak is proportional to the firing rate (fo) of the postsynaptic cell. This allows dynamic modulation of polysynaptic linkages. The greater the number of links, the more susceptible they are to modulation. Acknowledgements: The author thanks Mr. Garrett Kenyon for stimulating discussions and the cited colleagues for collaborative efforts. This work was supported in part by Nll-I grants NS 12542 and RR00166. REFERENCES 1. Bishop, B., Reyes, A.D., and Fetz E.E., Soc. for Neurosci Abst. 11:157 (1985). 2. Cope, T.C., Fetz, E.E., and Matsumura, M., J. Physiol. 390:161-18 (1987). 3. Fetz, E.E. and Cheney, P.D., J. Neurophysiol. 44:751-772 (1980). 4. Fetz, E.E. and Gustafsson, B., J. Physiol. 341:387-410 (1983). 5. Gustafsson, B., and McCrea, D., J. Physiol. 347:431-451 (1984). 6. Kirkwood, P.A., J. Neurosci. Meth. 1:107-132 (1979). 7. Kirkwood, P.A., and Sears, T._ J. Physiol. 275:103-134 (1978). 8. Knox, C.K., Biophys. J. 14: 567-582 (1974). 9. Moore, G.P., Segundo, J.P., Perkel, D.H. and Levitan, H., Biophys. J. 10:876900 (1970). 10. Reyes, A.D., Fetz E.E. and Schwindt, P.C., Soc. for Neurosci Abst. 13:157 (1987). 11. Surmeier, D.J. and Weinberg, R.J., Brain Res. 331:180-184 (1985).
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DISCOVERING STRUCfURE FROM MOTION IN MONKEY, MAN AND MACHINE Ralph M. Siegel· The Salk Institute of Biology, La Jolla, Ca. 92037 ABSTRACT 701 The ability to obtain three-dimensional structure from visual motion is important for survival of human and non-human primates. Using a parallel processing model, the current work explores how the biological visual system might solve this problem and how the neurophysiologist might go about understanding the solution. INTRODUcnON Psychophysical experiments have shown that monke1 and man are equally adept at obtaining three dimensional structure from motion . In the present work, much effort has been expended mimicking the visual system. This was done for one main reason: the model was designed to help direct physiological experiments in the primate. It was hoped that if an approach for understanding the model could be developed, the approach could then be directed at the primate's visual system. Early in this century, von Helmholtz2 described the problem of extracting three-dimensional structure from motion: Suppose, for instance, that a person is standing still in a thick woods, where it is impossible for him to distinguish, except vaguely and roughly, in the mass of foliage and branches all around him what belongs to one tree and what to another, or how far apart the separate trees are, etc. But the moment he begins to move forward, everything disentangles itself, and immediately he gets an apperception of the material content of the woods and their relation to each other in space, just as if he were looking at a good stereoscopic view of it. If the object moves, rather than the observer, the perception of threedimensional structure from motion is still obtained. Object-centered structure from motion is examined in this report. Lesion studies in monkey have demonstrated that two extra-striate visual cortices called the middle temporal area (abbreviated MT ·Current address: Laboratory of Neurobiology, The Rockefeller University, 1230 York Avenue, New York, NY 10021 © American Institute of Physics 1988 702 or V5) and the medial superior temporal area (MST)3,4 are involved in obtaining structure from motion. The present model is meant to mimic the V5-MST part of the cortical circuitry involved in obtaining structure from motion. The model attempts to determine ifthe visual image corresponds to a three-dimensional object. TIlE STRUCfURE FROM MOTION STIMULUS The problem that the model solved was the same as that posed in the studies of monkey and man 1. Structured and unstructured motion displays of a hollow, orthographically projected cylinder were computed (Figure 1). The cylinder rotates about its vertical axis. The unstructured stimulus was generated by shuffling the velocity vectors randomly on the display screen. The overall velocity and spatial distribution for the two displays are identical; only the spatial relationships have been changed in the unstructured stimulus. Human subjects report that the points are moving on the surface of a hollow cylinder when viewing the structured stimulus. With the unstructured stimulus, most subjects report that they have no sense of three-dimensional structure. A. Rotating Cylinder , ~ B. Orthographic Projection ~ .... +--~ +-.. ~~ ) ~ ~ + ~~ -:/' ~~ -+ ~ ~~ ~ ,. c. Unstructured Display ~ ~ ~ ~ +-.. ~~ ~ ~ ---+ + ~ ---+~ ~ ~ ~ ~ ~ Figure 1. The structured and unstructured motion stimulus. A) "N" pomts are randomly placed on the surface of a cylinder. B) The points are orthographically projected. The motion gives a strong percept of a hollow cylinder. C) The unstructured stimulus was generated by shuffling the velocity vectors randomly on the screen. FUNCTIONALARCHITECfUREOFTIlEMODEL As with the primate subjects, the model was required to only indicate whether or not the display was structured. Subjects were not required to describe the shape, velocity or size of the cylinder. Thus the output cell* of the model signaled "1" if *By cell, I mean a processing unit of the model which may correspond to a single neuron or group of neurons. The term neuron refers only to the actual wetware in the brain. 703 structured and "0" if not structured. This output layer corresponds to the cortical area MST of macaque monkey which appear to be sensitive to the global organization of the motion image5. It is not known if MST neurons will distinguish between structured and unstructured images. The input to the model was based on physiological studies in the maca~ue monkey. Neurons in area V5 have a retinotopic representation of visual space ,7. <l.I CIl C o 0CIl <l.I s.... 1 -O.~-+-~I--""-"""-+-"'" -3 -2 -1 0 1 2 3 retinal position (deg) Figure 2. The receptive field of an input layer cell. The optimal velocity is "vo". For each retinotopic location there is an encoding of a wide range of velocitiesS. Thus in the model's input representation, there were cells that represent different combinations of velocity and retinotopic spatial position. Furthermore motion velocity neurons in V5 have a center-surround opponent organization9. The width of the receptive fields was taken from the data of Albright et al.S. A typical receptive field of the model is shown in Figure 2. lt was possible to determine what the activity of the input cells would be for the rotating cylinder given this representation. The activation pattern of the set of input cells was computed by convolving the velocity points with the difference of gaussians. The activity of the 100 input cells for an image of 20 points, with an angular velocity of SO/sec is presented in Figure 3. >. -' ..... () o Q) > Relinotopic map Retinotopic map Structure = 1 Structure = 0 Figure 3. The input cell's activation pattern for a structured and unstructured stimuIus. The circles correspond to the cells of the input layer. The contours were com704 puted using a linear interpolation between the individual cells. The horizontal axis corresponds to the position along the horizontal meridian. The vertical axis corresponds to the speed along the horizontal meridian. Thus activation of a cell in the upper right hand corner of the graph correspond to a velocity of 300 / sec towards the right at a location of 30 to the right along the horizontal meridian. Inspection of this input pattern suggested that the problem of detecting three-dimensional structure from motion may be reduced to a pattern recognition task. The problem was then: "Given a sparsely sampled input motion flow field, determine whether it corresponds best to a structured or unstructured object." Itwas next necessary to determine the connections between the two input and output layers such that the model will be able to correctly signal structure or no structure (1 or 0) over a wide range of cylinder radii and rotational velocities. A parallel distributed network of the type used by Rosenberg and Sejnowski 10 provided the functional architecture (Figure 4). o M I Figure 4. The parallel architecture used to extract structure from motion. The input layer (I), corresponding to area V5, mapped the position and speed along the horizontal axis. The output layer (0) corresponded to area MST that, it is proposed, signals structure or not. The middle layer (M) may exist in either V5 or MST. The input layer of cells was fully connected to the middle layer of cells. The middle layer of cells represented an intermediate stage of processing and may be in either V5 or MST. All of the cells of the middle layer were then fully connected to the output cell. The inputs from cells of the lower layer to the next higher level were summed linearly and then "thresholded" using the Hill equation X3/(X3 + 0.53). The weights between the layers were initially chosen between.±.1. The values of the weights were then adjusted using back-propagation methods (steepest descent) so that the network would "learn" to correctly predict the structure of the input image. The model learned to correctly perform the task after about 10,000 iterations (Figure 5). o. o. Figure 5. The "education" of the network to perform the structure from motion problem. The iteration number is plotted against the mean square error. The error is defined as the difference between the model's prediction and the known structure. The model was trained on a set of structured and unstructured cylinders o 100002000030000 0000 with a wi?e range o~ ~adii, number of points, 4 and rotatlOnal velOCItIes. Iteration number 705 PSYCHOPHYSICAL PERFORMANCE OF THE MODEL The model's performance was comparable to that of monkey and man with respect to fraction of structure and number of points in the display (Figure 6). The model was indeed performing a global analysis as shown by allowing the model to view only a portion of the image. Like man and monkey, the model's performance suffers. Thus it appears that the model's performance was quite similar to known monkey and human psychophysics. 1 Output 1 0.8 -..- monkey .8 ..... man 0.6 machine 0.6 0.4 0.4 -.l- monkey -man 0.2 0.2 machine 0 0 0 0.2 0.4 0.6 0.8 1 0 32 64 96 128 Fraction structure Number of points Figure 6. Psychophysical performance of the model. A. The effect of varying the fraction of structure. As the fraction of structure increase, the model's performance improves. Thirty repetitions were averaged for each value of structure for the model. The fraction of structure is defined as (1-Rs/Rc), where Rs is the radius of shuffling of the motion vectors and Rc is the radius of the cylinder. The human and monkey data are taken from psychophysical studies 1. HOW IS IT DONE? The model has similar performance to monkey and man. It was next possible to examine this artificial network in order to obtain hints for studying the biological system. Following the approach of an electrophysiologist, receptive field maps for all the cells of the middle and ou tput layers were made by activating individual inpu t cells. The receptive field of some middle layer cells are shown in Figure 7. The layout of these maps are quite similar to that of Figure 4. However, now the activity of one cell in the middle layer is plotted as a function of the location and speed of a motion stimulus in the input layer. One could imagine that an electrode was placed in one of the cells of the middle layer while the experimentalist moved a bar about the 706 horizontal meridian with different locations and speeds. The activity of the cell is then plotted as a function of position and space. ~ -f Relinolopic map 30 rJ \ 00 (I') I ~ "'::>:->::::':-:-" ... ~~=-~~-L~~~~~~~~ Figure 7. The activity of two different cells in the middle layer. Activity is plotted as a contour map as a function of horizontal position and speed. Dotted lines indicate inhibition. These middle layer receptive field maps were interesting because they appear to be quite simple and symmetrical. In some, the inhibitory central regions of the receptive field were surrounded by excitatory regions (Figure 7A). Complementary cells were also found. In others, there are inhibitory bands adjacent to excitatory bands (Figure 7B). The above results suggest that neurons involved in extracting structure from motion may have relatively simple receptive fields in the spatial velocity domain. These receptive fields might be thought of as breaking the image down into component parts (i.e. a basis set). Correct recombination of these second order cells could then be used to detect the presence of a three-dimensional structure. The output cell also had a simple receptive field again with interesting symmetries (Figure 8). However, the receptive field analysis is insufficient to indicate the role of the cell. Therefore in order to properly understand the "meaning" of the cell's receptive field, it is necessary to use stimuli that are "real world relevant" - in this case the structure from motion stimuli. The output cell would give its maximal response only when a cylinder stimulus is presented. Figure 8. The receptive field map of the output layer cell. Nothing about this receptive field structure indicates the cell is involved in obtaining structure from motion. 707 This work predicts that neurons in cortex involved in extracting structure from motion will have relatively simple receptive fields. In order to test this hypothesis, it will be necessary to make careful maps of these cells using small patches of motion (Figure 9). Known qualitative results in areas V5 and MST are consistent with, but do not prove, this hypothesis. As well, it will be necessary to use "relevant" stimuli (e.g. three-dimensional objects). If such simple receptive fields are indeed used in structure from motion, then support will be found for the idea that a simple cortical circuit (e.g. center-surround) can be used for many different visual analyses. • ru Fix point Motion patches consisting of random dots with variable velocity . Figure 9. It may be necessary to make careful maps of these neurons using small patches of motion, in order to observe the postulated simple receptive field properties of cortical neurons involved in extracting structure from motion. Such structures may not be apparent using hand moved bar stimuli. DISCUSSION In conclusion, it is possible to extract the three-dimensional structure of a rotating cylinder using a parallel network based on a similar functional architecture as found in primate cortex. The present model has similar psychophysics to monkey and man. The receptive field structures that underlie the present model are simple when viewed using a spatial-velocity representation. It is suggested that in order to understand how the visual system extracts structure from motion, quantitative spatial-velocity maps of cortical neurons involved need to be made. One also needs to use stimuli derived from the "real world" in order to understand how they may be used in visual field analysis. There are similarities between the shapes of the receptive fields involved in analyzing structure from motion and receptive fields in striate cortex 11. It may be that similar cortical mechanisms and connections are used to perform different functions in different cortical areas. Lastly, this model demonstrates that the use of parallel architectures that are closely modeled on the cortical representation is a computationally efficient means to solve problems in vision. Thus as a final caveat, I would like to advise the creators of networks that solve ethologically realistic problems to use solutions that evolution has provided. 708 REFERENCES 1. R.M. Siegel and R.A. Andersen, Nature {Lond.} (1988). 2. H. von Helmholtz, Treatise on Physiological Optics {Dover Publications, N.Y., 1910}, p. 297. . 3. R.M. Siegel and R.A. Andersen, Soc. Neurosci. Abstr., 12, p. 1183 {1986}. 4. R.M. Siegel and R.A Andersen, Localization of function in extra-striate cortex: the effect of ibotenic acid lesions on motion sensitivity in Rhesus monkey, {in preparation}. 5. K. Tanaka, K. Hikosaka, H. Saito, M. Yukie, Y. Fukada, and E. Iwai, J., Neurosci., ~, pp. 134-144 {1986}. 6. S.M. Zeki, Brain Res.,~, pp. 528-532 {1971}. 7. J.H.R. Maunsell and D.C. VanEssen, J. Neurophysiol., 49, pp. 1127-1147 {1983}. 8. T.D. Albright, R. Desimone, and C.G. Gross, J. NeurophysioI., 51, pp. 16-31 {1984}. 9. J. Allman, F. Miezen, and E. McGuinness, Ann. Rev. Neurosci., 8, pp. 407-430 (1985). 10. C.R. Rosenberg and T.J. Sejnowski, in: Reports of the Cognitive Neuropsychology Laboratory, John-Hopkins University {1986}. 11. D.H. Hubel and T.N. Wiesel, Proc. R. Soc. Lond. B., 198, pp.I-59 {1977}. This work was supported by the Salk Institute for Biological Studies, The San Diego Supercomputer Center, and PHS NS07457-02.
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632 STATIC AND DYNAMIC ERROR PROPAGATION NETWORKS WITH APPLICATION TO SPEECH CODING A J Robinson, F Fallside Cambridge University Engineering Department Trumpington Street, Cambridge, England Abstract Error propagation nets have been shown to be able to learn a variety of tasks in which a static input pattern is mapped outo a static output pattern. This paper presents a generalisation of these nets to deal with time varying, or dynamic patterns, and three possible architectures are explored. As an example, dynamic nets are applied to tbe problem of speech coding, in which a time sequence of speech data are coded by one net and decoded by another. The use of dynamic nets gives a better signal to noise ratio than that achieved using static nets. 1. INTRODUCTION This paper is based upon the use of the error propagation algorithm of Rumelbart, Hinton and Williams l to train a connectionist net. The net is defined as a set of units, each witb an activation, and weights between units which determine the activations. The algorithm uses a gradient descent technique to calculate the direction by which each weight should be changed in order to minimise the summed squared difference between the desired output and the actual output. Using this algorithm it is believed that a net can be trained to make an arbitrary non-linear mapping of the input units onto the output units if given enough intermediate units. This 'static' net can be used as part of a larger system with more complex behaviour. The static net has no memory for past inputs, but many problems require the context of the input in order to c.ompute the answer. An extension to the static net is developed, the 'dynamic' net, which feeds back a section of the output to the input, so creating some internal storage for context, and allowing a far greater class of problems to be learned. Previously this method of training time dependence into uets has suffered from a computational requirement which increases linearly with the time span of the desired context. The three architectures for dynamic uets presented here overcome this difficulty. To illustrate the power of these networks a general coder is developed and applied to the problem of speech coding. The non-liuear solution found by training a dynamic net coder is compared with an established linear solution, and found to have an increased performance as measured by the signal to noise ratio. 2. STATIC ERROR PROPAGATION NETS A static Ret is defined by a set of units and links between the units. Denoting 0i as the value of the ith unit, and wi,l as the weight of the link between Oi and OJ, we may divide up the units into input units, hidden units and output units. If we assign 00 to a. constant to form a @ American Institute of Physics 1988 633 bias, the input units run from 01 up to on",\., followed by the hidden units to onh • .t and then the output units to On.".' The values of the input units are defined by the problem and the values of the remaining units are defined by: i-I neti ~1LJ' '0' ',1 J (2.1) j=O °i !(net;) (2.2) where !( x) is any continuous monotonic non-linear function and is known as the activation function. The function used the application is: !(x) 2 ----1 1 + e- z", (2.3) These equations define a net which has the maximum number of interconnections. This arrangement is commonly restricted to a layered structure in which units are only connected to the immediately preceding layer. The architecture of these nets is specified by the number of input, output and hidden units. Diagrammatically the static net is transformation of an input 'U, onto the output y, as in figure 1. static net figure 1 The net is trained by using a gradient descent algorithm which mlDlsmises an energy term, E, defined as the summed squared error between the actual outputs, ai, and the target outputs, ti . The algorithm also defines an error signal, Oi, for each unit: 1 "lint E ~ (ti -- od 2 (2.4) 2 i=nlw l+1 [Ii !' (netd(ti - 0;) nhid < i ::; nout (2.5 ) " lint .f' (net;) ~ OiWj,i ninp < i ::; nhid (2.6) j=i+l where f' (x) is the derivative of !( x). The error signal and the adivations of the units define the change in each weight, D. Wi,j' (2.7) where '1 is a constant of proportionality which determines the learning rate. The above equations define the error signal, 0;, for the input units as well as for the hidden units. Thus any number of static nets can be connected together, the values of Oi being passed from input units of one net to output units of the preceding net. It is this ability of error propagation nets to be 'glued' together in this way that enables the construction of dynamic nets. 3. DYNAMIC ERROR PROPAGATION NETS The essential quality of the dynamic net is is that its behaviour is determined both by the external input to the net, and also by its own internal state. This state is represented by the 634 activation of a group of units. These units form part of the output of a st.atic net and also part of the input to another copy of the same static net in the next time period. Thus the state units link multiple copies of static nets over time to form a dynamk net. 3.1. DEVELOPMENT FROM LINEAR CONTROL THEORY The analogy of a dynamic net in linear systems2 may be stated as: (3.1.1) (3.1.2) where up is the input vector, zp the state vector, and Yp the output vector at the integer time p. A, Band C are matrices. The structure of the linear systems solution may be implemented as a non-linear dynamic net by substituting the matrices A, Band C by statk nets, represented by the non-linear functions A[.]' B[.] and C[.]. The summation operation of Azp and Bup could be achieved using a net with one node for each element in z and u and with unity weights from the two inputs to the identity activation function f( x) = z. Alternatively this net can be incorporated into the A[.] net giving the architecture of figure 2. B [.] y(p+l) dynamic t---of A[.] e[.] y(p+l) net x(p+l) Time Time Delay Delay figure 2 figure 3 The three networks may be combined into one, as in figure 3. Simplicity of architecture is not just an aesthetic consideration. If three nets are used then each one must have enough computational power for its part of the task, combining the nets means that only the combined power must be sufficient and it allows common computations can be shared. The error signal for thf' output Yp+l, can be calculated by comparison with the desired output. However, the error signal for thf' state units, x P ' is only given by the net at time p+l, which is not known at time p. Thus it is impossible to use a single backward pass to train this net. It is this difficulty which introduces the variation in the architectures of dynamic nets. 3.2. THE FINITE INPUT DURATION (FID) DYNAMIC NET If the output of a dynamic net, YP' is df'pendf'nt on a finite number of previous inputs, up_p to up, or if this assumption is a good approximation, then it is possible to formulate the 635 learning algorithm by expansion of the dynamk net for a finite time, as in figure 4. This formulation is simlar to a restricted version of the recurrent net of Rumelhart, Hinton and Williams. 1 dynamic net (p-2) dynamic net (p-l) figure 4 yep) dynamic net (p) x(p+l) y(p+l) Consider only the component of the error signal in past instantiations of the nets which is the result of the error signal at time t. The errot signal for YP is calculated from the target output and the ('rror signal for xr is zero. This combined error signal is propagated back though the dynamic net at p to yield the error signals for up and xp' Similarly these error signals can then be propagated back through the net at t - P, and so on for all relevant inputs. The summed error signal is then used to change the weights as for a static net. Formalising the FID dynamic net for a general time q, q ~ p: n, °q,i tq,i 6'1,' Wi,j ~Wq,i,i ~wi,i is the number of state units is the output value of unit i at time q is the target value of unit i at time q is the error value of unit i at time q is the weight between 0; and OJ is the weight change for this iteration at time q is the total weight change for this iteration These values are calculated in the same way as in a static net, netq,i i-1 L Wi,jOq,j j=O f(net q,.) f' (netq,d( tq,i - 0'1,;) nullt !'(n('tq,;) L 6q,jWj,i j-=i+l nhid + n, < i :S nout nhid < i :S nhid + n, (3.2.1) (3.2.2) (3.2.3) (3.2.4) (3.2.5) (3.2.6) and the total weight change is given by the summation of the partial weight changes for all 636 previous times. p L Llu'q,i,j q=p-P p L 7]6q,iOq,j q=p-P (3.2.7) (3.2.8) Thus, it is possible to train a dynamic net to incorporate the information from any time period of finite length, and so l~arn any function which has a finite impulse response.· In some situations the approximation to a finite length may not be valid, or the storage and computational requirements of such a net may not be feasible. In such situations another approach is possible, the infinite input duration dynamic net. 3.3. THE INFINITE INPUT DURATION (lID) DYNAMIC NET Although the forward pass of the FID net of the previous section is a non-linear process, th .. backward pass computes the efred of small variations on the forward pass, and is a linear process. Thus the recursive learning procedure described in the previous section may be compressed into a single operation. Given the target values for the output of the net at time p, equations (3.2.3) and (3.2.4) define valu~s of 6p,i at the outputs. If we denote this set of 6p,i by Dp then equation (3.2.5) states that any 6p ,i in the net at time p is simply a linear transformation o( Dp. Writing the transformation matrix as S: (3.3.1) In particular the set of 6p ,i which is to be fed back into the network at time p - 1 is also a linear transformation of Dp (3.3.2) or for an arbitrary time q: (3.3.3) so substituting equations (3.3.1) and (3.3.3) into equation (3.2.8): p ( IT T,) D,o"j LlU'i,j 7]L Sq,i (3.3.4) q=-oo 7=q+l 7]Mp,i,i Dp (3.3.5) where: p ( IT T}"j M . , L Sq,i (3.3.6) p,',) q=-oo "=q+l • This is a restriction on the class of functions which can be learned, the output will always be affected in some way by all previous inputs giving an infinite impulse response performance. 637 and note that Mp,i,i can be written in terms of Mp-1,i,i : Sp,i ( IT T,.) 0p,i + (I: Sq,i ,.=p+l q=-oo (3.3.7) M ., P,- ,J Sp,iop,i + Mp-1,i,iTp (3.3 .8) Hence we can calculate the weight changes for an infinite recursion using only the finite matrix M, 3.3. THE STATE COMPRESSION DYNAMIC NET The previous architectures for dynamic nets rely on the propagation of the error signal hack ill time to define the format of the information in the state units. All alternative approach is to use another error propagation net to define the format of the state units. The overall architecture is given in figure 5. 1-----\1 Tranlllatort---""'" Bncoder x(p+1) net Decoder net figure 5 y(p+1) net The encoder net is trained to code the current input and current state onto the next state, while the decoder net is trained to do the reverse operation. The tran81ator net code8 the next state onto the desired output. This encoding/decoding attempts to represent the current input and the current state in the next state, and by the recursion, it will try to represent all previous inputs. Feeding errors back from the translator directs this coding of past inputs to those which are useful in forming the output. 3.4. COMPARISON OF DYNAMIC NET ARCHITECTURES III comparing the three architectures for dynamic nets, it is important to consider the computational and memory requirements, and how these requirements scale with increasing context. To train an FID net the net must store the past activations of the all the units within the time span of thel'necessary context, Using this minimal storage, the computational load scales proportiona.lly to the time span considered, as for every new input/output pair the net must propagate an error signal back though all the past nets. However, if more sets of past activations are stored in a buffer, then it is possible to wait until this buffer is full before computing the weight changes. As the buffer size increases the computational load in 638 calculating the weight changes tends to that of a single backward pass through the units, and so becomes independent of the amount of coutext. The largest matrix required to compute the 110 net is M, which requires a factor of the number of outputs of the net more storage than the weight matrix. This must be updated on each iteration, a computational requirement larger than that of the FlO net for smaJl problems3 . However, if this architecture were implemented on a paraJlel machine it would be possible to store the matrix M in a distributed form over the processors, and locally calculate the weight changes. Thus, whilst the FID net requires the error signal to be propagated back in time in a strictly sequential manner, the 110 net may be implemented in paraJld, with possible advantages on parallel machines. The state compression net has memory and computational requirements independent of the amount of context. This is achieved at the expense of storing recent information in the state units whether it is required to compute the output or not. This results in an increased computational and memory load over the more efficient FID net when implemented with a buffer for past outputs. However, the exclusion of external storage during training gives this architecture more biological plausibili ty, constrained of course by the plausibility of the error propagation algorithm itself. With these considerations in mind, the FlO net was chosen to investigate a 'real world' problem, that of the coding of the speech waveform. 4. APPLICATION TO SPEECH CODING The problem of speech coding is one of finding a suitable model to remove redundancy and hence reduce the data rate of the speech. The Boltzmann machine learning algorithm has already been extended to deal to the dynamic case and applied to speech recognition4. However, previous use of error propagation nets for speech processing has mainly been restricted to explicit presentation of the context 5,6 or explicit feeding back the output units to the input 7,8, with some work done in usillg units with feedback links to themselves9 . In a similar area, static error propagation nets have been used to perform image coding as well as cOllventional techniques1o. 4.1. THE ARCHITECTURE OF A GENERAL CODER The coding principle used in this section is not restricted to c.oding speech data. The general problem is one of encoding the present input using past input context to form the transmitted signal, and decoding this signal using the context ofthe coded signals to regenerate the original input. Previous sections have shown that dynamic nets are able to represent context, so two dynamic, nets in series form the architecture of the coder, as in figure 6. This architecture may be specified by the number of input, state, hidden and transmission units. There are as many output units as input units and, in this application, both the transmitter and receiver have the same number of state and hidden units. The input is combined with the internal state of the transmitter to form the coded signal, and then decoded by the receiver using its internal state. Training of the net involves the comparison of the input and output to form the error signal, which is thell propagated back through past instantiations of the receiver and transmitter in the same way as a for a FID dynamic net. It is useful to introduce noise into the coded signal during the training to reduce the information capacity of the transmission line. This forces the dynamic 11ets to incorporate time information, without this constraint both nets can learn a simple transformation without any time dependence. The noise can be used to simulate quantisation of the coded signal so 639 .. , , input J coded signal output I • • TX ax r-\ r-\ rI ~ rI I"" ioTime VITime I Delay ~ Delay \-figure 6 quantifying the transmission rate. Unfortunately, a straight implementation of quantisation violates tbe requirement of the activation function to be continuous, which is necessary to train the net. Instead quantisation to n levels may be simulated by adding a random value distributed uniformly in the range + 1/ n to -1/ n to each of the channels in the coded signal. 4.2. TRAINING OF THE SPEECH CODER The chosen problem was to present a sinJZ;le sample of digitised speech to the input, code to a single value quantised to fifteen levels, and then to reconstruct tile original speech at the output. Fifteen levels was chosen as the point where there is a marked loss in the intelligibility of the speech, so implementation of these coding schemes gives an audible improvement. Two version of the coder net were implemented, both nets had eight hidden units, with no state units for the static time independent case and four state units for the dynamic time dependent case. The data for this problem was 40 seconds of speech from a single male speaker, digit,ised to 12 bits at 10kHz and recorded in a laboratory environment. The speech was divided into two halves, the first was used for training and the second for testing. The static and the dynamic versions of the architecture were trained on about 20 passes through the training data. After training the weights were frozen and the inclusion of random noise was replaced by true quantisation of the coded representation. A further pass was then made through the test data to yield the performance measurements. The adaptive training algorithm of Chan 11 was used to dynamically alter the learning rates during training. Previously these machines were trained with fixed learning rates and weight update after every sample3 , and the use of the adaptive t.raining algorithm has been found to result in a substantially deeper energy minima. Weights were updated after every 1000 samples, that is about 200 times in one pass of the training data. 4.3. COMPARISON OF PERFORMANCE The performance of a coding schemes can be measured by defining the noise energy as half the summed squared difference between the actual output and the desired output. This energy is the quantity minimised by the error propagation algorithm. The lower the noise energy in relation to the energy of the signal, the higher the performance. Three non-connectionist coding schemes were implemented for comparison with the static 640 and dynamic net coders. In the first the signal is linearly quantised within the dynamic range of the original signal. In the second the quantiser is restricted to operate over a reduced dynamic range, with values outside that range thresholded to the maximuJn and minimum outputs of the quantiser. The thresholds of the quantiser were chosen to optimise the signal to noise ratio. The third scheme used the technique of Differential Pulse Code Modulation (DPCM)12 which involves a linear filter to predict the speech waveform, and the transmitted signal is the difference between the real signal and the predicted signal. Another linear filter reconstructs the original signal from the difference signal at the receiver. The filter order of the DPCM coder was chosen to be the same as the number of state units in the dynamic net coder, thus both coders can store the same amount of context enabling a comparison with this established technique. The resulting noise energy when the signal energy was normalised to unity, and the corresponding signal to noise ratio are given in table 1 for the five coding techniques. coding method normalised signal to noise nOise energy ratio in dB linear, original thresholds 0.071 11.5 linear, optimum thresholds 0.041 13.9 static net 0.049 13.1 DPCM, optimum thresholds 0.037 14.3 dynamic net 0.028 15.5 table 1 The static net may be compared with the two forms of the linear quantiser. Firstly note that a considerable improvemeut in the signal to noise ratio may be achieved by reducing the thresholds of the qllantiser from the extremes of the input. This improvement is achieved because the distribution of samples in the input is concentrated around the mean value, with very few values near the extremes. Thus many samples are represented with greater accuracy at the expense of a few which are thresholded. The static net has a poorer performance than the linear quantiser with optimum thresholds. The form of the linear quantiser solution is within the class of problems which the static net can represent. It's failure to do so can be attributed to finding a local minima, a plateau in weight space, or corruption of the true steepest descent direction by noise introduced by updating the weights more than once per pass through the training data. The dynamic net may be compared with the DPCM coding. The output from both these coders is no longer constrained to discrete signal levels and the resulting noise energy is lower than all the previous examples. The dynamic net has a significantly lower noise energy than any other coding scheme, although, from the static net example, this is unlikely to be an optimal solution. The dynamic net achieves a lower noise energy than the DPCM coder by virtue of the non-linear processing at each unit, and the flexibility of data storage in the state units. As expected from the measured noise energies, there is an improvement in signal quality and intelligibility from the linear quantised speech through to the DCPM and dynamic net quantised speech. 5. CONCLUSION This report has developed three architectures for dynamic nets. Each architecture can be formulated in a way where the computational requirement is independent of the degree of context necessary to learn the solution. The FID architecture appears most suitable for 641 implementation on a sf'rial processor, t.hf' nn archit.f'd,11fe has possihle a(lvant,ages for implementation on parallel processors, and the state compression net has a higher degree of biological plausibility. Two FID dynamic nets have been coupled together to form a coder, and this has been applied to speech coding. Although the dynamic net coder is unlikely to have learned the optimum coding strategy, it does delUonstrate that dynamic nets can be used to 8.Chieve an improved performance in a real world task over an estaBlished conventional technique. One of the authors, A J Robinson, is supported by a maintenance grant from the U.K. Science and Engineering Research Council, and gratefully acknowledges this support. References [1] D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning internal representations by error propagation. In D. E. Rumelhart and J. L. McClelland, editors, Parallel Distributed Processing: E2:plorations in the M1crostructure of Cognition, Vol. 1: Foundations., Bradford Books/MIT Press, Cambridge, MA, 1986, [2] O. L. R. Jacobs. IntroductIOn to Contml Theory. Clarendon Press, Oxford, 1974. [3J A. J. Robinson and F. Fallside. The Utility Drit'en Dynamic Error Propagation Network. Technical Report CUED/F-INFENG/TR.l, Cambridge University Engineering Department, 1987. [4J R. W. Prager, T. D. Harrison, and F. Fallside, Boltzmann machines for speech recognition. Compllter Speech and Language, 1:3-27, 1986, [5] J. L. Elman and D. Zipser. Learning the Hidden Structure of Speech. ICS Report 8701, University of California, San Diego, 1987. [6] A. J. Robinson. Speech Rerognition wIth Associatille Networks. M.Phil Computer Speech and Language Processing thesis, Cambridge University Engineering Department, 1986. [7] M. I. Jordan. Serial Order: A Parallel Distributed Processing Approach. ICS Report 8604, Institute for Cognitive Science, University of California, San Diego, May 1986. [8] D. J, C. MacKay. A Method of Increa,sing the Conte2:tual Input to Adaptive Pattern Recognition Systems. Technical Report RIPRREP /1000 /14/87, Research Initiative in Pattern Recognition, RSRE, Malvern, 1987. [9) R. L. Watrous, L. Shastri, and A. H. Waibel. Learned phonetic discrimination using connectionist networks. In J. Laver and M. A. Jack, editors, Proceedings of the Etl.ropea,n Conference on Speech Technology, CEP Consultants Ltd, Edinburgh, September 1987. (10) G. W. Cottrell, P. Munro, and D Zipser. Image Compression by Back Propagation: An E2:ample of Existential Programming. ICS Report 8702, Institute for Cognitive Science, University of California, San Diego, Febuary 1986. [11) L. W . Chan and F. Fallside. An Adaptive Learning Algori.thm for Back Propaga.tion Networks. Technical Report CUED / F-INFENG/TR.2, Cambridge University Engineering Department, 1987, submitted to Compute?' Speech and Language. [12] L, R. Rabiner and R. W, Schefer. DIgital Processmg of Speech Signals. Prentice Hall, Englewood Cliffs, New Jersey, 1978.
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367 SCHEMA I'OR MOTOR CONTROL OT ILl ZING A NETWORK MODEL 01' THE CEREBELLUM James C. Houk, Ph.D. Northwestern University Medical School, Chicago, Illinois 60201 ABSTRACT This paper outlines a schema for movement control based on two stages of signal processing. The higher stage is a neural network model that treats the cerebellum as an array of adjustable motor pattern generators. This network uses sensory input to preset and to trigger elemental pattern generators and to evaluate their performance. The actual patterned outputs, however, are produced by intrinsic circuitry that includes recurrent loops and is thus capable of self-sustained activity. These patterned outputs are sent as motor commands to local feedback systems called motor servos. The latter control the forces and lengths of individual muscles. Overall control is thus achieved in two stages: (1) an adaptive cerebellar network generates an array of feedforward motor commands and (2) a set of local feedback systems translates these commands into actual movements. INTRODUCTION There is considerable evidence that the cerebellum is involved in the adaptive control of movement1, although the manner in which this control is achieved is not well understood. As a means of probing these cerebellar mechanisms, my colleagues and I have been conducting microelectrode studies of the neural messages that flow through the intermediate division of the cerebellum and onward to limb muscles via the rubrospinal tract. We regard this cerebellorubrospinal pathway as a useful model system for studying general problems of sensorimotor integration and adaptive brain function. A summary of our findings has been published as a book chapter2 . On the basis of these and other neurophysiological results, I recently hypothesized that the cerebellum functions as an array of adjustable motor pattern generators3. The outputs from these pattern generators are assumed to function as motor commands, i.e., as neural control signals that are sent to lower-level motor systems where they produce movements. According to this hypothesis, the cerebellum uses its extensive sensory input to preset the © American Institute of Physics 1988 368 pattern generators, to trigger them to initiate the production of patterned outputs and to evaluate the success or failure of the patterns in controlling a motor behavior. However, sensory input appears not to playa major role in shaping the waveforms of the patterned outputs. Instead, these waveforms seem to be produced by intrinsic circuity. The initial purpose of the present paper is to provide some ideas for a neural network model of the cerebellum that might be capable of accounting for adjustable motor pattern generation. Several previous authors have described network models of the cerebellum that, like the present model, are based on the neuroanatomical organization of this brain structure4,5,6. While the present model borrows heavily from these previous models, it has some additional features that may explain the unique manner in which the cerebellum processes sensory input to produce motor commands. A second purpose of this paper is to outline how this network model fits within a broader schema for motor control that I have been developing over the past several years3,7. Before presenting these ideas, let me first review some basic physiology and anatomy of the cerebelluml . SIGNALS AND CIRCUITS IN TRB CBRBBBLLUM There are three main categories of input fibers to the cerebellum, called mossy fibers, climbing fibers and noradrenergic fibers. As illustrated in Fig. 1, the mossy fiber input shows considerable fan-out via granule cells and parallel fibers. The parallel fibers in turn are arranged to provide a high degree of fan-in to individual Purkinje cells (P). These P cells are the sole output elements of the cortical portion of the cerebellum. Via the parallel fiber input, each P cell is exposed to approximately 200,000 potential messages. In marked contrast, the climbing fiber input to P cells is highly focused. Each climbing fiber branches to only 10 P cells, and each cell receives input from only one climbing fiber. Although less is known about input via noradrenergic fibers, it appears to be diffuse and even more divergent than the mossy fiber input. Mossy fibers originate from several brain sites transmitting a diversity of information about the external world and the internal state of the body. Some mossy fiber inputs are clearly sensory. They come fairly directly from cutaneous, muscle or vestibular receptors. Others are routed via the cerebral cortex where they represent highly processed visual, auditory or somatosensory information. Yet another category of mossy fiber transmits information about central motor commands (Fig. 1 shows one such pathway, from collaterals of the rubrospinal tract relayed 369 through the lateral reticular nucleus (L». The discharge rates of mossy fibers are modulated over a wide dynamic range which permits them to transmit detailed parametric information about the state of the body and its external environment. noradrenergic fibers sensorimotor cortex Motor Sensory Inputs --o~Ll ______________________________ ~)--~C~o~m~m=M~d=s~ rubrospinal tract Figure 1: Pathways through the cerebellum. This diagram, which highlights the cerebellorubrospinal system, also constitutes a circuit diagram for the model of an elemental pattern generator. The sole source of climbing fibers is from cells located in the inferior olivary nucleus. Olivary neurons are selectively sensitive to sensory events. These cells have atypical electrical properties which limit their discharge to rates less than 10 impulses/sec, and usual rates are closer to 1 impulse/sec. As a consequence, 370 individual climbing fibers transmit very little parametric information about the intensity and duration of a stimulus; instead, they appear to be specialized to detect simply the occurrences of sensory events. There are also motor inputs to this pathway, but they appear to be strictly inhibitory. The motor inputs gate off responsiveness to self-induced (or expected) stimuli, thus converting olivary neurons into detectors of unexpected sensory events. Given the abundance of sensory input to P cells via mossy and climbing fibers, it is remarkable that these cells respond so weakly to sensory stimulation. Instead, they discharge vigorously during active movements. P cells send abundant collaterals to their neighbors, while their main axons project to the cerebellar nuclei and then onward to several brain sites that in turn relay motor commands to the spinal cord. Fig. 1 shows P cell projections to the intermediate cerebellar nucleus (I), also called the interpositus nucleus. The red nucleus (R) receives its main input from the interpositus nucleus, and it then transmits motor commands to the spinal cord via the rubrospinal tract. Other premotor nuclei that are alternative sources of motor commands receive input from alternative cerebellar output circuits. Fig. 1 thus specifically illustrates the cerebellorubrospinal system, the portion of the cerebellum that has been emphasized in my laboratory. Microelectrode recordings from the red nucleus have demonstrated signals that appear to represent detailed velocity commands for distal limb movements. Bursts of discharge precede each movement, the frequency of discharge within the burst corresponds to the velocity of movement, and the duration of the burst corresponds to the duration of movement. These velocity signals are not shaped by continuous feedback from peripheral receptors; instead, they appear to be produced centrally. An important goal of the modelling effort outlined here is to explain how these velocity commands might be produced by cerebellar circuits that function as elemental pattern generators. I will then discuss how an array of these pattern generators might serve well in an overall schema of motor control. ELEMENTAL PATTBRN GBNERATORS The motivation for proposing pattern generators rather than more conventional network designs derives from the experimental observation that motor commands, once initiated, are not affected, or are only minimally affected, by alterations in sensory input. This observation indicates that the temporal features of these motor commands are produced by self-sustained activity within the neural network rather than by the time courses of network inputs. 371 Two features of the intrinsic circuitry of the cerebellum may be particularly instrumental in explaining selfsustained activity. One is a recurrent pathway from cerebellar nuclei that returns back to cerebellar nuclei. In the case of the cerebellorubrospinal system in Fig. 1, the recurrent pathway is from the interpositus nucleus to red nucleus to lateral reticular nucleus and back to interpositus, what I will call the IRL loop. The other feature of intrinsic cerebellar circuitry that may be of critical importance in pattern generation is mutual inhibition between P cells. Fig. 1 shows how mutual inhibition results from the recurrent collaterals of P-cell axons. Inhibitory interneurons called basket and stellate cells (not shown in Fig. 1) provide additional pathways for mutual inhibition. Both the IRL loop and mutual inhibition between P cells constitute positive feedback circuits and, as such, are capable of self-sustained activity. Self-sustained activity in the form of high-frequency spontaneous discharge has been observed in the IRL loop under conditions in which the inhibitory P-cell input to I cells is blocked 3. Trace A in Fig. 2 shows this unrestrained discharge schematically, and the other traces illustrate how a motor command might be sculpted out of this tendency toward high-frequency, repetitive discharge. Trace B shows a brief burst of input presumed to be sent from the sensorimotor cortex to the R cell in Fig. 1. This burst serves as a trigger that initiates repetitive discharge in an IRL loop, and trace D illustrates the discharge of an I cell in the active loop. The intraburst discharge frequency of this cell is presumed to be determined by the summed magnitude of inhibitory input (shown in trace C) from the set of P cells that project to it (Fig. 1 shows only a few P cells from this set). Since the inhibitory input to I was reduced to an appropriate magnitude for controlling this intraburst frequency some time prior to the arrival of the trigger event, this example illustrates a mechanism for presetting the pattern generator. Note that the same reduction of inhibition that presets the intraburst frequency would bring the loop closer to the threshold for repetitive firing, thus serving to enable the triggering operation. The I-cell burst, after continuing for a duration appropriate for the desired motor behavior, is assumed to be terminated by an abrupt increase in inhibitory input from the set of P cells that project to I (trace C). The time course of bursting discharge illustrated in Fig. 2D would be expected to propagate throughout the IRL loop and be transmitted via the rubrospinal tract to the spinal cord where it could serve as a motor command. Bursts of R-cell discharge similar to this are observed to precede movements in trained monkey subjects2. 372 A. 111111111111111111111111111111111111111111111111111111111111111111111111 B. 1111 c. D. 111111111111111 time Figure 2: Signals Contributing to Pattern Generation. A. Repetitive discharge of I cell in the absence of Pcell inhibition. B. Trigger burst sent to the IRL loop from sensorimotor cortex. C. Summed inhibition produced by the set of P cells projecting to the I cell. D. Resultant motor pattern in I cell. The sculpting of a motor command out of a repetitive firing tendency in the IRL loop clearly requires timed transitions in the discharge rates of specific P cells. The present model postulates that the latter result from state transitions in the network of P cells. Bell and Grimm8 described spontaneous transitions in P-cell firing that occur intermittently, and I have frequently observed them as well. These transitions appear to be produced by intrinsic mechanisms and are difficult to influence with sensory stimulation. The mutual recurrent inhibition between P cells might explain this tendency toward state transitions. Recurrent inhibition between P cells is mediated by synapses near the cell bodies and primary dendrites of the P cells whereas parallel fiber input extends far out on the dendritic tree. This arrangement may explain why sensory input via parallel fibers does not have a strong, continuous effect on P cell discharge. This sensory input may serve mainly to promote state transitions in the network of P cells, perhaps by modulating the likelihood that a given P cell would participate in a state transition. Once the 373 transition starts, the activity of the P cell may be dominated by the recurrent inhibition close to the cell body. The mechanism responsible for the adaptive adjustment of these elemental pattern generators may be a change in the synaptic strengths of parallel fiber input to P cells9. Such alterations in the efficacy of sensory input would influence the state transitions discussed in the previous paragraph, thus mediating adaptive adjustments in the amplitude and timing of patterned output. Elsewhere I have suggested that this learning process is analogous to operant conditioning and includes both positive and negative reinforcement3. Noradrenergic fibers might mediate positive reinforcement, whereas climbing fibers might mediate negative reinforcement. For example, if the network were controlling a limb movement, negative reinforcement might occur when the limb bumps into an object in the work space (climbing fibers fire in response to unexpected somatic events such as this), whereas positive reinforcement might occur whenever the limb successfully acquires the desired target (the noradrenergic fibers to the cerebellum are thought to receive input from reward centers in the brain) . Positive reinforcement may be analogous to the associative reward-punishment algorithm described by BartolO which would fit with the diffuse projections of noradrenergic fibers. Negative reinforcement might be capable of a higher degree of credit assignment in view of the more focused projections of climbing fibers. In summary, the previous paragraphs outline some ideas that may be useful in developing a network model of the cerebellum. This particular set of ideas was motivated by a desire to explain the unique manner in which the cerebellum uses sensory input to control patterned output. The model deals explicitly with small circuits within a much larger network. The small circuits are considered elemental pattern generators, whereas the larger network can be considered an array of these pattern generators. The assembly of many elements into an array may give rise to some emergent properties of the network, due to interactions between the elements. However, the highly compartmentalized anatomical structure of the cerebellum fosters the notion of relatively independent elemental pattern generators as hypothesized in the schema for movement control presented in the next section. SCHEMA I'OR MOTOR CONTROL A major aim in developing the elemental pattern generator model described in the previous section was to explain the intriguing manner in which the cerebellum uses sensory input. Stated succinctly, sensory input is used to preset and to trigger each elemental pattern generator and 374 to evaluate the success of previous output patterns in controlling motor behavior. However, sensory input is not used to shape the waveform of an ongoing output pattern. This means that continuous feedback is not available, at the level of the cerebellum, for any immediate adjustments of motor commands. Is this kind of behavior actually advantageous in the control of movement? I would propose the affirmative, particularly on the grounds that this strategy seems to have withstood the test of evolution. Elsewhere I have reviewed the global strategies that are used to control several different types of body function11 • A common theme in each of these physiological control systems is the use of negative feedback only as a low-level strategy, and this coupled with a high-level stage of adaptive feedforward control. It was argued that this particular two-stage control strategy is well suited for utilizing the advantageous features of feedback, feedforward and adaptive control in combination. The adjustable pattern generator model of the cerebellum outlined in the previous section is a prime example of an adaptive, feedforward controller. In the subsequent paragraphs I will outline how this high-level feedforward controller communicates with low-level feedback systems called motor servos to produce limb movements (Fig. 3). The array of adjustable pattern generators (PGn) in the first column of .Fig. 3 produce an array of elemental commands that are transmitted via descending fibers to the spinal cord. The connectivity matrix for descending fibers represents the consequences of their branching patterns. Any given fiber is likely to branch to innervate several motor servos. Similarly, each member of the array of motor servos (MSm) receives convergent input from a large number of pattern generators, and the summed total of this input constitutes its overall motor command. A motor servo consists of a muscle, its stretch receptors and the spinal reflex pathways back to the same muscle12 • These reflex pathways constitute negative feedback loops that interact with the motor command to control the discharge of the motor neuron pool innervating the particular muscle. Negative feedback from the muscle receptors functions to maintain the stiffness of the muscle relatively constant, thus providing a spring-like interface between the body and its mechanical environment13 • The motor command acts to set the slack length of this equivalent spring and, in this way, influences motion of the limb. Feedback also gives rise to an unusual type of damping proportional to a low fractional power of velocity14. The individual motor servos interact with each other and with external loads via the trigonometric relations of the musculoskeletal matrix to produce resultant joint positions. Cerebellar Network PG1 ......... -PG2 ...... --...... PG3 .. --....... ....... -... PGN elemental commands r---motor commands (II ... CD ..c u: CI c '6 c CD &l CD 0 ... 0 u. )( °C a; ~ f ~ C C 8 --375 Motor joint Servos positions forces, lengths MS1 shoulder ._ .. ------MS2 .......... elbow )( °C a; ~ "ii CD wrist G) ~ a "3 &l :J ~ ........... finger MSM External Load ----Figure 3: Schema for Motor Control Utilizing Pattern Generator Model of Cerebellum. An array of elemental pattern generators (PGn ) operate in an adaptive, feedforward manner to produce motor commands. These outputs of the high-level stage are sent to the spinal cord where they serve as inputs to a low-level array of negative feedback systems called motor servos (MSm). The latter regulate the forces and lengths of individual muscles to control joint angles. While the schema for motor control presented here is based on a considerable body of experimental data, and it also seems plausible as a strategy for motor control, it will be important to explore its capabilities for human limb control with simulation studies. It may also be fruitful to apply this schema to problems in robotics. Since I am mainly an experimentalist, my authorship of this paper is meant as an entre for collaborative work with neural network modelers that may be interested in these problems. 376 RJ:I'J:RJ:HCJ:S 1. M. Ito, The Cerebellum and Neural Control (Raven Press, N. Y., 1984). 2. J. C. Houk & A. R. Gibson, In: J. S. King, New Concepts in Cerebellar Neurobiology (Alan R. Liss, Inc., N. Y., 1987), p. 387. 3. J. C. Houk, In: M. Glickstein & C. Yeo, Cerebellum and Neuronal Plasticity (Plenum Press, N. Y., 1988), in press. 4. D. Marr, J. Physiol. (London) 2D2, 437 (1969). 5. J. S. Albus, Math. Biosci. lQ, 25 (1971). 6. C. C. Boylls, A Theory of Cerebellar Function with Applications to Locomotion (COINS Tech. Rep., U. Mass. Amherst), 76-1. 7. J. C. Houk, In: J. E. Desmedt, Cerebral Motor Control in Man: Long Loop Mechanisms (Karger, Basel, 1978), p. 193. 8. C. C. Bell & R. J. Grimm, J. Neurophysiol., J2, 1044 (1969) . 9 C.-F. Ekerot & M. Kano, Brain Res., ~, 357 (1985). 10. A. G. Barto, Human Neurobiol., ~, 229 (1985). 11. J. C. Houk, FASEB J., Z, 97-107 (1988). 12. J. C. Houk & W. Z. Rymer, In: V. B. Brooks, Handbook of Physiology, Vol. 1 of Sect. 1 (American Physiological Society, Bethesda, 1981), p.257. 13. J. C. Houk, Annu. Rev. Physiol., ~, 99 (1979). 14. C. C. A. M. Gielen & J. C. Houk, BioI. Cybern., 52, 217 (1987).
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DISTRIBUTED NEURAL INFORMATION PROCESSING IN THE VESTIBULO-OCULAR SYSTEM Clifford Lau Office of Naval Research Detach ment Pasadena, CA 91106 Vicente Honrubia* UCLA Division of Head and Neck Surgery Los Angeles, CA 90024 ABSTRACT A new distributed neural information-processing model is proposed to explain the response characteristics 457 of the vestibulo-ocular system and to reflect more accurately the latest anatomical and neurophysiological data on the vestibular afferent fibers and vestibular nuclei. In this model, head motion is sensed topographically by hair cells in the semicircular canals. Hair cell signals are then processed by multiple synapses in the primary afferent neurons which exhibit a continuum of varying dynamics. The model is an application of the concept of "multilayered" neural networks to the description of findings in the bullfrog vestibular nerve, and allows us to formulate mathematically the behavior of an assembly of neurons whose physiological characteristics vary according to their anatomical properties. INTRODUCTION Traditionally the physiological properties of individual vestibular afferent neurons have been modeled as a linear time-invariant system based on Steinhausents description of cupular motion.1 The vestibular nerve input to different parts of the central nervous system is usually represented by vestibular primary afferents that have *Work supported by grants NS09823 and NS08335 from the National Institutes of Health (NINCDS) and grants from the Pauley Foundation and the Hope for Hearing Research Foundation. © American Institute of Physics 1988 458 response properties defined by population averages from individual neurons. 2 A new model of vestibular nerve organization is proposed to account for the observed variabilities in the primary vestibular afferent's anatomical and physiological characteristics. The model is an application of the concept of "multilayered" neural networks,3,4 and it attempts to describe the behavior of the entire assembly of vestibular neurons based on new physiological and anatomical findings in the frog vestibular nerve. It was found that primary vestibular afferents show systematic differences in sensitivity and dynamics and that there is a correspondence between the individual neuron's physiological properties and the location of innervation in the area of the crista and also the sizes of the neuron's fibers and somas. This new view of topological organization of the receptor and vestibular nerve afferents is not included in previous models of vestibular nerve function. Detailed findings from this laboratory on the anatomical and physiological properties of the vestibular afferents in the bullfrog have been published.5,6 REVIEW OF THE ANATOMY AND PHYSIOLOGY OF THE VESTIBULAR NERVE The most pertinent anatomical and physiological data on the bullfrog vestibular afferents are summarized here. In the vestibular nerve from the anterior canal four major branches (bundles) innervate different parts of the crista (Figure 1). From serial histological sections it has been shown that fibers in the central bundle innervate hair cells at the center of the crista, and the lateral bundles project to the periphery of the crista. I n each nerve there is an average of 1170 ± 171 (n = 5) fibers, of which the thick fibers (diameter > 7.0 microns, large dots) constitute 8% and the thin fibers « 4.0 microns, small dots) 76%. The remaining fibers (16%) fall into the range between 4.0 and 7.0 microns. We found that the thick fibers innervate only the center of the crista, and the thinner ones predominantly innervate the periphery. (f) ~ w ()) H LL LL o ~ w ()) L ::) Z 459 400 00 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20 DIAMETER (micron) Fig. 1. Number of fibers and their diameters in the anterior semicircular canal nerve in the bullfrog. There appears to be a physiological and anatomical correlation between fiber size and degree of regularity of spontaneous activity. By recording from individual neurons and subsequently labeling them with horseradish peroxidase intracellularly placed in the axon, it is possible to visualize and measure individual ganglion cells and axons and to determine the origin of the fiber in the crista as well as the projections in different parts of the vestibular nuclei. Figure 2 shows an example of three neurons of different sizes and degrees of regularity of spontaneous activity. In general, fibers with large diameters tend to be more irregular with large coefficients of variation (CV) of the interspike intervals, whereas thin fibers tend to be more regular. There is also a relationship for each neuron between CV and the magnitude of the response to physiological rotatory stimuli, that is, the response gain. (Gain is defined as the ratio of the response in spikes per second to the stimulus in degrees per second.) Figure 3 shows a plot of gain as a function of CV as well as of fiber diameter. For the more regular fibers (CV < 0.5), the gain tends to increase as the diameter of the fiber increases. 460 y -'x~ 300um THIN MEDIUM THICK c . V. = 0 . 25 c v = 0 39 c V = 0 61 11 o 200 ,l .. , o 200 ..... I I 200 MILLISECONDS Fig. 2. Examples of thin, medium and thick fibers and their spontaneous activity. CV - coefficient of variation. For the more irregular fibers (CV > 0.5), the gain tends to remain the same with increasing fiber diameter (4.9 ± 1.9 spikes/second/deg rees/seco nd). Figure 4 shows the location of projection of the afferent fibers at the vestibular nuclei from the anterior, posterior, and horizontal canals and saccule. There is an overall organization in the pattern of innervation from the afferents of each vestibular organ to the vestibular nuclei, with fibers from different receptors overlapping in various ... · CI) • C) CD "C ... · CI) • 10 ~ 1 .::t!! a. CI) c: as (!) • I • • • 3.8 • • • • • •• . . ~ .... . . I.. .. ••• • • •• • •• • • • • • • • • • • • • • • .. ~. . . . ... . ..... . .. ...... . ·:·,i ..... • • • • • 6. 1 8.4 10.7 13.0 15.3 Ff3ER DIAMETER 0.1r-~~~--~~~~--~~~~~ o 0.2 0.4 0.6 0.8 1 1.2 Coefficient of Variation Fig. 3. Gain versus fiber diameters and CV. Stimulus was a sinusoidal rotation of 0.05 Hz at 22 degrees/second peak velocity. parts of the vestibular nuclei. Fibers from the anterior semicircular canal tend to travel ventrally, from the horizontal canal dorsally, and from the posterior canal the most dorsally. For each canal nerve the thick fibers (indicated by large dots) tend to group together to travel lateral to the thin fibers (indicated by diffused shading); thus, the topographical segregation between thick and thin fibers at the periphery is preserved at the vestibular nuclei. In following the trajectories of individual neurons in the central nervous system, however, we found that each fiber innervates all parts of the vestibular nuclei, caudally to rostrally as well as transversely, and because of the spread of the large number of branches, as many as 200 from each neuron, there is a great deal of overlap among the projections. DISTRIBUTED NEURAL INFORMATION-PROCESSING MODEL Figure 5 represents a conceptual organization, based on the above anatomical and physiological data, of Scarpa's 461 462 ANT. -,--i!' HOIII. / ; :.::~~C --:-;~~ -;#'--'" -~ -~~POST. SAC. .. .. ..- • . ":r .. ~_ .. r"" --:-..,.;,~ •• <'::..• -;7" -~~ - - --~ ~ .;:.:::= =~~ --~ --..--~~ , Z~n_!=l. ~8 r 3OO)J ., Fig. 4. Three-dimensional reconstruction of the primary afferent fibers' location in the vestibular nuclei. ganglion cells of the vestibular nerve and their innervation of the hair cells and of the vestibular nuclei. The diagram depicts large Scarpa's ganglion cells with thick fibers in nervating restricted areas of hair cells near the center of the crista (top) and smaller Scarpa's ganglion cells with thin fibers on the periphery of the crista innervating multiple hair cells with a great deal of overlap among fibers. At the vestibular nuclei, both thick and thin fibers innervate large areas with a certain gradient of overlapping among fibers of different diameters. The new distributed neural information-processing model for the vestibular system is based on this anatomical organization, as shown in Figure 6. The response H. C. S. G. v. N. H. C. s.G. Fig. 5. ft.natomical organization of the vestibular nerve. H.C. - hair cells. S.G. - Scarpa's ganglion cells. V.N. - vestibular n ucle i. V. N. Fig. 6. Distributed neural information-processing model of the vestibular nerve. 463 464 characteristic of the primary afferent fiber is represented by the transfer function SGj(s). This transfer function serves as a description of the gain and phase response of individual neurons to angular rotation. The simplest model would be a first-order system with d.c. gain Kj (spikesl second over head acceleration) and a time constant Tj (seconds) for the jth fiber as shown in equation (1): SGj(s) = 1 + sT .. (1 ) J For the bullfrog, Kj can range from about 3 to 25 spikes/second/degree/second2, and Tj from about 10 to 0.5 second. The large and high-gain neurons are more phasic than the small neurons and tend to have shorter time constants. As described above, Kj and Tj for the jth neuron are functions of location and fiber diameter. Bode plots (gain and phase versus frequency) of experimental data seem to indicate, however, that a better transfer function would consist of a higher-order system that includes fractional power. This is not surprising since the afferent fiber response characteristic must be the weighted sum of several electromechanical steps of transduction in the hair cells. A plausible description of these processes is given in equation (2): SGj(s) = 1: Wjk 1 + s T k ' k where gain Kk and time constant T k are the electromechanical properties of the hair cell-cupula complex and are functions of location on the crista, and Wjk is the (2) synaptic efficacy (strength) between the jth neuron and the kth hair cell. In this context, the transfer function given in equation (1) provides a measure of the "weighted average" response of the multiple synapses given in equation (2). 465 We also postulate that the responses of the vestibular nuclei neurons reflect the weighted sums of the responses of the primary vestibular afferents, as follows: V N· = f ( l: T.. SG·) 1 IJ J' (3) j where f(.) is a sigmoid function describing the change in firing rates of individual neurons due to physiological stimulation. It is assumed to saturate between 100 to 300 spikes/second, depending on the neuron. Tij is the synaptic efficacy (strength) between the ith vestibular neuron and the jth afferent fiber. C(l\JCLUSIONS Based on anatomical and physiological data from the bullfrog we presented a description of the organization of the primary afferent vestibular fibers. The responses of the afferent fibers represent the result of summated excitatory processes. The information on head movement in the assemblage of neurons is codified as a continuum of varying physiological responses that reflect a sensoritopic organization of inputs from the receptor to the central nervous system. We postulated a new view of the organization in the peripheral vestibular organs and in the vestibular nuclei. This view does not require unnecessary simplification of the varying properties of the individual neurons. The model is capable of extracting the weighted average response from assemblies of large groups of neurons while the unitary contribution of individual neurons is preserved. The model offers the opportunity to incorporate further developments in the evaluation of the different roles of primary afferents in vestibular function. Large neurons with high sensitivity and high velocity of propagation are more effective in activating reflexes that require quick responses such as vestibulo-spinal and vestibulo-ocular reflexes. Small neurons with high thresholds for the generation of action potentials and lower sensitivity are more tuned to the maintenance of posture 466 and muscle tonus. We believe the physiological differences reflect the different physiological roles. I n this emerging scheme of vestibular nerve organization it appears that information about head movement, topographically filtered in the crista, is distributed through multiple synapses in the vestibular centers. Consequently, there is also reason to believe that different neurons in the vestibular nuclei preserve the variability in response characteristics and the topological discrimination observed in the vestibular nerve. Whether this idea of the organization and function of the vestibular system is valid remains to be proven experimentally. REFERENCES 1. W. Steinhausen, Arch. Ges. Physio!. 217,747 (1927). 2. J. M. Goldberg and C. Fernandez, in: Handbook of Physiology, Sect. 1, Vol. III, Part 2 (I. Darian-Smith, ed., Amer. Physio!. Soc., Bethesda, MD, 1984), p. 977. 3. D. E. Rumelhart, G. E. Hinton and J. L. McClelland, in: Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vol. 1: Foundations (D. E. Rumelhart, J. L. McClelland and the PDP Research Group, eds., MIT Press, Cambridge, MA, 1986), p. 45. 4. J. Hopfield, Proc. Nat!. Acad. Sci. la, 2554 (1982). 5. V. Honrubia, S. Sitko, J. Kimm, W. Betts and I. Schwartz, Intern. J. Neurosci. ~, 197 (1981). 6. V. Honrubia, S. Sitko, R. Lee, A. Kuruvilla and I. Schwartz, Laryngoscope .aA., 464 (1984).
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TIME-SEQUENTIAL SELF-ORGANIZATION OF HIERARCHICAL NEURAL NETWORKS Ronald H. Silverman Cornell University Medical College, New York, NY 10021 Andrew S. Noetzel polytechnic University, Brooklyn, NY 11201 ABSTRACT Self-organization of multi-layered networks can be realized by time-sequential organization of successive neural layers. Lateral inhibition operating in the surround of firing cells in each layer provides for unsupervised capture of excitation patterns presented by the previous layer. By presenting patterns of increasing complexity, in co-ordination with network selforganization, higher levels of the hierarchy capture concepts implicit in the pattern set. INTRODUCTION A fundamental difficulty in self-organization of hierarchical, multi-layered, networks of simple neuron-like cells is the determination of the direction of adjustment of synaptic link weights between neural layers not directly connected to input or output patterns. Several different approaches have been used to address this problem. One is to provide teaching inputs to the cells in internal layers of the hierarchy. Another is use of back-propagated error signals1,2 from the uppermost neural layer, which is fixed to a desired outfut pattern. A third is the "competitive learning" mechanism, in which a Hebbian synaptic modification rule is used, with mutual inhibition among cells of each layer preventing them from becoming conditioned to the same patterns. The use of explicit teaching inputs is generally felt to be undesirable because such signals must, in essence, provide individual direction to each neuron in internal layers of the network. This requires extensive control signals, and is somewhat contrary to the notion of a self-organizing system. Back-propagation provides direction for link weight modification of internal layers based on feedback from higher neural layers. This method allows true self-organization, but at the cost of specialized neural pathways over which these feedback signals must travel. In this report, we describe a simple feed-forward method for self-organization of hierarchical neural networks. The method is a variation of the technique of competitive learning. It calls for successive neural layers to initiate modification of their afferent synaptic link weights only after the previous layer has completed its own self-organization. Additionally, the nature of the patterns captured can be controlled by providing an organized © American Institute of Physics 1988 709 710 group of layer of layers. pattern sets which would excite the lowermost (input) the network in concert with training of successive such a collection of pattern sets might be viewed as a "lesson plan." MODEL The network is composed of neuron-like cells, organized in hierarchical layers. Each cell is excited by variably weighted afferent connections from the outputs of the previous (lower) layer. Cells of the lowest layer take on the values of the input pattern. The cells themselves are of the McCulloch-pitts type: they fire only after their excitation exceeds a threshold, and are otherwise inactive. Let Si(t) do,,} be the state of cell i at time t. Let Wij' a real number ranging from 0 to " be the weight, or strength, of the synapse connecting cell i to cell j. Let eij be the local excitation of cell i at the synaptic connect~on from cell j. The excitation received along each synaptic connection is integrated locally over time as follows: e· . (t) = e .. ( t-l) + w· . S· (t) ( 1 ) ~J ~J ~J ~ Synaptic connections may, therefore be viewed as capacitive. The total excitation, Ej , is the sum of the local excitations of cell j. = Ee .. (t) ~ ~J The use of the time-integrated activity of connection between two neurons, instead of the ( 2) a synaptic more usual instantaneous classification of neurons as "active" or "inactive", permits each synapse to provide a statistical measure of the activity of the input, which is assumed to be inherently stochastic. It also embodies the principle of learning based on locally available information and allows for implementations of the synapse as a capacitive element. Over time, the total excitation of individual neurons on a give layer will increase. When excitation exceeds a threshold, a, then the neuron fires, otherwise it is inactive. = if else Sj (t) = Ej (t) > a o ( 3) During a neuron I s training phase, a modified Hebbian rule results in changes in afferent synaptic link weights such that, upon firing, synapses with integrated activity greater than mean activity are reinforced, and those with less than mean activity are weakened. More formally, if Sj(t) = 1 then the synapse weights are modified by (4 ) Here, n represents the fan-in to a cell, and k is a small, positive constant. The "sign" function specifies the direction of change and the "sine" function determines the magnitude of change. The sine curve provides the property that intermediate link weights are subject to larger modifications than weights near zero or saturation. This helps provide for stable end-states after learning. Another effect of the integration of synaptic activity may be seen. A synapse of small weight is allowed to contribute to the firing of a cell (and hence have its weight incremented) if a series of patterns presented to the network consistently excite that synapse. The sequence of pattern presentations, therefore, becomes a factor in network self-organization. Upon firing, the active cell inhibits other cells in its vicinity (lateral inhibi tion) • This mechanism supports unsupervised, competi ti ve learning. By preventing cells in the neighborhood of an active cell from modifying their afferent connections in response to a pattern, they are left available for capture of new patterns. Suppose there are n cells in a particular level. The lateral inhibitory mechanism is specified as follows: If S . (t) = 1 then eik(t) = 0 for all i, tor k = (j-m)mod(n) to (j+m)mod(n) (5) Here, m specifies the size of a "neighborhood." A neighborhood significantly larger than a pattern set will result in a number of untrained cells. A neighborhood smaller than the pattern set will tend to cause cells to attempt to capture more than one pdttern. Schematic representations of an individual cell and the network organization are provided in Figures 1 and 2. It is the pattern generator, or "instructor", that controls the form that network organization will take. The initial set of patterns are repeated until the first layer is trained. Next, a new pattern set is used to excite the lowermost (trained) level of the network, and so, induce training in the next layer of the hierarchy. Each of the patterns of the new set is composed of elements (or subpatterns) of the old set. The structure of successive pattern sets is such that each set is either a more complex combination of elements from the previous set (as words are composed of letters) or a generaliza tlon of some concept implicit in the previous set (such as line orientation). Network organization, as described above, requires some exchange of control signals between the network and the instructor. The instructor requires information regarding firing of cells during training in order to switch to a new patterns appropriately. Obviously, if patterns are switched before any cells fire, learning will either not take place or will be smeared over a number of patterns. If a single pattern excites the network until one or more cells are fully trained, subsequent presentation of a non-orthogonal pattern could cause the trained cell to fire before any naive cell because of its saturated link weights. The solution is simply to allow gradual training over the full complement of the pattern set. After a few firings, a new pattern should be provided. After a layer has been trained, the instructor provides a control signal to that layer which permanently fixes the layer's afferent synaptic link weights. 711 712 Lateral Inhibtio~n __ ~ Excitation Excitatory Inputs Lateral Inhibtion Fig. 1. Schematic of neuron. Shading of afferent synaptic connections indicates variations in levels of local time-integrated excitation. Fig. 2. Schematic of network showing lateral inhibition and forward excitation. Shading of neurons, indicating degree of training, indicates time-sequential organization of successive neural layers. SIMULATIONS As an example, simulations were run in which a network was taught to differentiate vertical from horizontal line orientation. This problem is of interest because it represents a case in which pattern sets cannot be separated by a single layer of connections. This is so because the set of vertical (or horizontal) lines has activity at all positions within the input matrix. Two variations were simulated. input was a 4x4 matrix. This was unidirectional links to 25 cells. In the first simulation, the completely connected with These cell$ had fixed inhibi tory connections to the nearest (using a circular arrangement), and connectivity, a ring of eight cells, nearest neighbor on either side. five cells on either side excited, using complete wi th inhibition over the Ini tially, all excitatory link weights were small, random numbers. Each pattern of the initial input consisted of a single active row or column in the input matrix. Active elements had, during any clock cycle, a probability of 0.5 of being "on", while inactive elements had a 0.05 probability of being "on." After exposure to the initial pattern set, all cells on the first layer captured some input pattern, and all eight patterns had been captured by two or more cells. The next pattern set consisted of two subsets of four vertical and four horizontal lines. The individual lines were presented until a few firings took place within the trained layer, and then another line from the same subset was used to excite the network. After the upper layer responed with a few firings, and some training occured, the other set was used to excite the network in a similar manner. After five cycles, all cells on the uppermost layer had become sensitive, in a postionally independent manner, to lines of a vertical or a horizontal orientation. Due to lateral inhibition, adj acent cells developed opposite orientation specificities. In the second simulation, a 6x6 input matrix was connected to six cells, which were, in turn, connected to two cells. For this network, the lateral inhibitory range extended over the entire set of cells of each layer. The initial input set consisted of six patterns, each of which was a pair of either vertical lines or horizontal lines. After excitation by this set, each of the six middle level cells became sensitized to one of the input patterns. Next, the set of vertical and horizontal patterns were grouped into two sUDsets: vertical lines and horizontal lines. Individual patterns from one subset were presented until a cell, of the previously trained layer, fired. After one of the two cells on the uppermost layer fired, the procedure was repeated with the pattern set of opposite orientation. After 25 cycles, the two cells on the uppermost layer had developed opposite orientation specificities. Each of these cells was shown to be responsive, in a positionally independent manner, to any single 713 714 line of appropriate orientation. CONCLUSION Competitive learning mechanisms, when applied sequentially to successive layers in a hierarchical structure, can capture pattern elements, at lower levels of the hierarchy, and their generalizations, or abstractions, at higher levels. In the above mechanism, learning is externally directed, not by explicit teaching signals or back-propagation, but by provision of instruction sets consisting of patterns of increasing complexity, to be input to the lowermost layer of the network in concert with successive organization of higher neural layers. The central difficulty of this method involves the design of pattern sets a procedure whose requirements may not be obvious in all cases. The method is, however, attractive due to its simplicity of concept and design, providing for multi-level selforganization without direction by elaborate control signals. Several research goals suggest themselves: 1) simplification or elimination of control signals, 2) generalization of rules for structuring of pattern sets, 3) extension of this learning principle to recurrent networks, and 4) gaining a deeper understanding of the role of time as a factor in network selforganization. REFERENCES t. D. E. Rumelhart and G.E. Hinton, Nature 323, 533 (1986). 2. K. A. Fukushima, BioI. Cybern. 55, 5 (1986). 3. D. E. Rumelhart and D. Zipser, Cog. Sci. 9, 75 (1985).
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860 A METHOD FOR THE DESIGN OF STABLE LATERAL INHIBITION NETWORKS THAT IS ROBUST IN THE PRESENCE OF CIRCUIT PARASITICS J.L. WYATT, Jr and D.L. STANDLEY Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge, Massachusetts 02139 ABSTRACT In the analog VLSI implementation of neural systems, it is sometimes convenient to build lateral inhibition networks by using a locally connected on-chip resistive grid. A serious problem of unwanted spontaneous oscillation often arises with these circuits and renders them unusable in practice. This paper reports a design approach that guarantees such a system will be stable, even though the values of designed elements and parasitic elements in the resistive grid may be unknown. The method is based on a rigorous, somewhat novel mathematical analysis using Tellegen's theorem and the idea of Popov multipliers from control theory. It is thoroughly practical because the criteria are local in the sense that no overall analysis of the interconnected system is required, empirical in the sense that they involve only measurable frequency response data on the individual cells, and robust in the sense that unmodelled parasitic resistances and capacitances in the interconnection network cannot affect the analysis. I. INTRODUCTION The term "lateral inhibition" first arose in neurophysiology to describe a common form of neural circuitry in which the output of each neuron in some population is used to inhibit the response of each of its neighbors. Perhaps the best understood example is the horizontal cell layer in the vertebrate retina, in which lateral inhibition simultaneously enhances intensity edges and acts as an automatic lain control to extend the dynamic range of the retina as a whole. The principle has been used in the design of artificial neural system algorithms by Kohonen2 and others and in the electronic design of neural chips by Carver Mead et. al. 3 ,4. In the VLSI implementation of neural systems, it is convenient to build lateral inhibition networks by using a locally connected on-chip resistive grid. Linear resistors fabricated in, e.g., polysilicon, yield a very compact realization, and nonlinear resistive grids, made from MOS transistors, have been found useful for image segmentation. 4 ,5 Networks of this type can be divided into two classes: feedback systems and feedforward-only systems. In the feedforward case one set of amplifiers imposes signal voltages or © American Institute of Physics 1988 861 currents on the grid and another set reads out the resulting response for subsequent processing, while the same amplifiers both "write" to the grid and "read" from it in a feedback arrangement. Feedforward networks of this type are inherently stable, but feedback networks need not be. A practical example is one of Carver Meadls retina chips3 that achieves edge enhancement by means of lateral inhibition through a resistive grid. Figure 1 shows a single cell in a continuous-time version of this chip. Note that the capacitor voltage is affected both by the local light intensity incident on that cell and by the capacitor voltages on neighboring cells of identical design. Any cell drives its neighbors, which drive both their distant neighbors and the original cell in turn. Thus the necessary ingredients for instability--active elements and signal feedback--are both present in this system, and in fact the continuous-time version oscillates so badly that the original design is scarcely usable in practice with the lateral inhibition paths enabled. 6 Such oscillations can incident light v out I Figure 1. This photoreceptor and signal processor Circuit, using two MOS transconductance amplifiers, realizes lateral inhibition by communicating with similar units through a resistive grid. readily occur in any resistive grid circuit with active elements and feedback,even when each individual cell is quite stable. Analysis of the conditions of instability by straightforward methods appears hopeless, since any repeated array contains many cells, each of which influences many others directly or indirectly and is influenced by them in turn, so that the number of simultaneously active feedback loops is enormous. This paper reports a practical design approach that rigorously guarantees such a system will be stable. The very simplest version of the idea is intuitively obvious: design each individual cell so that, although internally active, it acts like a passive system as seen from the resistive grid. In circuit theory language, the design goal here is that each cellis output impedance should be a positive-real? function. This is sometimes not too difficult in practice; we will show that the original network in Fig. 1 satisfies this condition in the absence of certain parasitic elements. More important, perhaps, it is a condition one can verify experimentally 862 by frequency-response measurements. It is physically apparent that a collection of cells that appear passive at their terminals will form a stable system when interconnected through a passive medium such as a resistive grid. The research contributions, reported here in summary form, are i) a demonstration that this passivity or positive-real condition is much stronger than we actually need and that weaker conditions, more easily achieved in practice, suffice to guarantee stability of the linear network model, and ii) an extension of i) to the nonlinear domain that furthermore rules out large-signal oscillations under certain conditions. II. FIRST-ORDER LINEAR ANALYSIS OF A SINGLE CELL We begin with a linear analysis of an elementary model for the circuit in Fig. 1. For an initial approximation to the output admittance of the cell we simplify the topology (without loss of relevant information) and use a naive'model for the transconductance amplifiers, as shown in Fig. 2. e + Figure 2. Simplified network topology and transconductance amplifier model for the circuit in Fig. 1. The capacitor in Fig. 1 has been absorbed into CO2 • Straightforward calculations show that the output admittance is given by yes) (1) This is a positive-real, i.e., passive, admittance since it can always be realized by a network of the form shown in Fig. 3, where -1 -1 -1 Rl = (gm2+ Ro2 ) , R2= (gmlgm2Rol) , and L = COI/gmlgm2· Although the original circuit contains no inductors, the realization has both capacitors and inductors and thus is capable of damped oscillations. Nonetheless, if the transamp model in Fig. 2 were perfectly accurate, no network created by interconnecting such cells through a resistive grid (with parasitic capacitances) could exhibit sustained oscillations. For element values that may be typical in practice, the model in Fig. 3 has a lightly damped resonance around I KHz with a Q ~ 10. This disturbingly high Q suggests that the cell will be highly sensitive to parasitic elements not captured by the simple models in Fig. 2. Our preliminary 863 yes) Figure 3. Passive network realization of the output admittance (eq. (1) of the circuit in Fig. 2. analysis of a much more complex model extracted from a physical circuit layout created in Carver Mead's laboratory indicates that the output impedance will not be passive for all values of the transamp bias currents. But a definite explanation of the instability awaits a more careful circuit modelling effort and perhaps the design of an on-chip impedance measuring instrument. III. POSITIVE-REAL FUNCTIONS, e-POSITlVE FUNCTIONS, AND STABILITY OF LINEAR NETWORK MODELS In the following discussion s = cr+jw is a complex variable, H(s) is a rational function (ratio of polynomials) in s with real coefficients, and we assume for simplicity that H(s) has no pure imaginary poles. The term closed right halE plane refers to the set of complex numbers s with Re{s} > o. Def. I The function H(s) is said to be positive-real if a) it has no poles in the right half plane and b) Re{H(jw)} ~ 0 for all w. If we know at the outset that H(s) has no right half plane poles, then Def. I reduces to a simple graphical criterion: H1s} is positivereal if and only if the Nyquist diagram of H(s) (i.e. the plot of H(jW) for w ~ 0, as in Fig. 4) lies entirely in the closed right half plane. Note that positive-real functions are necessarily stable since they have no right half plane poles, but stable functions are not necessarily positive-real, as Example 1 will show. A deep link between positive real functions, physical networks and passivity is established by the classical result7 in linear circuit theory which states that H(s) is positive-real if and only if it is possible to synthesize a 2-terminal network of positive linear resistors, capacitors, inductors and ideal transformers that has H(s) as its driving-point impedance or admittance. 864 Oef. 2 The function H(s) is said to be a-positive for a particular value of e(e ~ 0, e ~ ~), if a) H{s) has no poles in the right half plane, and b) the Nyquist plot of H(s) lies strictly to the right of the straight line passing through the origin at an angle a to the real positive axis. Note that every a-positive function is stable and any function that is e-positive with e = ~/2 is necessarily positive-real. I {G(jw)} m Re{G(jw) } Figure 4. Nyquist diagram for a fUnction that is a-positive but not positive-real. Example 1 The function (s+l) (s+40) G (s) = (s+5) (s+6) (s+7) (2) is a-positive (for any e between about 18° and 68°) and stable, but it is not positive-real since its Nyquist diagram, shown in Fig. 4, crosses into the left half plane. The importance of e-positive functions lies in the following observations: 1) an interconnection of passive linear resistors and capacitors and cells with stable linear impedances can result in an unstable network, b) such an instability cannot result if the impedances are also positive-real, c) a-positive impedances form a larger class than positive-real ones and hence a-positivity is a less demanding synthesis goal, and d) Theorem 1 below shows that such an instability cannot result if the impedances are a-positive, even if they are not positive-real. Theorem 1 Consider a linear network of arbitrary topology, consisting of any number of passive 2-terminal resistors and capacitors of arbitrary value driven by any number of active cells. If the output impedances 'II" of all the active cells are a-positive for some common a, 0<a22, then the network is stable. The proof of Theorem 1 relies on Lemma 1 below. Lemma 1 865 If H(s) is a-positive for some fixed a, then for all So in the closed first quadrant of the complex plane, H(so) lies strictly to the right of the straight line passing through the origin at an angle a to the real positive axis, i.e., Re{so} ~ 0 and Im{so} ~ 0 ~ a-'II" < L H (so) < a. Proof of Lemma 1 (Outline) Let d be the function that assigns to each s in the closed right half plane the perpendicular distance des) from H(s) to the line defined in Def. 2. Note that des) is harmonic in the closed right half plane, since H is analytic there. It then follows, by application of the maximum modulus principle8 for harmonic functions, that d takes its minimum value on the boundary of its domain, which is the imaginary axis. This establishes Lemma 1. Proof of Theorem 1 (OUtline) The network is unstable or marginally stable if and only if it has a natural frequency in the closed right half plane, and So is a natural frequency if and only if the network equations have a nonzero solution at so. Let {Ik} denote the complex branch currents Of such a solution. By Tellegen I s theorern9 the sum of the complex powers absorbed by the circuit elements must vanish at such a solution, i.e., ~ IIk12/s0Ck + L capac~tances cell terminal pairs (3) where the second term is deleted in the special case so=O, since the complex power into capacitors vanishes at so=O. If the network has a natural frequency in the closed right half plane, it must have one in the closed first quadrant since natural frequencies are either real or else occur in complex conjugate pairs. But (3) cannot be satisfied for any So in the closed first quadrant, as we can see by dividing both sides of (3) by k IIkI2, where the sum is taken over all network branches. After this division, (3) asserts that zero is a convex combination of terms of the form Rk, terms of the form (CkSo)-I, and terms of the form Zk(So). Visualize where these terms lie in the complex plane: the first set lies on the real positive axis, the second set lies in the closed 4-th ~adrant since So lies in the closed 1st quadrant by assumption, and the third set lies to the right of a line passing through the origin at an angle a by Lemma 1. Thus all these terms lie strictly to the right of this line, which implies that no convex combination of them can equal zero. Hence the network is stable! 866 IV. STABILITY RESULT FOR NETWORKS WITH NONLINEAR RESISTORS AND CAPACITORS The previous result for linear networks can afford some limited insight into the behavior of nonlinear networks. First the nonlinear equations are linearized about an equilibrium point and Theorem 1 is applied to the linear model. If the linearized model is stable, then the equilibrium point of the original nonlinear network is locally stable, i.e., the network will return to that equilibrium point if the initial condition is sufficiently near it. But the result in this section, in contrast, applies to the full nonlinear circuit model and allows one to conclude that in certain circumstances the network cannot oscillate even if the initial state is arbitrarily far from the equilibrium point. Def. 3 A function H(s) as described in Section III is said tc satisfy the Popov criterionlO if there exists a real number r>O such that Re{(l+jwr) H(jw)} ~ 0 for all w. Note that positive real functions satisfy the Popov criterion with r=O. And the reader can easily verify that G(s) in Exam~le I satisfies the Popov criterion for a range of values of r. The important effect of the term (l+jwr) in Def. 3 is to rotate the Nyquist plot counterclockwise by progressively greater amounts up to 90° as w increases. Theorem 2 Consider a network consisting of nonlinear 2-terminal resistors and capacitors, and cells with linear output impedances ~(s). Suppose i) the resistor curves are characterized by continuously diffefentiable functions i k = gk(vk ) where gk(O) = 0 and o < gk(vk ) < G < 00 for all values of k and vk' ii) the capacitors are characterized by i k = Ck(Vk)~k with o < CI < Ck(vk ) < C2 < 00 for all values of k and vk' iii) the impedances Zk(s) have no poles in the closed right half plane and all satisfy the Popov criterion for some common value of r. If these conditions are satisfied, then the network is stable in the sense that, for any initial condition, foo( ) I i~(t) dt o all branches < 00 • (4) The proof, based on Tellegen's theorem, is rather involved. It will be omitted here and will appear elsewhere. 867 ACKNOWLEDGEMENT We sincerely thank Professor Carver Mead of Cal Tech for enthusiastically supporting this work and for making it possible for us to present an early report on it in this conference proceedings. This work was supportedJ::¥ Defense Advanced Research Projects Agency (DoD), through the Office of Naval Research under ARPA Order No. 3872, Contract No. N00014-80-C-0622 and Defense Advanced Research Projects Agency (DARPA) Contract No. N00014-87-R-0825. REFERENCES 1. F.S. Werblin, "The Control of Sensitivity on the Retina," Scientific American, Vol. 228, no. 1, Jan. 1983, pp. 70-79. 2. T. Kohonen, Self-Organization and Associative Memory, (vol. 8 in the Springer Series in Information Sciences), Springer Verlag, New York, 1984. 3. M.A. Sivilotti, M.A. Mahowald, and C.A. Mead, "Real Time Visual Computations Using Analog CMOS processing Arrays," Advanced Research in VLSI - Proceedings of the 1987 Stanford Conference, P. Losleben, ed., MIT Press, 1987, pp. 295-312. 4. C.A. Mead, Analog VLSI and Neural Systems, Addison-Wesley, to appear in 1988. 5. J. Hutchinson, C. Koch, J. Luo and C. Mead, "Computing Motion Using Analog and Binary Resistive Networks," submitted to IEEE Transactions on Computers, August 1987. 6. M. Mahowald, personal communication. 7. B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and synthesis - A Modern Systems Theory Approach, Prentice-Hall, Englewood Cliffs, NJ., 1973. 8. L.V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1966, p. 164. 9. P. penfield, Jr., R. Spence, and S. Duinker, Tellegen's Theorem and Electrical Networks, MIT Press, Cambridge, MA,1970. 10. M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1970, pp. 211-217.
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612 Constrained Differential Optimization John C. Platt Alan H. Barr California Institute of Technology, Pasadena, CA 91125 Abstract Many optimization models of neural networks need constraints to restrict the space of outputs to a subspace which satisfies external criteria. Optimizations using energy methods yield "forces" which act upon the state of the neural network. The penalty method, in which quadratic energy constraints are added to an existing optimization energy, has become popular recently, but is not guaranteed to satisfy the constraint conditions when there are other forces on the neural model or when there are multiple constraints. In this paper, we present the basic differential multiplier method (BDMM), which satisfies constraints exactly; we create forces which gradually apply the constraints over time, using "neurons" that estimate Lagrange multipliers. The basic differential multiplier method is a differential version of the method of multipliers from Numerical Analysis. We prove that the differential equations locally converge to a constrained minimum. Examples of applications of the differential method of multipliers include enforcing permutation codewords in the analog decoding problem and enforcing valid tours in the traveling salesman problem. 1. Introduction Optimization is ubiquitous in the field of neural networks. Many learning algorithms, such as back-propagation,18 optimize by minimizing the difference between expected solutions and observed solutions. Other neural algorithms use differential equations which minimize an energy to solve a specified computational problem, such as associative memory, D differential solution of the traveling salesman problem,s,lo analog decoding,lS and linear programming.1D Furthennore, Lyapunov methods show that various models of neural behavior find minima of particular functions.4,D Solutions to a constrained optimization problem are restricted to a subset of the solutions of the corresponding unconstrained optimization problem. For example, a mutual inhibition circuitS requires one neuron to be "on" and the rest to be "off". Another example is the traveling salesman problem,ls where a salesman tries to minimize his travel distance, subject to the constraint that he must visit every city exactly once. A third example is the curve fitting problem, where elastic splines are as smooth as possible, while still going through data points.s Finally, when digital decisions are being made on analog data, the answer is constrained to be bits, either 0 or 1.14 A constrained optimization problem can be stated as minimize / (~), subject to g(~) = 0, (1) where ~ is the state of the neural network, a position vector in a high-dimensional space; f(~) is a scalar energy, which can be imagined as the height of a landscape as a function of position~; g(~) = 0 is a scalar equation describing a subspace of the state space. During constrained optimization, the state should be attracted to the subspace g(~) = 0, then slide along the subspace until it reaches the locally smallest value of f(~) on g(~) = O. In section 2 of the paper, we describe classical methods of constrained optimization, such as the penalty method and Lagrange multipliers. Section 3 introduces the basic differential multiplier method (BDMM) for constrained optimization, which calcuIates a good local minimum. If the constrained optimization problem is convex, then the local minimum is the global minimum; in general, finding the global minimum of non-convex problems is fairly difficult. In section 4, we show a Lyapunov function for the BDMM by drawing on an analogy from physics. © American Institute of Physics 1988 613 In section 5, augmented Lagrangians, an idea from optimization theory, enhances the convergence properties of the BDMM. In section 6, we apply the differential algorithm to two neural problems, and discuss the insensitivity of BDMM to choice of parameters. Parameter sensitivity is a persistent problem in neural networks. 2. Classical Methods of Constrained Optimization This section discusses two methods of constrained optimization, the penalty method and Lagrange multipliers. The penalty method has been previously used in differential optimization. The basic differential multiplier method developed in this paper applies Lagrange multipliers to differential optimization. 2.l. The Penalty Method The penalty method is analogous to adding a rubber band which attracts the neural state to the subspace g(~) = o. The penalty method adds a quadratic energy term which penalizes violations of constraints. 8 Thus, the constrained minimization problem (1) is converted to the following unconstrained minimization problem: (2) Figure 1. The penalty method makes a trough in state space The penalty method can be extended to fulfill multiple constraints by using more than one rubber band. Namely, the constrained optimization problem minimize f (.~), 8ubject to go (~) = OJ a = 1,2, ... , n; (3) is converted into unconstrained optimization problem n minimize l'pena1ty(~) = f(~) + L Co(go(~))2. (4) 0:::1 The penalty method has several convenient features. First, it is easy to use. Second, it is globally convergent to the correct answer as Co 00.8 Third, it allows compromises between constraints. For example, in the case of a spline curve fitting input data, there can be a compromise between fitting the data and making a smooth spline. 614 However, the penalty method has a number of disadvantages. First, for finite constraint strengths COl' it doesn't fulfill the constraints exactly. Using multiple rubber band constraints is like building a machine out of rubber bands: the machine would not hold together perfectly. Second, as more constraints are added, the constraint strengths get harder to set, especially when the size of the network (the dimensionality of .u gets large. In addition, there is a dilemma to the setting of the constraint strengths. If the strengths are small, then the system finds a deep local minimum, but does not fulfill all the constraints. If the strengths are large, then the system quickly fulfills the constraints, but gets stuck in a poor local minimum. 2.2. Lagrange Multipliers Lagrange multiplier methods also convert constrained optimization problems into unconstrained extremization problems. Namely, a solution to the equation (1) is also a critical point of the energy (5) ). is called the Lagrange multiplier for the constraint g(~) = 0.8 A direct consequence of equation (5) is that the gradient of f is collinear to the gradient of 9 at the constrained extrema (see Figure 2). The constant of proportionality between 'i1 f and 'i1 9 is -).: 'i1 'Lagrange = 0 = 'i1 f + ). 'i1 g. (6) We use the collinearity of 'i1 f and 'i1 9 in the design of the BDMM. Figure 2. At the constrained minimum, 'i1 f = -). 'i1 9 A simple example shows that Lagrange multipliers provide the extra degrees of freedom necessary to solve constrained optimization problems. Consider the problem of finding a point (x, y) on the line x + y = 1 that is closest to the origin. Using Lagrange multipliers, 'Lagrange = x2 + y2 + ).(x + y - 1) Now, take the derivative with respect to all variables, x, y, and A. aeLagrange = 2x + A = 0 ax a'Lagrange = 2y + A = 0 ay a'Lagrange = x + y - 1 = 0 a). (7) (8) 615 With the extra variable A, there are now three equations in three unknowns. In addition, the last equation is precisely the constraint equation. 3. The Basic Differential Multiplier Method for Constrained Optimization This section presents a new "neural" algorithm for constrained optimization, consisting of differential equations which estimate Lagrange multipliers. The neural algorithm is a variation of the method of multipliers, first presented by Hestenes9 and Powell 16 • 3.1. Gradient Descent does not work with Lagrange Multipliers The simplest differential optimization algorithm is gradient descent, where the state variables of the network slide downhill, opposite the gradient. Applying gradient descent to the energy in equation (5) yields x. - _ a!Lagrange = _ al _ A ag , ax· ax· ax' ' , " \. a!Lagrange ( ) J\ = = -g * aA . (9) Note that there is a auxiliary differential equation for A, which is an additional "neuron" necessary to apply the constraint g(~) = O. Also, recall that when the system is at a constrained extremum, VI = -AVg, hence, x. = O. Energies involving Lagrange multipliers, however, have critical points which tend to be saddle points. Consider the energy in equation (5). If ~ is frozen, the energy can be decreased by sending A to +00 or -00. Gradient descent does not work with Lagrange multipliers, because a critical point of the energy in equation (5) need not be an attractor for (9). A stationary point must be a local minimum in order for gradient descent to converge. 3.2. The New Algorithm: the Basic Differential Multiplier Method We present an alternative to differential gradient descent that estimates the Lagrange multipliers, so that the constrained minima are attractors of the differential equations, instead of "repulsors." The differential equations that solve (1) is . al ag X' =---A, ax, ax.' i = +g(*). (10) Equation (10) is similar to equation (9). As in equation (9), constrained extrema of the energy (5) are stationary points of equation (10). Notice, however, the sign inversion in the equation for i, as compared to equation (9). The equation (10) is performing gradient ascent on A. The sign flip makes the BDMM stable, as shown in section 4. Equation (10) corresponds to a neural network with anti-symmetric connections between the A neuron and all of the ~ neurons. 3.3. Extensions to the Algorithm One extension to equation (10) is an algorithm for constrained minimization with multiple constraints. Adding an extra neuron for every equality constraint and summing all of the constraint forces creates the energy !multiple = !(~) + I: Ao<ga(~), which yields differential equations 0< x' - _ al _ "" A agcr. ,ax' ~ 0< ax' ) '0< ' (11) (12) 616 Another extension is constrained minimization with inequality constraints. As in traditional optimization theory.8 one uses extra slack variables to convert inequality constraints into equality constraints. Namely. a constraint of the form h(~) ~ 0 can be expressed as (13) Since Z2 must always be positive, then h(~) is constrained to be positive. The slack variable z is treated like a component of ~ in equation (10). An inequality constraint requires two extra neurons, one for the slack variable % and one for the Lagrange multiplier ~. Alternatively, the inequality constraint can be represented as an equality constraint For example, if h(~) ~ 0, then the optimization can be constrained with g(~) = h(.~), when h(~) ~ 0; and g(.~) = 0 otherwise. 4. Why the algorithm works The system of differential equations (10) (the BDMM) gradually fulfills the constraints. Notice that the function g(~) can be replaced by kg(~), without changing the location of the constrained minimum. As k is increased, the state begins to undergo damped oscillation about the constraint subspace g(~) = o. As k is increased further, the frequency of the oscillations increase, and the time to convergence increases. constraint subspace ./ • .,./' initial state path of algorithm " \ \ Figure 3. The state is attracted to the constraint subspace The damped oscillations of equation (10) can be explained by combining both of the differential equations into one second-order differential equation. (14) Equation (14) is the equation for a damped mass system, with an inertia term Xi. a damping matrix (15) and an internal force, gOg/O%i, which is the derivative of the internal energy (16) 617 If the system is damped and the state remains bounded, the state falls into a constrained minima. As in physics, we can construct a total energy of the system, which is the sum of the kinetic and potential energies. E = T + U = L i(xd2 + i(g(~))2 . , (17) If the total energy is decreasing with time and the state remains bounded, then the system will dissipate any extra energy, and will settle down into the state where which is a constrained extremum of the original problem in equation (1). The time derivative of the total energy in equation (17) is = - Lx,A,jxj. ',i If damping matrix Aii is positive definite, the system converges to fulfill the constraints. (18) (19) BDMM always converges for a special case of constrained optimization: quadratic programming. A quadratic programming problem has a quadratic function f(~) and a piecewise linear continuous function g(~) such that (20) Under these circumstances, the damping matrix Aii is positive definite for all ~ and A, so that the system converges to the constraints. 4.1. Multiple constraints For the case of multiple constraints, the total energy for equation (12) is E = T + U = L i(Xi)2 + L igo(~)2. i 0 (21) and the time derivative is (22) Again, BDMM solves a quadratic programming problem, if a solution exists. However, it is possible to pose a problem that has contradictory constraints. For example, gdx) = x = 0, g2(X) = x-I = 0 (23) In the case of conflicting constraints, the BDMM compromises, trying to make each constraint go as small as possible. However, the Lagrange multipliers Ao goes to ±oo as the constraints oppose each other. It is possible, however, to arbitrarily limit the Ao at some large absolute value. 618 LaSalle's invariance theorem12 is used to prove that the BDMM eventually fulfills the constraints. Let G be an open subset of Rn. Let F be a subset of G*, the closure of G, where the system of differential equations (12) is at an equilibrium. (24) If the damping matrix a2f a2g -----:;_ + '" A a ax, ax; ~ a ax,ax; (25) is positive definite in G, if xa{ t) and Aa (t) are bounded, and remain in G for all time, and ~f F is non-empty, then F is the largest invariant set in G*, hence, by LaSalle's invariance theorem, the system x, (t), Aa (t) approaches Fast -+ 00. 5. The Modified Differential Method of Multipliers This section presents the modified differemiaI multiplier method (MDMM), which is a modification of the BDMM with more robust convergence properties. For a given constrained optimization problem, it is frequently necessary to alter the BDMM to have a region of positive damping surrounding the constrained minima. The non-differential method of multipliers from Numerical Analysis also has this difficulty. 2 Numerical Analysis combines the multiplier method with the penalty method to yield a modified multiplier method that is locally convergent around constrained minima. 2 The BDMM is completely compatible with the penalty method. If one adds a penalty force to equation (10) corresponding to an quadratic energy Epenalty = ~(g(~))2. then the set of differential equations for MDMM is . af ag ag x, = -- - A- - cg-, ax, ax, ax, j = g(~). (26) (27) The extra force from the penalty does not change the position of the stationary points of the differential equations, because the penalty force is 0 when g(~) = O. The damping matrix is modified by the penalty force to be (28) There is a theorem 1 that states that there exists a c* > 0 such that if c > c*, the damping matrix in equation (28) is positive definite at constrained minima. Using continuity, the damping matrix is positive definite in a region R surrounding each constrained minimum. If the system starts in the region R and remains bounded and in R, then the convergence theorem at the end of section 4 is applicable, and MDMM will converge to a constrained minimum. The minimum necessary penalty strength c for the MDMM is usually much less than the strength needed by the penalty method alone.2 6. Examples This section contains two examples which illustrate the use of the BDMM and the MDMM. First, the BDMM is used to find a good solution to the planar traveling salesman problem. Second, the MDMM is used to enforcing mutual inhibition and digital results in the task of analog decoding. 6.1. Planar Traveling Salesman The traveling salesman problem (fSP) is, given a set of cities lying in the plane, find the shortest closed path that goes through every city exactly once. Finding the shortest path is NP-complete. 619 Finding a nearly optimal path, however, is much easier than finding a globally optimal path. There exist many heuristic algorithms for approximately solving the traveling salesman problem.5,10,11,13 The solution presented in this section is moderately effective and illustrates the independence of BDMM to changes in parameters. Following Durbin and Willshaw,5 we use an elastic snake to solve the TSP. A snake is a discretized curve which lies on the plane. The elements of the snake are points on the plane, (Xi, Yd. A snake is a locally connected neural network, whose neural outputs are positions on the plane. The snake minimizes its length 2:)Xi+1 - x,)2 - (Yi+l - Yi)2, i subject to the constraint that the snake must lie on the cities: (29) k(x* - xc) = 0, k(y* - Yc) = 0, (30) where (x*, y*) are city coordinates, (xc, Yc) is the closest snake point to the city, and k is the constraint strength. The minimization in equation (29) is quadratic and the constraints in equation (30) are piecewise linear, corresponding to a CO continuous potential energy in equation (21). Thus, the damping is positive definite, and the system converges to a state where the constraints are fulfilled. In practice, the snake starts out as a circle. Groups of cities grab onto the snake, deforming it As the snake gets close to groups of cities, it grabs onto a specific ordering of cities that locally minimize its length (see Figure 4). The system of differential equations that solve equations (29) and (30) are piecewise linear. The differential equations for Xi and Yi are solved with implicit Euler's method, using tridiagonal LV decomposition to solve the linear system.17 The points of the snake are sorted into bins that divide the plane, so that the computation of finding the nearest point is simplified. Figure 4. The snake eventually attaches to the cities The constrained minimization in equations (29) and (30) is a reasonable method for approximately solving the TSP. For 120 cities distributed in the unti square, and 600 snake points, a numerical step size of 100 time units, and a constraint strength of 5 x 10-3 , the tour lengths are 6% ± 2% longer than that yielded by simulated annealing11 . Empirically, for 30 to 240 cities, the time needed to compute the final city ordering scales as N1.6, as compared to the Kernighan-Lin method13, which scales roughly as N 2.2 • The constraint strength is usable for both a 30 city problem and a 240 city problem. Although changing the constraint strength affects the performance, the snake attaches to the cities for any nonzero constraint strength. Parameter adjustment does not seem to be an issue as the number of cities increases, unlike the penalty method. 620 6.2. Analog Decoding Analog decoding uses analog signals from a noisy channel to reconstruct codewords. Analog decoding has been performed neurally,15 with a code space of permutation matrices, out of the possible space of binary matrices. To perform the decoding of permutation matrices, the nearest permutation matrix to the signal matrix must be found. In other words, find the nearest matrix to the signal matrix, subject to the constraint that the matrix has on/off binary elements, and has exactly one "on" per row and one "on" per column. If the signal matrix is Ii; and the result is Vi;, then minimize - "v.. ,1-. L..J ., ., (31) i ,; subject to constraints Vi,,(l- Vi;) = OJ LVi" -1 = O. (32) ; In this example, the first constraint in equation (32) forces crisp digital decisions. The second and third constraints are mutual inhibition along the rows and columns of the matrix. The optimization in equation (31) is not quadratic, it is linear. In addition, the first constraint in equation (32) is non-linear. Using the BDMM results in undamped oscillations. In order to converge onto a constrained minimum, the MDMM must be used. For both a 5 x 5 and a 20 x 20 system, a c = 0,2 is adequate for damping the oscillations. The choice of c seems to be reasonably insensitive to the size of the system, and a wide range of c, from 0.02 to 2.0, damps the oscillations . . . · ' •....•. . ..•...... • • ... e· ... . · .. . e· ... · .. ••• • ••••• • • • •• • ••••••• • • ••• • •••• .•. ' . . ... • • ••• .. ... . .. , .. . • ••••• • • • ••• • •••••• • •••• .. . . . .•. ' :~:.:.: • •••••••• ••• ••• • • • • • • ···::r:::::::: • • • • .... . . . ... • • • • • : :e&:.:: ....•. ••••• ••• .. . . . . , ... . ... .•... . . •.... • ••• • ..' . • •• • •• • •• ..' . ••• •• • •• ••• • •• ••• ••• • •• • •• • •• . •.... . ..... Figure 5. The decoder finds the nearest permutation matrix In a test of the MDMM, a signal matrix which is a permutation matrix plus some noise, with a signal-to-noise ratio of 4 is supplied to the network. In figure 5, the system has turned on the correct neurons but also many incorrect neurons. The constraints start to be applied, and eventually the system reaches a permutation matrix. The differential equations do not need to be reset. If a new signal matrix is applied to the network, the neural state will move towards the new solution. 7. ConClusions In the field of neural networks, there are differential optimization algorithms which find local solutions to non-convex problems. The basic differential multiplier method is a modification of a standard constrained optimization algorithm, which improves the capability of neural networks to perform constrained optimization. The BDMM and the MDMM offer many advantages over the penalty method. First, the differential equations (10) are much less stiff than those of the penalty method. Very large quadratic terms are not needed by the MDMM in order to strongly enforce the constraints. The energy terrain for the 621 penalty method looks like steep canyons, with gentle floors; finding minima of these types of energy surfaces is numerically difficult In addition, the steepness of the penalty tenns is usually sensitive to the dimensionality of the space. The differential multiplier methods are promising techniques for alleviating stiffness. The differential multiplier methods separate the speed of fulfilling the constraints from the accuracy of fulfilling the constraints. In the penalty method, as the strengths of a constraint goes to 00, the constraint is fulfilled, but the energy has many undesirable local minima. The differential multiplier methods allow one to choose how quickly to fulfill the constraints. The BDMM fulfills constraints exactly and is compatible with the penalty method. Addition of penalty tenns in the MDMM does not change the stationary points of the algorithm, and sometimes helps to damp oscillations and improve convergence. Since the BDMM and the MDMM are in the form of first-order differential equations, they can be directly implemented in hardware. Performing constrained optimization at the raw speed of analog VLSI seems like a promising technique for solving difficult perception problems. 14 There exist Lyapunov functions for the BDMM and the MDMM. The BDMM converges globally for quadratic programming. The MDMM is provably convergent in a local region around the constrained minima Other optimization algorithms, such as Newton's method,17 have similar local convergence properties. The global convergence properties of the BDMM and the MDMM are currently under investigation. In summary, the differential method of multipliers is a useful way of enforcing constraints on neural networks for enforcing syntax of solutions, encouraging desirable properties of solutions, and making crisp decisions. Acknowledgments This paper was supported by an AT&T Bell Laboratories fellowship (JCP). References 1. K. J. Arrow, L. Hurwicz, H. Uzawa, Studies in Linear and Nonlinear Programming. (Stanford University Press, Stanford, CA, 1958). 2. D. P. Bertsekas, Automatica, 12, 133-145, (1976). 3. C. de Boor, A Practical Guide to Splines. (Springer-Verlag, NY, 1978). 4. M. A. Cohen, S. Grossberg, IEEE Trans. Systems. Man. and Cybernetics, ,815-826, (1983). 5. R. Durbin, D. Willshaw, Nature, 326, 689-691, (1987). 6. J. C. Eccles, The Physiology of Nerve Cells, (Johns Hopkins Press, Baltimore, 1957). 7. M. R. Hestenes, J. Opt. Theory Appl., 4, 303-320, (1969). 8. M. R. Hestenes, Optimization Theory, (Wiley & Sons, NY, 1975). 9. J. J. Hopfield, PNAS, 81, 3088, (1984). 10. J. J. Hopfield, D. W. Tank, Biological Cybernetics, 52, 141, (1985). 11. S. Kirkpatrick, C. D. Gelatt, C. M. Vecchi, Science, 220, 671-680, (1983). 12. J. LaSalle, The Stability of Dynamical Systems, (SIAM, Philadelphia, 1976). 13. S. Lin, B. W. Kernighan, Oper. Res., 21,498-516 (1973). 14. C. A. Mead, Analog VLSI and Neural Systems, (Addison-Wesley, Reading. MA, TBA). 15. J. C. Platt, J. J. Hopfield, in AlP Con/. Proc.151: Neural Networksfor Computing (1. Denker ed.) 364-369, (American Institute of PhysiCS, NY, 1986). 16. M. 1. Powell, in Optimization, (R. Fletcher, ed.), 283-298, (Academic Press, NY, 1969). 17. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, (Cambridge University Press, Cambridge, 1986). 18. D. Rumelhart, G. Hinton, R. Williams, in Parallel Distributed Processing, (D. Rumelhart, ed), 1, 318-362, (MIT Press, Cambridge, MA, 1986). 19. D. W. Tank, J. J. Hopfield, IEEE Trans. Cir. & Sys., CAS-33, no. 5,533-541 (1986).
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ENCODING GEOMETRIC INVARIANCES IN HIGHER-ORDER NEURAL NETWORKS C.L. Giles Air Force Office of Scientific Research, Bolling AFB, DC 20332 R.D. Griffin Naval Research Laboratory, Washington, DC 20375-5000 T. Maxwell Sachs-Freeman Associates, Landover, MD 20785 ABSTRACT 301 We describe a method of constructing higher-order neural networks that respond invariantly under geometric transformations on the input space. By requiring each unit to satisfy a set of constraints on the interconnection weights, a particular structure is imposed on the network. A network built using such an architecture maintains its invariant performance independent of the values the weights assume, of the learning rules used, and of the form of the nonlinearities in the network. The invariance exhibited by a firstorder network is usually of a trivial sort, e.g., responding only to the average input in the case of translation invariance, whereas higher-order networks can perform useful functions and still exhibit the invariance. We derive the weight constraints for translation, rotation, scale, and several combinations of these transformations, and report results of simulation studies. INTRODUCTION A persistent difficulty for pattern recognition systems is the requirement that patterns or objects be recognized independent of irrelevant parameters or distortions such as orientation (position, rotation, aspect), scale or size, background or context, doppler shift, time of occurrence, or signal duration. The remarkable performance of humans and other animals on this problem in the visual and auditory realms is often taken for granted, until one tries to build a machine with similar performance. Thoufh many methods have been developed for dealing with these problems, we have classified them into two categories: 1) preprocessing or transformation (inherent) approaches, and 2) case-specific or "brute force" (learned) approaches. Common transformation techniques include: Fourier, Hough, and related transforms; moments; and Fourier descriptors of the input signal. In these approaches the signal is usually transformed so that the subsequent processing ignores arbitrary parameters such as scale, translation, etc. In addition, these techniques are usually computationally expensive and are sensitive to noise in the input signal. The "brute force" approach is exemplified by training a device, such as a perceptron, to classify a pattern independent of it's position by presenting the @ American Institute of Physics 1988 302 training pattern at all possible positions. MADALINE machines2 have been shown to perform well using such techniques. Often, this type of invariance is pattern specific, does not easily generalize to other patterns, and depends on the type of learning algorithm employed. Furthermore, a great deal of time and energy is spent on learning the invariance, rather than on learning the signal. We describe a method that has the advantage of inherent invariance but uses a higher-order neural network approach that must learn only the desired signal. Higher-order units have been shown to have unique computational strengths and are quite amenable to the encoding of a priori know1edge. 3-7 MATHEMATICAL DEVELOPMENT Our approach is similar to the group invariance approach,8,10 although we make no appeal to group theory to obtain our results. We begin by selecting a transformation on the input space, then require the output of the unit to be invariant to the transformation. The resulting equations yield constraints on the interconnection weights, and thus imply a particular form or structure for the network architecture. For the i-th unit Yi of order M defined on a discrete input space, let the output be given by Yi[YiM(X),P(x)] - f( WiO + ~ Wi1 (X1) P(x1) + ~~ Wi2(X1,X2) P(x1) P(x2) + ... +~ ... ~ WiM(X1,· ·XM) P(x1)· ·P(XM) ), (1) where p(x) is the input pattern or signal function (sometimes called a pixel) evaluated at position vector x, wim(xl, ... Xm) is the weight of order m connecting the outputs of units at Xl, x2, .. Xm to the ith unit, i.e., it correlates m values, f(u) is some threshold or sigmoid output function, and the summations extend over the input space. YiM(X) represents the entire set of weights associated with the i-th unit. These units are equivalent to the sigma-pi unitsa defined by Rumelhart, Hinton, and Williams. 7 Systems built from these units suffer from a combinatorial explosion of terms, hence are more complicated to build and train. To reduce the severity of this problem, one can limit the range of the interconnection weights or the number of orders, or impose various other constraints. We find that, in addition to the advantages of inherent invariance, imposing an invariance constraint on Eq. (1) reduces the number of allowed aThe sigma-pi neural networks are multi-layer networks with higher-order terms in any layer. As such, most of the neural networks described here can be considered as a special case of the sigma-pi units. However, the sigma-pi units as originally formulated did not have invariant weight terms, though it is quite simple to incorporate such invariances in these units. weights, thus simplifying the architecture and training time. shortening the 303 We now define what we mean by invariance. is invariant with respect to the transformation pattern if9 The output of a unit T on the input (2) An example of the class of invariant response defined by Eq. (2) would be invariant detection of an object in the receptive field of a panning or zooming camera. An example of a different class would be invariant detection of an object that is moving within the field of a fixed camera. One can think of this latter case as consisting of a fixed field of "noise" plus a moving field that contains only the object of interest. If the detection system does not respond to the fixed field, then this latter case is included in Eq. (2). To illustrate our method we derive the weight constraints for one-dimensional translation invariance. We will first switch to a continuous formulation, however, for reasons of simplicity and generality, and because it is easier to grasp the physical significance of the results, although any numerical simulation requires a discrete formulation and has significant implications for the implementation of our results. Instead of an index i, we now keep track of our units with the continuous variable u. With these changes Eq. (2) now becomes y[u;wM(x),p(X)] = f( wO + JrdXl Wl(U;Xl) P(xl) + ... + f·· Jr dXl· .dXM wM(U;Xl,· ·XM) P(Xl)· .P(XM) ), (3) The limits on the integrals are defined by the problem and are crucial in what follows. Let T be a translation of the input pattern by -xO, so that T[p(x)] - p(x+XO) (4) where xo is the translation of the input pattern. Then, from eq (2), Ty[u;wM(x) ,p(x)] - y[u;YM(x),p(x+XO») = y[u;wM(x),p(x)] (5) Since p(x) is arbitrary we must impose term-by-term equality in the argument of the threshold function; i.e., f dXl Wl(U;Xl) P(xl) = f dxl Wl(U;Xl) P(xl+XO), (Sa) Jr fdxl dX2 W2(U;Xl,X2) P(xl) P(x2) = Jr f dXl dX2 W2(U;Xl,X2) P(xl+XO) P(x2+XO), etc. (Sb) 304 Making the substitutions xl. xl-XO, x2 ",x2-XO, etc, we find that f dXl Wl(U;Xl) P(xl) - f dxl WI(U;Xl-XO) P(XI) , (6a) f f dxl dX2 W2(U;XI,X2) P(xI) P(x2) f f dXI dX2 W2(U;XI-XO,X2-XO) P(xI) P(x2), (6b) etc. Note that the limits of the integrals on the right hand side must be adjusted to satisfy the change-of-variables. If the limits on the integrals are infinite or if one imposes some sort of periodic boundary condition, the limits of the integrals on both sides of the equation can be set equal. We will assume in the remainder of this paper that these conditions can be met; normally this means the limits of the integrals extend to infinity. (In an implementation, it is usually impractical or even impossible to satisfy these requirements, but our simulation results indicate that these networks perform satisfactorily even though the regions of integration are not identical. This question must be addressed for each class of transformation; it is an integral part of the implementation design.) Since the functions p(x) are arbitrary and the regions of integration are the same, the weight functions must be equal. This imposes a constraint on the functional form of the weight functions or, in the discrete implementation, limits the allowed connections and thus the number of weights. In the case of translation invariance, the constraint on the functional form of the weight functions requires that w1(U;XI) - wl(u;X].-XO), w2(U;XI,X2) - w2(U;XI-XO,X2-XO), etc. (7a) (7b) These equations imply that the first order weight is independent of input position, and depends only on the output position u. The second order weight is a function only of vector differences,IO i.e., w1(u;Xj) - J..(u), w2(U;X].,X2) - w2(u:X]. -Xl)· (8a) (8b) For a discrete implementation with N input units (pixels) fully connected to an output unit, this requirement reduces the number of second-order weights from order N2 to order N, i.e., only weights for differences of indexes are needed rather than all unique pair combinations. Of course, this advantage is multiplied as the number of fully-connected output units increases. FURTHER EXAMPLES We have applied these techniques to several other transformations of interest. For the case of transformation of scale 305 define the scale operator S such that Sp(x) aIlp(ax) (9) where a is the scale factor, and x is a vector of dimension n. The factor an is used for normalization purposes, so that a given figure always contains the same "energy" regardless of its scale. Application of the same procedure to this transformation leads to the following constraints on the weights: wl(u;Xjfa) -= wl(u;~, w2(u;X1Ia,xv'a) .. w2(u;'X.l.'~)' w3(u;xlla,x2/a ,x3/a) ... w3(U;X].,X2,X3), etc. (lOa) (lOb) (lOc) Consider a two-dimensional problem viewed in polar coordinates (r,t). A set of solutions to these constraints is J.(u;q,tI) - w1(u;Q), w2(u;rl,r2;tl,t2) - w2(u;rllr2;tl,t2). w3(u;rl,r2,r3;tl,t2,t3) - w3(u;(rl-r2)/r3;tl,t2,t3). (lla) (llb) (llc) Note that with increasing order comes increasing freedom in the selection of the functional form of the weights. Any solution that satisfies the constraint may be used. This gives the designer additional freedom to limit the connection complexity, or to encode special behavior into the net architecture. An example of this is given later when we discuss combining translation and scale invariance in the same network. Now consider a change of scale for a two-dimensional system in rectangular coordinates, and consider only the second-order weights. A set of solutions to the weight constraint is: W2(U;Xl,Yl;X2,Y2) W2(U;Xl/Yl;X2/Y2), W2(U;Xl,Yl;X2,Y2) W2(U;Xl/X2;Yl/Y2), W2(U;Xl,Yl;X2,Y2) - w2(U;(Xl-X2)/(Yl-Y2)), etc. (12a) (l2b) (12c) We have done a simulation using the form of Eq. (12b). The simulation was done using a small input space (8x8) and one output unit. A simple least-mean-square (back-propagation) algorithm was used for training the network. When taught to distinguish the letters T and C at one scale, it distinguished them at changes of scale of up to 4X with about 15 percent maximum degradation in the output strength. These results are quite encouraging because no special effort was required to make the system work, and no corrections or modifications were made to account for the boundary condition requirements as discussed near Eq. (6). This and other simulations are discussed further later. As a third example of a geometric transformation, consider the case of rotation about the origin for a two-dimensional space in polar coordinates. One can readily show that the weight constraints 306 are satisfied if wl(u;rl,tl) ~ wl(u;rl), w2(u;rl,r2;tl,t2) - w2(u;rl,r2;tl-t2), etc. (13a) (l3b) These results are reminiscent of the results for translation invariance. This is not uncommon: seemingly different problems often have similar constraint requirements if the proper change of variable is made. This can be used to advantage when implementing such networks but we will not discuss it further here. An interesting case arises when one considers combinations of invariances, e.g., scale and translation. This raises the question of the effect of the order of the transformations, i.e., is scale followed by translation equivalent to translation followed by scale? The obvious answer is no, yet for certain cases the order is unimportant. Consider first the case of change-of-scale by a, followed by a translation XC; the constraints on the weights up to second order are: Wl(U;Xl) - wl(u; (xl-xo)/a), w2 (u; Xl ,x2) 0= w2(u; (xl-xo)/a, (x2-xo)/a) , and for translation followed by scale the constraints are: wl(u;Xl) - wl(u; (xl/a)-xo). and w2(U;Xl,X2) = w2(u;(xl/a)-xo,(x2Ia)-XO) . (14a) (l4b) (lSa) (lSb) Consider only the second-order weights for the two-dimensional case. Choose rectangular coordinate variables (x,y) so that the translation is given by (xO,YO). Then W2(U;Xl,Yl;X2,Y2) = w2(u;(xl/a)-xO,(Yl/a)-YO;(x2/a)-xO'(Y2/a)-yO)' (l6a) or W2(U;Xl,Yl;X2,Y2) w2(U;(Xl-xo)/a, (Yl-yo)/a; (x2- xo)/a, (Y2-Yo)/a). If we take as our solution w2(U;Xl,Yl;X2,Y2) = w2(U;(X1-X2)/(Yl-Y2», then w2 is invariant to scale and translation, and unimportant. With higher-order weights one can be adventurous. the order is even more (16b) (17) As a final example consider the case factor a and rotation about the origin by dimensional system in polar coordinates. transformation makes no difference.) The second order are: of a change of scale by a an amount to for a two(Note that the order of weight constraints up to (18a) 307 (18b) The first-order constraint requires that wI be independent of the input variables, but for the second-order term one can obtain a more useful solution: (19) This implies that with second-order weights, one can construct a unit that is insensitive to changes in scale and rotation of the input space. How useful it is depends upon the application. SIMULATION RESULTS We have constructed several higher-order neural networks that demonstrated invariant response to transformations of scale and of translation of the input patterns. The systems were small, consisting of less than 100 input units, were constructed from second-and first-order units, and contained only one, two, or three layers. We used a back-propagation algorithm modified for the higher-order (sigma-pi) units. The simulation studies are still in the early stages, so the performance of the networks has not been thoroughly investigated. It seems safe to say, however, that there is much to be gained by a thorough study of these systems. For example, we have demonstrated that a small system of second-order units trained to distinguish the letters T and C at one scale can continue to distinguish them over changes in scale of factors of at least four without retraining and with satisfactory performance. Similar performance has been obtained for the case of translation invariance. Even at this stage, some interesting facets of this approach are becoming clear: 1) Even with the constraints imposed by the invariance, it is usually necessary to limit the range of connections in order to restrict the complexity of the network. This is often cited as a problem with higher-order networks, but we take the view that one can learn a great deal more about the nature of a problem by examining it at this level rather than by simply training a network that has a general-purpose architecture. 2) The higher-order networks seem to solve problems in an elegant and simple manner. However, unless one is careful in the design of the network, it performs worse than a simpler conventional network when there is noise in the input field. 3) Learning is often "quicker" than in a conventional approach, although this is highly dependent on the specific problem and implementation design. It seems that a tradeoff can be made: either faster learning but less noise robustness, or slower learning with more robust performance. DISCUSSION We have shown a simple way to encode geometric invariances into neural networks (instead of training them), though to be useful the networks must be constructed of higher-order units. The invariant encoding is achieved by restricting the allowable network 308 architectures and is independent of learning rules and the form of the sigmoid or threshold functions. The invariance encoding is normally for an entire layer, although it can be on an individual unit basis. It is easy to build one or more invariant layers into a multi-layer net, and different layers can satisfy different invariance requirements. This is useful for operating on internal features or representations in an invariant manner. For learning in such a net, a multi-layered learning rule such as generalized backpropagation7 must be used. In our simulations we have used a generalized back-propagation learning rule to train a two-layer system consisting of a second-order, translation-invariant input layer and a first-order output layer. Note that we have not shown that one can not encode invariances into layered first-order networks, but the analysis in this paper implies that such invariance would be dependent on the form of the sigmoid function. When invariances are encoded into higher-order neural networks, the number of interconnections required is usually reduced by orders of powers of N where N is the size of the input. For example, a fully connected, first-order, single-layer net with a single output unit would have order N interconnections; a similar second-order net, order N2 . If this second-order net (or layer) is made shift invariant, the order is reduced to N. The number of multiplies and adds is still of order N2 . We have limited our discussion in this paper to geometric invariances, but there seems to be no reason why temporal or other invariances could not be encoded in a similar manner. REFERENCES 1. D.H. Ballard and C.M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, NJ, 1982). 2. B. Widrow, IEEE First Int1. Conf. on Neural Networks, 87TH019l7, Vol. 1, p. 143, San Diego, CA, June 1987. 3. J.A. Feldman, Biological Cybernetics 46, 27 (1982). 4. C.L. Giles and T. Maxwell, App1. Optics 26, 4972 (1987). 5. G.E. Hinton, Proc. 7th IntI. Joint Conf. on Artificial Intelligence, ed. A. Drina, 683 (1981). 6. Y.C. Lee, G. Doolen, H.H. Chen, G.Z. Sun, T. Maxwell, H.Y. Lee, C.L. Giles, Physica 22D, 276 (1986). 7. D.E. Rume1hart, G.E. Hinton, and R.J. Williams, Parallel Distributed Processing, Vol. 1, Ch. 8, D.E. Rume1hart and J.L. McClelland, eds., (MIT Press, Cambridge, 1986). 8. T. Maxwell, C.L. Giles, Y.C. Lee, and H.H. Chen, Proc. IEEE IntI. Conf. on Systems, Man, and Cybernetics, 86CH2364-8, p. 627, Atlanta, GA, October 1986. 9. W. Pitts and W.S. McCulloch, Bull. Math. Biophys. 9, 127 (1947). 10. M. Minsky and S, Papert, Perceptrons (MIT Press, Cambridge, Mass., 1969). 309
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A NEURAL-NETWORK SOLUTION TO THE CONCENTRATOR ASSIGNNlENT PROBLEM Gene A. Tagliarini Edward W. Page Department of Computer Science, Clemson University, Clemson, SC 29634-1906 ABSTRACT 775 Networks of simple analog processors having neuron-like properties have been employed to compute good solutions to a variety of optimization problems. This paper presents a neural-net solution to a resource allocation problem that arises in providing local access to the backbone of a wide-area communication network. The problem is described in terms of an energy function that can be mapped onto an analog computational network. Simulation results characterizing the performance of the neural computation are also presented. INTRODUCTION This paper presents a neural-network solution to a resource allocation problem that arises in providing access to the backbone of a communication network. 1 In the field of operations research, this problem was first known as the warehouse location problem and heuristics for finding feasible, suboptimal solutions have been developed previously.2. 3 More recently it has been known as the multifacility location problem4 and as the concentrator assignment problem.1 THE HOPFIELD NEURAL NETWORK MODEL The general structure of the Hopfield neural network model5 • 6,7 is illustrated in Fig. 1. Neurons are modeled as amplifiers that have a sigmoid input! output curve as shown in Fig. 2. Synapses are modeled by permitting the output of any neuron to be connected to the input of any other neuron. The strength of the synapse is modeled by a resistive connection between the output of a neuron and the input to another. The amplifiers provide integrative analog summation of the currents that result from the connections to other neurons as well as connection to external inputs. To model both excitatory and inhibitory synaptic links, each amplifier provides both a normal output V and an inverted output V. The normal outputs range between 0 and 1 while the inverting amplifier produces corresponding values between 0 and -1. The synaptic link between the output of one amplifier and the input of another is defined by a conductance Tij which connects one of the outputs of amplifier j to the input of amplifier i. In the Hopfield model, the connection between neurons i and j is made with a resistor having a value Rij = 1rrij . To provide an excitatory synaptic connection (positive Tij ), the resistor is connected to the normal output of This research was supported by the U.S. Army Strategic Defense Command. © American Institute of Physics 1988 776 13 14 inputs VI V2 V3 V4 outputs Fig. 1. Schematic for a simplified Hopfield network with four neurons. 1 V o -u o +u Fig. 2. Amplifier input/output relationship amplifier j. To provide an inhibitory connection (negative Tij), the resistor is connected to the inverted output of amplifier j. The connections among the neurons are defined by a matrix T consisting of the conductances Tij . Hopfield has shown that a symmetric T matrix (Tij = Tji ) whose diagonal entries are all zeros, causes convergence to a stable state in which the output of each amplifier is either 0 or 1. Additionally, when the amplifiers are operated in the high-gain mode, the stable states of a network of n neurons correspond to the local minima of the quantity n n E = (-112) L L i=l j=l T·V.V· IJ 1 J n L V.I· I 1 (1) where Vi is the output of the ith neuron and Ii is the externally supplied input to the ph neuron. Hopfield refers to E as the computational energy of the system. THE CONCENTRATOR ASSIGNMENT PROBLEM Consider a collection of n sites that are to be connected to m concentrators as illustrated in Fig. 3(a). The sites are indicated by the shaded circles and the concentrators are indicated by squares. The problem is to find an assignment of sites to concentrators that minimizes the total cost of the assignment and does not exceed the capacity of any concentrator. The constraints that must be met can be summarized as follows: a) Each site i ( i = 1, 2, ... , n ) is connected to exactly one concentrator; and b) Each concentrator j (j = 1, 2, ... , m ) is connected to no more than kj sites (where kj is the capacity of concentrator D. 777 Figure 3(b) illustrates a possible solution to the problem represented in Fig. 3(a). 0 0 • • • • • • 0 • • • • • • 0 o Concentrators • Sites (a). Site/concentrator map (b). Possible assignment Fig. 3. Example concentrator assignment problem If the cost of assigning site i to concentrator j is cij , then the total cost of a particular assignment is total cost = n m L L i=l j=l x ·· c·· IJ IJ (2) where Xij = 1 only if we actually decide to assign site i to concentrator j and is 0 otherwise. There are mn possible assignments of sites to concentrators that satisfy constraint a). Exhaustive search techniques are therefore impractical except for relatively small values of m and n. THE NEURAL NETWORK SOLUTION This problem is amenable to solution using the Hopfield neural network model. The Hopfield model is used to represent a matrix of possible assignments of sites to concentrators as illustrated in Fig. 4. Each square corresponds 778 CONCENTRATORS 1 2 j m r,;------;-, /r 1 ,II 11---III ---III, 2 ,~ .---~ ---~I S •••• ITES ~~ ,.. • • I The darkly shaded neui III 11---II ---II I ron corresponds to the :: : : hypothesis that site i '~n 'Ii • ---Ii ---Ii ' should be as~igned to ~ ~ :.J concentrator J. " n+l II 111---11---11 SLACK .... < n+2 III II ---~ ---• • • • ,~n+k j II 11---III ---III Fig. 4. Concentrator assignment array to a neuron and a neuron in row i and column j of the upper n rows of the array represents the hypothesis that site i should be connected to concentrator j. If the neuron in row i and column j is on, then site i should be assigned to concentrator j; if it is off, site i should not be assigned to concentrator j. The neurons in the lower sub-array, indicated as "SLACK", are used to implement individual concentrator capacity constraints. The number of slack neurons in a column should equal the capacity (expressed as the number sites which can be accommodated) of the corresponding concentrator. While it is not necessary to assume that the concentrators have equal capacities, it was assumed here that they did and that their cumulative capacity is greater than or equal to the number of sites. To ena~le the neurons in the network illustrated above to compute solutions to the concentrator problem, the network must realize an energy function in which the lowest energy states correspond to the least cost assignments. The energy function must therefore favor states which satisfy constraints a) and b) above as well as states that correspond to a minimum cost assignment. The energy function is implemented in terms of connection strengths between neurons. The following section details the construction of an appropriate energy function. THE ENERGY FUNCTION Consider the following energy equation: n m 2 E = A L ( L y .. - 1 ) . 1 . 1 1J 1= J= + m n+k· B L ( L J y .. k . )2 j=1 i=1 IJ J m n+kj + C L L y.. ( 1 - Yij ) j=1 i=1 1J 779 (3~ where Yij is the output of the amplifier in row i and column j of the neuron matrix, m and n are the number of concentrators and the number of sites respectively, and kj is the capacity of concentrator j. The first term will be minimum when the sum of the outputs in each row of neurons associated with a site equals one. Notice that this term influences only those rows of neurons which correspond to sites; no term is used to coerce the rows of slack neurons into a particular state. The second term of the equation will be minimum when the sum of the outputs in each column equals the capacity kj of the corresponding concentrator. The presence of the kj slack neurons in each column allows this term to enforce the concentrator capacity restrictions. The effect of this term upon the upper sub-array of neurons (those which correspond to site assignments) is that no more than kj sites will be assigned to concentrator j. The number of neurons to be turned on in column j is kj ; consequently, the number of neurons turned on in column j of the assignment sub-array will be less than or equal to kj . The third term causes the energy function to favor the "zero" and "one" states of the individual neurons by being minimum when all neurons are in one or the other of these states. This term influences all neurons in the network. In summary, the first term enforces constraint a) and the second term enforces constraint b) above. The third term guarantees that a choice is actually made; it assures that each neuron in the matrix will assume a final state near zero or one corresponding to the Xij term of the cost equation (Eq. 2). After some algebraic re-arrangement, Eq. 3 can be written in the form of Eq. 1 where ., {A * 8(i,k) * (1-8U,I) + B * 8U,1) * (1-8(i,k», if i<n and k<n T IJ kl = (4) , C * 8U,I) * (1-8(i,k», if i>n or k>n. Here quadruple subscripts are used for the entries in the matrix T. Each entry indicates the strength of the connection between the neuron in row i and column j and the neuron in row k and column I of the neuron matrix. The function delta is given by 780 8( i , j ) = { 1, if i = j 0, otherwise. (5) The A and B terms specify inhibitions within a row or a column of the upper sub-array and the C term provides the column inhibitions required for the neurons in the sub-array of slack neurons. Equation 3 specifies the form of a solution but it does not include a term that will cause the network to favor minimum cost assignments. To complete the formulation, the following term is added to each Tij,kl: D • 8( j , I ) • ( 1 - 8( i , k ) ) (cost [ i , j ] + cost [ k , I ]) where cost[ i , j ] is the cost of assigning site i to concentrator j. The effect of this term is to reduce the inhibitions among the neurons that correspond to low cost assignments. The sum of the costs of assigning both site i to concentrator j and site k to concentrator I was used in order to maintain the symmetry of T. The external input currents were derived from the energy equation (Eq.3) and are given by I .. _ {2. k j , if i < n IJ 2 • k j - 1, otherwise. (6) This exemplifies a teChnique for combining external input currents which arise from combinations of certain basic types of constraints. AN EXAMPLE The neural network solution for a concentrator assignment problem consisting of twelve sites and five concentrators was simulated. All sites and concentrators were located within the unit square on a randomly generated map. For this problem, it was assumed that no more than three sites could be assigned to a concentrator. The assignment cost matrix and a typical assignment resulting from the simulation are shown in Fig. 5. It is interesting to notice that the network proposed an assignment which made no use of concentrator 2. Because the capacity of each concentrator kj was assumed to be three sites, the external input current for each neuron in the upper sub-array was I ij = 6 while in the sub-array of slack neurons it was I ij = 5. The other parameter values used in the simulation were A = B = C =-2 and D = 0.1 . 781 CONCENTRATORS SITES 1 2 3 4 5 A .47 .28 .55 @ .46 @ B .72 .75 .40 .63 C .95 .71 @ .39 .92 D .88 .78 @ .38 .82 E .31 .62 .81 .56 @ F .25 .51 .76 .46 G G .17 .39 .77 .41 G H @ .81 .54 .52 .56 I .60 .67 .44 G .51 @ J .84 .76 .66 .48 K .42 .33 .55 B .38 @ L .60 1.05 .71 .18 Fig. 5. The concentrator assignment cost matrix with choices circled. Since this choice of parameters results in a T matrix that is symmetric and whose diagonal entries are all zeros, the network will converge to the minima of Eq. 3. Furthermore, inclusion of the term which is weighted by the parameter D causes the network to favor minimum cost assignments. To evaluate the performance of the simulated network, an exhaustive search of all solutions to the problem was conducted using a backtracking algorithm. A frequency distribution of the solution costs associated with the assignments generated by the exhaustive search is shown in Fig. 6. For comparison, a histogram of the results of one hundred consecutive runs of the neural-net simulation is shown in Fig. 7. Although the neural-net simulation did not find a global minimum, ninety-two of the one hundred assignments which it did find were among the best 0.01 % of all solutions and the remaining eight were among the best 0.3%. CONCLUSION Neural networks can be used to find good, though not necessarily optimal, solutions to combinatorial optimization problems like the concentrator 782 Frequency 4000000 3500000 3000000 250000 1500000 100000 500000 OL---3.2 4.2 5.2 Cost 6.2 7.2 8.2 Fig. 6. Distribution of assignment costs resulting from an exhaustive search of all possible solutions. Frequency 25 20 15 10 5 o Fig. 7. Distribution of assignment costs resulting from 100 consecutive executions of the neural net simulation. assignment problem. In order to use a neural network to solve such problems, it is necessary to be able to represent a solution to the problem as a state of the network. Here the concentrator assignment problem was successfully mapped onto a Hopfield network by associating each neuron with the hypothesis that a given site should be assigned to a particular concentrator. An energy function was constructed to determine the connections that were needed and the resulting neural network was simulated. While the neural network solution to the concentrator assignment problem did not find a globally minimum cost assignment, it very effectively rejected poor solutions. The network was even able to suggest assignments which would allow concentrators to be removed from the communication network. REFERENCES 1. A. S. Tanenbaum, Computer Networks (Prentice-Hall: Englewood Cliffs, New Jersey, 1981), p. 83. 2. E. Feldman, F. A. Lehner and T. L. Ray, Manag. Sci. V12, 670 (1966). 3. A. Kuehn and M. Hamburger, Manag. Sci. V9, 643 (1966). 4. T. Aykin and A. 1. G. Babu, 1. of the Oper. Res. Soc. V38, N3, 241 (1987). 5. J. 1. Hopfield, Proc. Natl. Acad. Sci. U. S. A., V79, 2554 (1982). 6. J. 1. Hopfield and D. W. Tank, Bio. Cyber. V52, 141 (1985). 7. D. W. Tank and 1. 1. Hopfield, IEEE Trans. on Cir. and Sys. CAS-33, N5, 533 (1986).
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760 A NOVEL NET THAT LEARNS SEQUENTIAL DECISION PROCESS G.Z. SUN, Y.C. LEE and H.H. CHEN Department of PhYJicJ and AJtronomy and InJtitute for Advanced Computer StudieJ UNIVERSITY OF MARYLAND,COLLEGE PARK,MD 20742 ABSTRACT We propose a new scheme to construct neural networks to classify patterns. The new scheme has several novel features : 1. We focus attention on the important attributes of patterns in ranking order. Extract the most important ones first and the less important ones later. 2. In training we use the information as a measure instead of the error function. 3. A multi-percept ron-like architecture is formed auomatically. Decision is made according to the tree structure of learned attributes. This new scheme is expected to self-organize and perform well in large scale problems. © American Institute of Physics 1988 761 1 INTRODUCTION It is well known that two-layered percept ron with binary connections but no hidden units is unsuitable as a classifier due to its limited power [1]. It cannot solve even the simple exclusive-or problem. Two extensions have been prop'osed to remedy this problem. The first is to use higher order connections l2]. It has been demonstrated that high order connections could in many cases solve the problem with speed and high accuracy [3], [4]. The representations in general are more local than distributive. The main drawback is however the combinatorial explosion of the number of high-order terms. Some kind of heuristic judgement has to be made in the choice of these terms to be represented in the network. A second proposal is the multi-layered binary network with hidden units r5]. These hidden units function as features extracted from the bottom input layer to facilitate the classification of patterns by the output units. In order to train the weights, learning algorithms have been proposed that backpropagate the errors from the visible output layer to the hidden layers for eventual adaptation to the desired values. The multi-layered networks enjoy great popularity in their flexibility. However, there are also problems in implementing the multi-layered nets. Firstly, there is the problem of allocating the resources. Namely, how many hidden units would be optimal for a particular problem. If we allocate too many, it is not only wasteful but also could negatively affect the performance of the network. Since too many hidden units implies too many free parameters to fit specifically the training patterns. Their ability to generalize to noval test patterns would be adversely affected. On the other hand, if too few hidden units were allocated then the network would not have the power even to represent the trainig set. How could one judge beforehand how many are needed in solving a problem? This is similar to the problem encountered in the high order net in its choice of high order terms to be represented. Secondly, there is also the problem of scaling up the network. Since the network represents a parallel or coorperative process of the whole system, each added unit would interact with every other units. This would become a serious problem when the size of our patterns becomes large. Thirdly, there is no sequential communication among the patterns in the conventional network. To accomplish a cognitive function we would need the patterns to interact and communicate with each other as the human reasoning does. It is difficult to envision such an interacton in current systems which are basically input-output mappings. 2 THE NEW SCHEME In this paper, we would like to propose a scheme that constructs a network taking advantages of both the parallel and the sequential processes. We note that in order to classify patterns, one has to extract the intrinsic features, which we call attributes. For a complex pattern set, there may be a large number of attributes. But differnt attributes may have different 762 ranking of importance. Instead of ext racing them all simultaneously it may be wiser to extract them sequentially in order of its importance [6], [7]. Here the importance of an attribute is determined by its ability to partition the pattern set into sub-categories. A measure of this ability of a processing unit should be based on the extracted information. For simplicity, let us assume that there are only two categories so that the units have only binary output values 1 and ° ( but the input patterns may have analog representations). We call these units, including their connection weights to the input layer, nodes. For given connection weights, the patterns that are classified by a node as in category 1 may have their true classifications either 1 or 0. Similarly, the patterns that are classified by a node as in category 0 may also have their true classifications either 1 or o. As a result, four groups of patterns are formed: (1,1), (0,0), (1,0), (0,1). We then need to judge on the efficiency of the node by its ability to split these patterns optimally. To do this we shall construct the impurity fuctions for the node. Before splitting, the impurity of the input patterns reaching the node is given by (1) where pt = Nf / N is the probability of being truely classified as in category 1, and P~ = N~/N is the probability of being truely classified as in category o. After splitting, the patterns are channelled into two branches, the impurity becomes 1(1 = -Pt L P(j, 1) logP(j, 1) - P; L P(j, O)logP(j, 0) (2) j=O,1 j=O,1 where Pi = Ni / N is the probability of being classified by the node as in category 1, P; = N8/N is the probability of being classified by the node as in category 0, and P(j, i) is the probability of a pattern, which should be in category j, but is classified by the node as in category i. The difference (3) represents the decrease of the impurity at the node after splitting. It is the quantity that we seek to optimize at each node. The logarithm in the impurity function come from the information entropy of Shannon and Weaver. For all practical purI?ose, we found the. optimization of (3) the same as maximizing the entropy l6] where Ni is the number of training patterns classified by the node as in category i, Nij is the number of training patterns with true classification in category i but classified by the node as in category j. Later we shall call the terms in the first bracket SI and the second S2. Obviously, we have i = 0,1 763 After we trained the first unit, the training patterns were split into two branches by the unit. If the classificaton in either one of these two branches is pure enough, or equivalently either one of Sl and S2 is fairly close to 1, then we would terminate that branch ( or branches) as a leaf of the decision tree, and classify the patterns as such. On the other hand, if either branch is not pure enough, we add additional node to split the pattern set further. The subsequent unit is trained with only those patterns channeled through this branch. These operations are repeated until all the branches are terminated as leaves. 3 LEARNING ALGORITHM We used the stochastic gradient descent method to learn the weights of each node. The training set for each node are those patterns being channeled to this node. As stated in the previous section, we seek to maximize the entropy function S. The learning of the weights is therefore conducted through oS 1:::. Wj = 11 ow(5) J Where 11 is the learning rate. The gradient of S can be calculated from the following equation oS = ~ [(1 _ 2NJ1) oNn + (1 _ 2 Nil ) oNOl + oWj N Nl oW; Nl oWj (1 _ 2NJo) ONIO + (1 _ 2N'fO) ONoo] NJ oWj NJ oWj Using analog units we have or = 1 1 + exp( - Lj WjII) oor = orC1 _ or)!'; owJ J (6) (7) (8) Furthermore, let Ar = 1 or 0 being the true answer for the input pattern r , then N;; = t. [iA' + (1 - i)(1 - A') 1 [i O' + (1 - j)(1 - 0') 1 (9) Substituting these into equation (5), we get 1:::.Wj = 2T} :L[2Ar(NU - NlO) + Ni~ - Ni;]or(l - or)IJ (10) r Nl No No Nl In applying the formula (10),instead of calculating the whole summation at once, we update the weights for each pattern individually. Meanwhile we update Nij in accord with equation (9). 764 Figure 1: The given classification tree, where 01 , O'l and 03 are chosen to be all zeros in the numerical example. 4 AN EXAMPLE To illustrate our method, we construct an example which is itself a decision tree. Assuming there are three hidden variables ai, a'l, a3, a pattern is given by a ten-dimensional vector II, I'l, ... , 110 , constructed from the three hidden variables as follows II al + a3 16 2a3 1'l 2al - a'l 17 a3 - al 13 a3 - 2a'l 18 2al + 3a3 I" al + 2a'l + 3a3 19 4a3 - 3a l Is 5al - 4a" 110 2al + 2a'l + 2a3· A given pattern is classified as either 1 (yes) or 0 (no) according to the corresponding values of the hidden variables ai, a'l, a3. The actual decision is derived from the decision tree in Fig.1. In order to learn this classification tree, we construct a training set of 5000 patterns generated by randomly chosen values ai, a'l, a3 in the interval -1 to + 1. We randomly choose the initial weights for each node, and terminate 5=0.79 G 51 =0.60/ ~=0.87 " (fIg (2519/35) G 51 =0.65/ ,S2= 0.88 VI ~OCE:S] (16171114) G 51 = 0.85/ ~= 0.73 (SS/S)Wi 5. =0.90/ (92/S)rul 52=0.96 ffQ](548/12) Figure 2: The learned classification tree structure 765 a branch as a leaf whenever the branch entropy is greater than 0.80. The entropy is started at S = 0.65, and terminated at its maximum value S = 0.79 for the first node. The two branches of this node have the entropy fuction valued at SI = 0.61, S2 = 0.87 respectively. This corrosponds to 2446 patterns channeled to the first branch and 2554 to the second. Since S2 > 0.80 we terminate the second branch. Among 2554 patterns channeled to the second branch there are 2519 patterns with true classification as no and 35 yes which are considered as errors. After completing the whole training process, there are totally four nodes automatically introduced. The final result is shown in a tree structure in Fig.2. The total errors classified by the learned tree are 3.4 % of the 5000 trainig patterns. After trainig we have tested the result using 10000 novel patterns, the error among which is 3.2 %. 5 SUMMARY We propose here a new scheme to construct neural network that can automatically learn the attributes sequentially to facilitate the classification of patterns according to the ranking importance of each attribute. This scheme uses information as a measure of the performance of each unit. It is 766 self-organized into a presumably optimal structure for a specific task. The sequential learning procedure focuses attention of the network to the most important attribute first and then branches out' to the less important attributes. This strategy of searching for attributes would alleviate the scale up problem forced by the overall parallel back-propagation scheme. It also avoids the problem of resource allocation encountered in the high-order net and the multi-layered net. In the example we showed the performance of the new method is satisfactory. We expect much better performance in problems that demand large size of units. 6 acknowledgement This work is partially supported by AFOSR under the grant 87-0388. References [1] M. Minsky and S. Papert, Perceptron, MIT Press Cambridge, Ma(1969). [2] Y.C. Lee, G. Doolen, H.H. Chen, G.Z. Sun, T. Maxwell, H.Y. Lee and C.L. Giles, Machine Learning Using A High Order Connection Netweork, Physica D22,776-306 (1986). [3] H.H. Chen, Y.C. Lee, G.Z. Sun, H.Y. Lee, T. Maxwell and C.L. Giles, High Order Connection Model For Associate Memory, AlP Proceedings Vol.151,p.86, Ed. John Denker (1986). [4] T. Maxwell, C.L. Giles, Y.C. Lee and H.H. Chen, Nonlinear Dynamics of Artificial Neural System, AlP Proceedings Vol.151,p.299, Ed. John Denker(1986). [5] D. Rummenlhart and J. McClelland, Parallel Distributit'e Processing, MIT Press(1986). [6] L. Breiman, J. Friedman, R. Olshen, C.J. Stone, Classification and Regression Trees,Wadsworth Belmont, California(1984). [7] J.R. Quinlan, Machine Learning, Vol.1 No.1(1986).
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164 MATHEMATICAL ANALYSIS OF LEARNING BEHAVIOR OF NEURONAL MODELS By JOHN Y. CHEUNG MASSOUD OMIDVAR SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE UNIVERSITY OF OKLAHOMA NORMAN, OK 73019 Presented to the IEEE Conference on "Neural Information Processing SystemsNatural and Synthetic," Denver, November ~12, 1987, and to be published in the Collection of Papers from the IEEE Conference on NIPS. Please address all further correspondence to: John Y. Cheung School of EECS 202 W. Boyd, CEC 219 Norman, OK 73019 (405)325-4721 November, 1987 © American Institute of Physics 1988 MATHEMATICAL ANALYSIS OF LEARNING BEHAVIOR OF NEURONAL MODELS John Y. Cheung and Massoud Omidvar School of Electrical Engineering and Computer Science ABSTRACT 165 In this paper, we wish to analyze the convergence behavior of a number of neuronal plasticity models. Recent neurophysiological research suggests that the neuronal behavior is adaptive. In particular, memory stored within a neuron is associated with the synaptic weights which are varied or adjusted to achieve learning. A number of adaptive neuronal models have been proposed in the literature. Three specific models will be analyzed in this paper, specifically the Hebb model, the Sutton-Barto model, and the most recent trace model. In this paper we will examine the conditions for convergence, the position of convergence and the rate at convergence, of these models as they applied to classical conditioning. Simulation results are also presented to verify the analysis. INTRODUCTION A number of static models to describe the behavior of a neuron have been in use in the past decades. More recently, research in neurophysiology suggests that a static view may be insufficient. Rather, the parameters within a neuron tend to vary with past history to achieve learning. It was suggested that by altering the internal parameters, neurons may adapt themselves to repetitive input stimuli and become conditioned. Learning thus occurs when the neurons are conditioned. To describe this behavior of neuronal plasticity, a number of models have been proposed. The earliest one may have been postulated by Hebb and more recently by Sutton and Barto 1. We will also introduce a new model, the most recent trace (or MRT) model in this paper. The primary objective of this paper, however, is to analyze the convergence behavior of these models during adaptation. The general neuronal model used in this paper is shown in Figure 1. There are a number of neuronal inputs x,(t), i = 1, ... , N. Each input is scaled by the corresponding synaptic weights w,(t), i = 1, ... , N. The weighted inputs are arithmetically summed. N y(t) = L x,(t)w,(t) - 9(t) (1) ,=1 where 9(t) is taken to be zero. 166 Neuronal inputs are assumed to take on numerical values ranging from zero to one inclusively. Synaptic weights are allowed to take on any reasonable values for the purpose of this paper though in reality, the weights may very well be bounded. Since the relative magnitude of the weights and the neuronal inputs are not well defined at this point, we will not put a bound on the magnitude of the weights also. The neuronal output is normally the result of a sigmoidal transformation. For simplicity, we will approximate this operation by a linear transformation. Sigmodial Transfonution rilure 1. A leneral aeuronal .adel. neuronal output H+-+y For convergence analysis, we will assume that there are only two neuronal inputs in the traditional classical conditioning environment for simplicity. Of course, the analysis techniques can be extended to any number of inputs. In classical conditioning, the two inputs are the conditioned stimulus Xc (t) and the unconditioned stimulus xu(t). THE SUTTON-BARTO MODEL More recently, Sutton and Barto 1 have proposed an adaptive model based on both the signal trace x,(t) and the output trace y(t) as given below: w,(t + 1) =w,(t) + cx,(t)(y(t)) - y(t) y(t + 1) ={Jy(t) + (1 - {J)y(t) Xi(t + 1) =axi(t) + Xi(t) where both a and {J are positive constants. (2a) (2b) (2c) 167 Condition of Convergence In order to simplify the analysis, we will choose Q = 0 and (3 = 0, i.e.: %,(t) = x,(t - 1) and y(t) = y(t - 1) In other words, (2a) becomes: Wi(t + 1) = Wi(t) + CXi(t)(y(t) - y(t - I)} (3) The above assumption only serves to simplify the analysis and will not affect the convergence conditions because the boundedness of %i(t) and y(t) only depends on that for Xi(t) and y(t - 1) respectively. As in the previous section, we recognize that (3) is a recurrence relation so convergence can be checked by the ratio test. It is also possible to rewrite (3) in matrix format. Due to the recursion of the neuronal output in the equation, we will include the neuronal output y(t) in the parameter vector also: (4) or To show convergence, we need to set the magnitude of the determinant of A (S-B) to be less than unity. (5) Hence, the condition for convergence is: (6) From (6), we can see that the adaptation constant must be chosen to be less than the reciprocal of the Euclidean sum of energies of all the inputs. The same techniques can be extended to any number of inputs. This can be proved merely by following the same procedures outlined above. Position At Convergence 168 Having proved convergence of the Sutton-Barto model equations of neuronal plasticity, we want to find out next at what location the system remains when converged. We have seen earlier that at convergence, the weights cease to change and so does the neuronal output. We will denote this converged position as (W(S-B»- = W(S-B) (00). In other words: (7) Since any arbitrary parameter vector can always be decomposed into a weighted sum of the eigenvectors, i.e. (8) The constants Ql, Q2, and Q3 can easily be found by inverting A(5-B). The eigenvalues of A(5-B) can be shown to be 1, 1, and c(%j + %~}. When c is within the region of convergence, the magnitude of the third eigenvalue is less than unity. That means that at convergence, there will be no contribution from the third eigenvector. Hence, (9) From (9), we can predict precisely what the converged position would be given only with the initial conditions. Rate of Convergence We have seen that when c is carefully chosen, the Sutton-Barto model will converge and we have also derived an expression for the converged position. Next we want to find out how fast convergence can be attained. The rate of convergence is a measure of how fast the initial parameter approaches the optimal position. The asymptotic rate of convergence is2: (10) where SeA (5-B» is the spectral radius and is equalled to c(%~ + %~) in this case. This completes the convergence analysis on the Sutton-Barto model of neuronal plasticity. THE MRT MODEL OF NEURONAL PLASTICITY The most recent trace (MRT) model of neuronal plasticity 3 developed by the authors can be considered as a cross between the Sutton-Barto model and the Klopf's model ". The adaptation of the synaptic weights can he expressed as follows: (11) 169 A comparison of (11) and the Sutton-Barto model in (3) ahOWl that the .cond term on the right hand aide contains an extra factor, Wi(t), which iI used to apeed up the convergence as ahoWD later. The output trace hu been replaced by If(t - 1), the most recent output, hence the name, the most recent trace model. The input trace is also replaced by the most recent input. Condition of Convergence We can now proceed to analyze the condition of convergence for the MRT model. Due to the presence of the Wi(t) factor in the second term in (31), the ratio test cannot be applied here. To analyze the convergence behavior further, let us rewrite (11) in matrix format: 0) ( WI(t) ) o W2(t) o y(t - 1) (12) or The superscript T denotes the matrix transpose operation. The above equation is quadratic in W(MRT)(t). Complete convergence analysis of this equation is extremely difficult. In order to understand the convergence behavior of (12), we note that the dominant term that determines convergence mainly relates to the second quadratic term. Hence for convergence analysis only, we will ignore the first term: (13) We can readily see from above that the primary convergence factor is BT c. Since C is only dependent on %,(t), convergence can be obtained if the duration of the synaptic inputs being active is bounded. It can be shown that the condition of convergence is bounded by: (14) 170 We can readily see that the adaptation constant c can be chosen according to (14) to ensure convergence for t < T. SIMULATIONS To verify the theoretical analysis of these three adaptive neuronal models based on classical conditioning, these models have been simulated on the mM 3081 mainframe using the FORTRAN language in single precision. Several test scenarios have been designed to compare the analytical predictions with actual simulation results. To verify the conditions for convergence, we will vary the value of the adaptation constant c. The conditioned and unconditioned stimuli were set to unity and the value of c varies between 0.1 to 1.0. For the Sutton-Barto model the simulation given in Fig. 2 shows that convergence is obtained for c < 0.5 as expected from theoretical analysis. For the MRT model, simulation results given in Fig. 3 shows that convergence is obtained for c < 0.7, also as expected from theoretical analysis. The theoretical location at convergence for the Sutton and Barto model is also shown in Figure 2. It is readily seen that the simulation results confirm the theoretical expectations. "'r.al Output , I .• I.' ... ,v ....... · · .. · · · · .. · .. · .. · · · .. i /r · · · · · · · · · · · · · · · · · · · · · · · · ,. '..,...._....-_--------1: c • 0.1 2: c • 0.2 3: c • 0.3 4: c • 0.4 s: c ·0.5 6: c • 0.6 1: c • 0.7 .. ·~----~--~M----~JI-----.~--~a~--~. Figure 2. 'lou or MuroD&l _tpuu YeT.US Ule .... er of 1urat1011& for the Suttoa-Barto ~el witb '1frerent .alues of ~aptat1on CODstant c. lleuroul Output ., ... 1.1 I.' ... ~ ""1"" • •.•• I •• , I I 1 ................................. ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . 1: e - 0.1 2: e - 0.2 3: e - 0.3 4: e - 0.4 S: e - 0.5 6: e - 0.6 ... I~, ____ ~ ____ ~, ____ ~ ____ ~1~·~,~-~o~,7~ __ ~, • . " . a • Ju.ber of iteratiOGa Figure 3. Plotl of oeuroaal outputl .craus the uuaber of iteratious for the MaT ~el with different .alues of adantatlon I:DDStaut c. 171 To illustrate the rate of convergence, we will plot the trajectory of the deviation in synaptic weights from the optimal values in the logarithmic scale since this error is logarithmic as found earlier. The slope of the line yields the rate of convergence. The trajectory for the Sutton-Barto Model is given in Figure 4 while that for the MRT model is given in Figure 5. It is clear from Figure 4 that the trajectory in the logarithmic form is a straight line. The slope Rn(A(S-B)) can readily be calculated. The curve for the MRT model given in Figure 5 is also a straight line but with a much larger slope showing faster convergence. SUMMARY In this paper, we have sought to discover analytically the convergence behavior of three adaptive neuronal models. From the analysis, we see that the Hebb model does not converge at all. With constant active inputs, the output will grow exponentially. In spite of this lack of convergence the Hebb model is still a workable model realizing that the divergent behavior would be curtailed by the sigmoidal transformation to yield realistic outputs. The 172 'II ,._) .... t "uroul I Output Dniatiotl 1 Lto I I .. .~ \' \\ " 1: 2: 3: 4: e -.0.1 C - 0.2 e • 0.3 e - 0.4 \~ '--\ \ '\ \ \ \ \ " \ '\ .. \ \ \ ',,-II " • .u.ber of iterationa .. • Figure 4. Trajectories of Deuronal output deviationa froa atatic .alues for the Sutton-"rt~ ~el with ~lfferent value. ~f adaptation cOIIstallt C. I.80 .. lleuroD&l. Output Deviation Ltl ... . "~ ~ \.\ (\ \ "' \ \\ \ \\ \ .' , ! ~ \ 'I \ \\ \ \ \ " : " ~ , , \ \ \ \ '\ \ '. i , 1: C· 0.1 2: C· 0.2 3: c· 0.3 4: C. 0.4 \ ~ \ \ .. \ i ) , " n .. Nuaber of iterations , .. '" Figure 5. Trajectories of neuronal output deviations fra. atatic values for tbe KRT ~el witb different values of adaptation constant c. 173 analysis on the Sutton and Barto model shows that this model will converge when the adaptation constant c is carefully chosen. The bounds for c is also found for this model. Due to the structure of this model, both the location at convergence and the rate of convergence are also found. We have also introduced a new model of neuronal plasticity called the most recent trace (MRT) model. Certain similarities exist between the MRT model and the Sutton-Barto model and also between the MRT model and the Klopf model. Analysis shows that the update equations for the synaptic weights are quadratic resulting in polynomial rate of convergence. Simulation results also show that much faster convergence rate can be obtained with the MRT model. REFERENCES 1. Sutton, R.S. and A.G. Barto, Psychological Review, vol. 88, p. 135, (1981). 2. Hageman, L. A. and D.M. Young. Applied Interactive Methods. (Academic Press, Inc. 1981). 3. Omidvar, Massoud. Analysis of Neuronal Plasticity. Doctoral dissertation, School of Electrical Engineering and Computer Science, University of Oklahoma, 1987. 4. Klopf, A.H. Proceedings of the American Institute of Physics Conference #151 on Neural Networks for Computing, p. 265-270, (1986).
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201 NEW HARDWARE FOR MASSIVE NEURAL NETWORKS D. D. Coon and A. G. U. Perera Applied Technology Laboratory University of Pittsburgh Pittsburgh, PA 15260. ABSTRACT Transient phenomena associated with forward biased silicon p + - n - n + structures at 4.2K show remarkable similarities with biological neurons. The devices play a role similar to the two-terminal switching elements in Hodgkin-Huxley equivalent circuit diagrams. The devices provide simpler and more realistic neuron emulation than transistors or op-amps. They have such low power and current requirements that they could be used in massive neural networks. Some observed properties of simple circuits containing the devices include action potentials, refractory periods, threshold behavior, excitation, inhibition, summation over synaptic inputs, synaptic weights, temporal integration, memory, network connectivity modification based on experience, pacemaker activity, firing thresholds, coupling to sensors with graded signal outputs and the dependence of firing rate on input current. Transfer functions for simple artificial neurons with spiketrain inputs and spiketrain outputs have been measured and correlated with input coupling. INTRODUCTION Here we discuss the simulation of neuron phenomena by electronic processes in silicon from the point of view of hardware for new approaches to electronic processing of information which parallel the means by which information is processed in intelligent organisms. Development of this hardware basis is pursued through exploratory work on circuits which exhibit some basic features of biological neural networks. Fig. 1 shows the basic circuit used to obtain spiketrain outputs. A distinguishing feature of this hardware basis is the spontaneous generation of action potentials as a device physics feature. )!-__ ,----_O_u-f)t put JLJLL R Figure 1: Spontaneous, neuronlike spiketrain generating circuit. The spikes are nearly equal in amplitude so that information is contained in the frequency and temporal pattern of the spiketrain generation. © American Institute of Physics 1988 202 TWO-TERMINAL SWITCHING ELEMENTS The use of transistor based circuitry 1 is avoided because transistor electrical characteristics are not similar to neuron characteristics. The use of devices with fundamentally non-neuronlike character increases the complexity of artificial neural networks. Complexity would be an important drawback for massive neural networks and most neural networks in nature achieve their remarkable performance through their massive size. In addition) transistors have three terminals whereas the switching elements of Hodgkin-Huxley equivalent circuits have two terminals. Motivated in part by Hodgkin-Huxley equivalent circuit diagrams) we employ two-terminal p+ n - n+ devices which execute transient switching between low conductance and high conductance states. (See Fig. 2) We call these devices injection mode devices (IMDs). In the "OFF-STATE", a typical current through the devices is '" 100fA/mm2) and in the "ON-STATE" a typical current is '" 10mA/mm2. Hence this device is an extremely good switch with a ON / 0 F F ratio of 1011. As in real neurons2, the current in the device is a function of voltage and time, not only voltage. The devices require cryogenic cooling but this results in an advantageously low quiescent power drain of < 1 nanowatt/cm2 of chip area and the very low leakage currents mentioned above. In addition, the highly unique ability of the neural networks described here to operate in a cryogenic environment is an important advantage for infrared image processing at the focal plane (see Fig. 3 and further discussion below). Vision systems begin processing at the focal plane and there are many benefits to be gained from the vision system approach to IR image processing. / \ -----/ ...-. ----I I( V, t) I R VD C ~--~--VV~--~------~ IR ;;SS:Ulse Output 1----0 +Q C - Q Figure 2: Switching element in Hodgkin-Huxley equivalent circuits. Figure 3: Single stage conversion of infrared intensity to spiketrain frequency with a neuron-like semiconductor device. No pre-amplifiers are necessary. Coding of graded input signals (see Fig. 4) such as photocurrents into action potential spike trains with millimeter scale devices has been experimentally demonstrated3 with currents from 1 IlA down to about 1 picoampere with coding noise referred to input of < 10 femtoamperes. Coding of much smaller current levels should be possible with smaller devices. Figure 5 clearly shows the threshold behavior of the IMD. For devices studied to date, a transition from action potential output to graded signal output is observed for input currents of the order of 0.5 picoamperes 1~ --.. o 4 Z 10 o U w (f) 203 CURRENT (AMPERES) Figure 4: Coding of NIR-VISmLE-UV intensity into firing frequency of a spiketrain and the experimentally determined firing rate vs. the input current for one device. Note that the dynamic range is about 107. > '0 '-> o E2 UBI) ---PL 500 fLS/ div Figure 5: mustration of the threshold firing of the device in response to input step functions. This transition is remarkably well described in von Neumann's discussion5,6 of the mixed character of neural elements which he relates to the concept of subliminal stimulation levels which are too low to produce the stereotypical all-or-nothing response. Neural network modelers frequently adopt viewpoints which ignore this interesting mixed character. The von Neumann viewpoint links the mixed character to concepts of nonlinear dynamics in a way which is not apparent in recent neural network modeling literature. The scaling down of IMD size should result in even lower current requirements for all-or-nothing response. DEVICE PHYSICS Recently, neuronlike action potential transients in IMDs have been the subject of considerable research3,4,7,8,9,1O,1l,12,13. In the simple circuits of Fig. 1, the IMD gives rise to a spontaneous neuronlike spiketrain output. Between pulses, the IMD is polarized in the sense that it is in a low conductance state with a substantial voltage occurring across it, even though it is forward biased. The low conductance has been attributed to small interfacial work functions due to band offsets at the n+ -n and p+ -n interfaces8 • Low temperatures inhibit thermionic injection of electrons and holes into the n-region from the n+ -layer and p+ -layer impurity bands14 . Pulses are caused by 204 switching to depolarized states with low diode potential drops and large injection currents which are believed to be triggered by the slow buildup of a small thermionic injection current from the n+ -layer into the n-region. The injection current can cause impact ionization of n-region donor impurities resulting in an increasingly positive space charge which further enhances the injection current to the point where the IMD abruptly switches to the low conductance state with large injection current. Switching times are typically under lOOns. Charging of the load capacitance CL cuts off the large injection current and resets the diode to its low conductance state. The load capacitor CL then discharges through RL. During the CL discharging time constant RLCL the voltage across the IMD itself is low and therefore the bias voltage would have to be raised substantially to cause further firing. Thus, RLCL is analogous to the refractory period of a neuron. The output pulses of an IMD generally have about the same amplitude while the rate of pulsing varies over a wide range depending on the bias voltage and the presence of electromagnetic radiation.7,8,10 ~ DETECTOR ARRAY ¢=::I TRANSIENT SENSING ¢=::I MOTION SENSING TRACKING 2-D PARALLEL OUTPUT LAMINAR NEURAL NETWORK Figure 6: lllustrative laminar architecture showing stacked wafers in 3-dimensions. REAL TIME PARALLEL ASYNCHRONOUS PROCESSING The devices described here could form the hardware basis for a parallel asynchronous processor in much the same way that transistors form the basis for digital computers. The devices could be used to construct networks which could perform real time signal processing. Pulse propagation through silicon chips (parallel firethrough, see Fig. 7) as opposed to the lateral planar propagation in conventional integrated circuits has been proposed.1S This would permit the use of laminar, stacked wafer architectures. See Fig. 6. Such architectures would eliminate the serial processing limitations of standard processors which utilize multiplexing and charge transfer. There are additional advantages in terms of elimination of pre-amplifiers and reduction in power consumption. The approach would utilize the low power, low noise deviceslO described here to perform input signal-to-frequency conversion in every processing channel. POWER CONSUMPTION FOR A BRAIN SCALE SYSTEM The low power and low current requirements together with the electronic simplicity (lower parts-count as compared with transistor and op-amp approaches) and INPUTS 111111111111 ;1"*"*"* *'"*"* '*"* '* '*"* "*1 Siwafer ;1* * * * '* *"*"*"* *' * *1 Siwafer ;1* * * * * * * * ** * *1 Siwaf.r ;1*"*"*"* * ** *"* '*"* "*1 Siwaf.r ;1* * *"* * * *"*"* '*"* "*1 Siwaf.r ;IIIIIIIIIIII ! ! ! ! ! ! ! ! ! ! ! OUTPUTS I 1 Si wafer ! 205 Figure 7: Schematic illustration of the signal flow pattern through a real time parallel asynchronous processor consisting of stacked silicon wafers. the natural emulation of neuron features means that the approach described here would be especially advantageous for very large neural networks, e.g. systems comparable to supercomputers in which power dissipation and system complexity are important considerations. The power consumption of large scale analog16 and digital17 systems is always a major concern. For example, the power consumption of the CRAY XMP-48 is of the order of 300 kilowatts. For the devices described here, the power consumption is very low. For these devices, we have observed quiescent power drains of about 1 n W /cm2 and pulse power consumption of about 500 nJ/pulse/cm2 • We estimate that a system with 1011 active 10~m x 10~m elements (comparable to the number of neurons in the brain18) all firing with an average pulse rate of 1 KHz (corresponding to a high neuronal firing rateS) would consume about 50 watts. The quiescent power drain for this system would be 0.1 milliwatts. Thus, power (P) requirements for such an artificial neural network with the size scale (1011 pulse generating elements) of the human brain and a range of activity between zero and the maximum conceivable sustained activity for neurons in the brain would be 0.1 milliwatts < P < 50 watts for 10 micron technology. For comparison, we note that von Neumann's estimate for the power dissipation of the brain is of order 10 to 25 watts. S,6 Fabrication of a 1011 element 10 ~m artificial neural network would require processing of about 1500 four inch wafers. NETWORK CONNECTIVITY For a network with coupling between many IMD's3 we have shown" that (1) where Vj is the voltage across the diode and the input capacitance Cj of the i-th network node, Rj represents a leakage resistance in parallel with Cil and Ij represents an external current input to the i-th diode. iJ=1,2,3, ..... label different network nodes and Tij incoporates coupling between network elements. Equation 1 has the same form as equations which occur in the Hopfield modeI2o,21,22,23 for neural networks. Sejnowski has also discussed similar equations in connection with skeleton filters in 206 INPUTS c); o---j R o---j~~~~~'fi-~ o---j R ~: TRANSMISSION LINE OUTPUTS ;~ t----o ~f---r----t---t t----o t----o :P---o Figure 8: a) Main features of a typical neuron from Kandel and Schwartz.19 b) Our artificial neuron) which shows the summation over synaptic inputs and fan-out. the brain.24•25 Nonlinear threshold behavior of IMD)s enters through F(V) as it does in the neural network models. In Fig. 8-b a range of input capacitances is possible. This range of capacitances is related to the range of possible synaptic weights. The circuit in Fig. 8 accomplishes pulse height discrimination and each pulse can contribute to the charge stored on the central node capacitance C. The charge added to C during each input pulse is linearly related to the input capacitance except at extreme limits. The range of input capacitances for a particular experiment was .002 J-lF to .2 J-lF which differ by a factor of about 100. The effect of various input capacitance values (synaptic weights) on input-output firing rates is shown in Fig. 9. Also the Fig. 8-b shows many capacitive inputs/outputs to/from a single IMD. i.e. fan-in and fan-out. For pulses which arrive at different inputs at about the same time) the effect of the pulses is additive. The time within which inputs are summed is just the stored charge lifetime. Summation over many inputs is an important feature of neural information processing. EXCITATION) INHIBITION) MEMORY Both excitatory and inhibitory input circuits are shown in Fig. 10. Input pulses cause the accumulation of charge on C in excitatory circuits and the depletion of charge on C in inhibitory circuits. Charge associated with input spiketrains is integrated/stored on C. The temporally integrated charge is depleted by the firing of the IMD. Thus) the storage time is related to the firing rate. After an input spiketrain raises the potential across C to a value above the firing threshold) the resulting IMD 5 ,--.,. N I 4 .;,< W t~ 3 w Vl ---1 ::J (L 2 t::J CL t-6 1 (0) (b) O.2}J F O.03}JF 20 40 60 80 100 207 Figure 9: Output pulse rate vs. the input pulse rate for different input capacitance values Ci values INPUT PULSE RATE (Hz) R INP~ I----'-~-r_{)l__--,----<> OUTPUT R INP~ c· L R' R' L Figure 10: Circuits which incorporate rectifying synaptic inputs. a) an excitatory input. b) an inhibitory input. output spiketrain codes the input information. The output firing rate is linearly related to the input firing rate times the synaptic coupling strength (linearly related to Ci). See Fig. 9. If the input ceases, then the potential across C relaxes back to a value just below the firing threshold. When not firing, the IMD has a high impedance. If there is negligible leakage of charge from C, then V can remain near V T (threshold voltage) for a long time and a new input signal will quickly take the IMD over the firing threshold. See Fig. 11. We have observed stored charge lifetimes of 56 days and longer times may be acheivable. The lifetime of charge stored on C can be reduced by adding a resistance in parallel with C. From the discussion of integration, we see that long term storage of charge on C is equivalent to long term memory. The memory can be read by seeing if a new input pulse or spiketrain produces a prompt output pulse or spiketrain. The read signal input channel in Fig. 8-b can be the same as or different from the channel which resulted in the charge storage. In either case memory would produce a change in the pattern of connectivity if the circuit was imbedded in a neural network. Changes in patterns of connectivity are similar to Hebb's ruie considerations26 in which memory is associated with increases in the strength (weight) of synaptic couplings. Frequently, 208 13 QJ o a:: 11 Input Potential Figure 11: Firing rate vs. the bias voltage. The region where the firing is negligible is associated with memory. The state of the memory is associated with the proximity to the firing threshold. the increase in synaptic weights is modeled by increased conductance whereas in the circuits in Figs. lO(a) and 8-b memory is achieved by integration and charge storage. Note that for these particular circuits, the memory is not eraseable although volatile (short term) memory can easily be constructed by adding a resistor in parallel with C. Thus, a continuous range of memory lifetimes can be achieved. 2-D PARALLEL ASYNCHRONOUS CHIP-TO-CHIP TRANSMISSION For many IMD's the output pulse heights for a circuit like that in Fig. 1 are >3 volts. Thus, output from the first stage or any later stage of the network could easily be transmitted to other parts of an overall system. Two-dimensional arrays of devices on different chips could be coupled by indium bump bonding to form the laminar architecture described above. Planar technology could be used for local lateral interconnections in the processor. (See Fig. 7) In addition to transmission of electrical pulses, optical transmission is possible because the pulses can directly drive LED's. Emerging GaAs-on-Si technology is interesting as a means of fabricating two dimensional emitter arrays. Optical transmission is not necessary but it might be useful (A) for processed image data transfer, (B) for coupling to an optical processor, or (C) to provide 2-0 optical interconnects between chips bearing 2-D arrays of p+ - n - n+ diodes. Note that with optical interconnects between chips, the circuits employed here would be internal receivers. The p-i-n diodes employed in the present work would be well suited to the receiver role. An interesting possibility would entail the use optical interconnects between chips to achieve local, lateral interaction. This would be accomplished by having each optical emitter in a 2-D array broadcast locally to multiple receivers rather than to a single receiver. Similarly, each receiver would have a reeeptive field extending over multiple transmitters. It is also possible that an optical element could be placed in the gap between parallel transmitter and receiver planes to structure, control or alter 2-D patterns of interconnection. This would be an alternative to a planar technology approach to lateral interconnection. IT the optical elements were active then the system would constitute a hybrid optical/electronic processor, whereas if passive optical elements were employed, we would regard the system as an optoelectronic processor. In either case, we picture the processing functions of temporal integration, spatial summation over inputs, coding and pulse generation as residing on-chip. 209 ACKNOWLEDGEMENTS The work was supported in part by U.S. DOE under contract #DE-AC0280ER10667 and NSF under grant # ECS-8603075. References [1] L. D. Harmon, Kybernetik 1,89 (1961). [2] A. L. Hodgkin and A. F. Huxley, J. Physioll17, 500 (1952). [3] D. D. Coon and A. G. U. Perera, Int. J. Electronics 63, 61 (1987). [4] K. M. S. V. Bandara, D. D. Coon and R. P. G. Karunasiri, Infrared 'lransient Sensing, to be published. [5] J. von Neumann, The Computer and the Brain, Yale University Press, New Haven and London, 1958. [6] J. von Neumann, Collected Works, Pergamon Press, New York, 1961. [7] D. D. Coon and A. G. U. Perera, Int. J. Infrared and Millimeter Waves 7, 1571 (1986). [8] D. D. Coon and S. D. Gunapala, J. Appl. Phys 57, 5525 (1985). [9] D. D. Coon, S. N. Ma and A. G. U. Perera, Phys. Rev. Let. 58, 1139 (1987). [10] D. D. Coon and A. G. U. Perera, Applied Physics Letters 51, 1711 (1987). [11] D. D. Coon and A. G. U. Perera, Solid-State Electronics 29, 929 (1986). [12] D. D. Coon and A. G. U. Perera, Applied Physics Letters 51, 1086 (1987). [13] K. M. S. V. Bandara, D.D. Coon and R. P. G. Karunasiri, Appl. Phys. Lett 51, 961 (1987). [14] Y. N. Yang, D. D. Coon and P. F. Shepard, Applied Physics Letters 45, 752 (1984). [15] D. D. Coon and A. G. U. Perera, Int. J. IR and Millimeter Waves 8, 1037 (1987). [16] M. A. Sivilotti, M. R. Emerling and C. A. Mead, VLSI Arcbitectures for Implementation of Neural Networks, Neural Networks for Computing, A.J.P., 1986, pp. 408-413. [17] R. W. Keyes, Proc. IEEE 63, 740 (1975). [18] E. R. Kandel and J. H. Schwartz, Principles of Neural Science, Elsevier, New York, 1985. 210 [19] E. R. Kandel and J. H. Schwartz, Principles of Neural Science, Elsevier, New York, 1985, page 15, Reproduced by permission of Elsevier Science Publishing Co., N.Y .. [20] J. J. Hopfield, Proc. Nat!. Acad. Sci. U.S.A 81, 3088 (1984). [21] J. J. Hopfield and D. W. Tank, BioI. Cybern 52, 141 (1985). [22] J. J. Hopfield and D. W. Tank, Science 233,625 (1986). [23] D. W. Tank and J. J. Hopfield, IEEE. Circuits Syst. CAS-33, 533 (1986). [24] T. J. Sejnowski, J. Math. Biology 4, 303 (1977). [25] T. J. Sejnowski, Skeleton Filters in tbe Brain, Lawrence Erlbaum, New Jersey, 1981, pp. 189-212, edited by G. E. Hinton and J. A. Anderson. [26] J. L. McClelland, D. E. Rumelhart and the PDP research group, Parallel Distributed Processing, The MIT Press, Cambridge, Massachusetts, 1986, two volumes.
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662 AN ADAPTIVE AND HETERODYNE FILTERING PROCEDURE FOR THE IMAGING OF MOVING OBJECTS F. H. Schuling, H. A. K. Mastebroek and W. H. Zaagman Biophysics Department, Laboratory for General Physics Westersingel 34, 9718 eM Groningen, The Netherlands ABSTRACT Recent experimental work on the stimulus velocity dependent time resolving power of the neural units, situated in the highest order optic ganglion of the blowfly, revealed the at first sight amazing phenomenon that at this high level of the fly visual system, the time constants of these units which are involved in the processing of neural activity evoked by moving objects, are -roughly spokeninverse proportional to the velocity of those objects over an extremely wide range. In this paper we will discuss the implementation of a two dimensional heterodyne adaptive filter construction into a computer simulation model. The features of this simulation model include the ability to account for the experimentally observed stimulus-tuned adaptive temporal behaviour of time constants in the fly visual system. The simulation results obtained, clearly show that the application of such an adaptive processing procedure delivers an improved imaging technique of moving patterns in the high velocity range. A FEW REMARKS ON THE FLY VISUAL SYSTEM The visual system of the diptera, including the blowfly Calliphora erythrocephala (Mg.) is very regularly organized and allows therefore very precise optical stimulation techniques. Also, long term electrophysiological recordings can be made relatively easy in this visual system, For these reasons the blowfly (which is well-known as a very rapid and 'clever' pilot) turns out to be an extremely suitable animal for a systematic study of basic principles that may underlie the detection and further processing of movement information at the neural level. In the fly visual system the input retinal mosaic structure is precisely mapped onto the higher order optic ganglia (lamina, medulla, lobula), This means that each neural column in each ganglion in this visual system corresponds to a certain optical axis in the visual field of the compound eye. In the lobula complex a set of wide-field movement sensitive neurons is found, each of which integrates the input signals over the whole visual field of the entire eye, One of these wide field neurons, that has been classified as H I by Hausen J has been extensively studied both anatomically2, 3, 4 as well as electrophysiologically5, 6, 7, The obtained results generally agree very well with those found in behavioral optomotor experiments on movement detection8 and can be understood in terms of Reichardts correlation model9, 10. The H I neuron is sensitive to horizontal movement and directionally selective: very high rates of action potentials (spikes) up to 300 per second can be recorded from this element in the case of visual stimuli which move horizontally inward, i.e. from back to front in the visual field (pre/erred direction), whereas movement horizontally outward, i.e, from front to back (null direction) suppresses its activity, © American Institute of Physics 1988 663 EXPERIMENTAL RESULTS AS A MODELLING BASE When the H I neuron is stimulated in its preferred direction with a step wise pattern displacement, it will respond with an increase of neural activity. By repeating this stimulus step over and over one can obtain the averaged response: after a 20 ms latency period the response manifests itself as a sharp increase in average firing rate followed by a much slower decay to the spontaneous activity level. Two examples of such averaged responses are shown in the Post Stimulus Time Histograms (PSTH's) of figure 1. Time to peak and peak height are related and depend on modulation depth, stimulus step size and spatial extent of the stimulus. The tail of the responses can be described adequately by an exponential decay toward a constant spontaneous firing rate: R(t)=c+a -e( -t/1) (1) For each setting of the stimulus parameters, the response parameters, defined by equation (1), can be estimated by a least-squares fit to the tail of the PSTH. The smooth lines in figure 1 are the results of two such fits. OJ ~ I'JO 0\ ~ tf 100 w= 11'/s 1.00 600 800 tlmsl 300 !~, '00 )0 10 o o " " . • M:O.'O o MoO IO " Mdl05 o " 8 __ .L..._--'-_ _ ---', __ L-' _-----L, _ _ -'--_ _ 0.3 I 10)0 100 )00 W ("lsI Fig.l A veraged responses (PSTH's) obtained from the H I neuron, being adapted to smooth stimulus motion with velocities 0.36 0 /s (top) and 11 0 /s (bottom) respectively. The smooth lines represent least-squares fits to the PSTH's of the form R(t)=c+a-e(-t/1). Values of f for the two PSTH's are 331 and 24 ms respectively (de Ruyter van Steveninck et al.7). Fig.2 Fitted values of f as a function of adaptation velocity for three modulation depths M. The straight line is a least-squares fit to represent the data for M=0.40 in the region w:0.3-100 o/s. It has the form f=Q - w-13 with Q=150 ms and 13=0.7 (de Ruyter van Steveninck et al.7). 664 Figure 2 shows fitted values of the response time constant T as a function of the angular velocity of a moving stimulus (a square wave grating in most experiments) which was presented to the animal during a period long enough to let its visual system adapt to this moving pattern and before the step wise pattern displacement (which reveals 1') was given. The straight line, described by (2) (with W in 0 Is and T in ms) represents a least-squares fit to the data over the velocity range from 0.36 to 125 0 Is. For this range, T varies from 320 to roughly 10 ms, with a=150±1O ms and ~=0.7±0.05. Defining the adaptation range of 1 as that interval of velocities for which 1 decreases with increasing velocity, we may conclude from figure 2 that within the adaptation range, 1 is not very sensitive to the modulation depth. The outcome of similar experiments with a constant modulation depth of the pattern (M=0.40) and a constant pattern velocity but with four different values of the contrast frequency fc (Le. the number of spatial periods per second that traverse an individual visual axis as determined by the spatial wavelength As of the pattern and the pattern velocity v according to fc=v lAs) reveal also an almost complete independency of the behaviour of 1 on contrast frequency. Other experiments in which the stimulus field was subdivided into regions with different adaptation velocities, made clear that the time constants of the input channels of the H I neuron were set locally by the values of the stimulus velocity in each stimulus sub-region. Finally, it was found that the adaptation of 1 is driven by the stimulus velocity, independent of its direction. These findings can be summarized qualitatively as follows: in steady state, the response time constants 1 of the neural units at the highest level in the fly visual system are found to be tuned locally within a large velocity range exclusively by the magnitude of the velocity of the moving pattern and not by its direction, despite the directional selectivity of the neuron itself. We will not go into the question of how this amazing adaptive mechanism may be hard-wired in the fly visual system. Instead we will make advantage of the results derived thus far and attempt to fit the experimental observations into an image processing approach. A large number of theories and several distinct classes of algorithms to encode velocity and direction of movement in visual systems have been suggested by, for example, Marr and Ullman I I and van Santen and Sperling12. We hypothesize that the adaptive mechanism for the setting of the time constants leads to an optimization for the overall performance of the visual system by realizing a velocity independent representation of the moving object. In other words: within the range of velocities for which the time constants are found to be tuned by the velocity, the representation of that stimulus at a certain level within the visual circuitry, should remain independent of any variation in stimulus velocity. OBJECT MOTION DEGRADATION: MODELLING Given the physical description of motion and a linear space invariant model, the motion degradation process can be represented by the following convolution integral: co co g(x,y)= J J (h(x - u,y-v) • flu, v» dudv -00 -00 (3) 665 where f(u,v) is the object intensity at position (u,v) in the object coordinate frame, h(x-u,y-v) is the Point Spread Function (PSF) of the imaging system, which is the response at (x,y) to a unit pulse at (u,v) and g(x,y) is the image intensity at the spatial position (x,y) as blurred by the imaging system. Any possible additive white noise degradation of the already motion blurred image is neglected in the present considerations. For a review of principles and techniques in the field of digital image degradation and restoration, the reader is referred to Harris 13, Sawchuk 14, Sondhi 15, Nahi 16, A boutalib et al. 1 7, 18, Hildebrand 19, Rajala de Figueiredo20 . It has been demonstrated first by Aboutalib et al.17 that for situations in which the motion blur occurs in a straight line along one spatial coordinate, say along the horizontal axis, it is correct to look at the blurred image as a collection of degraded line scans through the entire image. The dependence on the vertical coordinate may then be dropped and eq. (3) reduces to: g~~ J ~x-u) -f(u)du (4) Given the mathematical description of the relative movement, the corresponding PSF can be derived exactly and equation (4) becomes: g(x)= k b(x - u) - f(u)du (5) where R is the extent of the motion blur. Typically, a discrete version of (5), applicable for digital image processing purposes, is described by: L g(k)=l: h(k-I)· f(l) I ; k=I, ... ,N where k and I take on integer values and L is related to the motion blur extent. (6) According to Aboutalib et al. 18 a scalar difference equation model (M,a,b,c) can then be derived to model the motion degradation process: x(k+l) = M· x(k)+a· f(k) g(k) = b· x(k)+c • f(k) ; k=I, ... ,N (7) h(i) = cof1(i)+Cl~(i-l)+ ...... +cmA(i-m) where x(k) is the m-dimensional state vector at position k along a scan line, f(k) is the input intensity at position k, g(k) is the output intensity, m is the blur extent, N is the number of elements in a line, c is a scalar, M, a and b are constant matrices of order (mxm), (mxl) and (lxm) respectively, containing the discrete values Cj of the blurring PSF h(j) for j=O, ... ,m and 1::.(.) is the Kronecker delta function. 666 INFLUENCE OF BOTH TIME CONSTANT AND VELOCITY ON THE AMOUNT OF MOTION BLUR IN AN ARTIFICIAL RECEPTOR ARRAY To start with, we incorporate in our simulation model a PSF, derived from equation (1), to model the performance of all neural columnar arranged filters in the lobula complex, with the restriction that the time constants f remain fixed throughout the whole range of stimulus velocities. Realization of this PSF can easily be achieved via the just mentioned state space model. 1 300 250 200 150 :3 100 . <{ 50 w 0 0 :::> ..... ::i 250 a.. ~ <{ 200 150 100 50 \ \ \ . . \ 7 \ , \ , " , , , , I. I. \ . \ . \ . \ \ \ \ \ \ \ \ \ \ \ " " , " , "" O~--~~----~~~~--~~--~ o 5 10 15 20 POSITION IN ARTIFICIAL RECEPTOR ARRAY • Fig.3 upper part. Demonstration of the effect that an increase in magnitude of the time constants of an one-dimensional array of filters will result in increase in motion blur (while the pattern velocity remains constant). Original pattern shown in solid lines is a square-wave grating with a spatial wavelength equal to 8 artificial receptor distances. The three other wave forms drawn, show that for a gradual increase increase in magnitude of the time constants, the representation of the original square-wave will consequently degrade. lower part. A gradual increase in velocity of the moving square-wave (while the filter time constants are kept fixed) results also in a clear increase of degradation. 667 First we demonstrate the effect that an increase in time constant (while the pattern velocity remains the same) will result in an increase in blur. Therefore we introduce an one dimensional array of filters all being equipped with the same time constant in their impulse response. The original pattern shown in square and solid lines in the upper part of figure 3 consists of a square wave grating with a spatial period overlapping 8 artificial receptive filters. The 3 other patterns drawn there show that for the same constant velocity of the moving grating, an increase in the magnitude of the time constants of the filters results in an increased blur in the representation of that grating. On the other hand, an increase in velocity (while the time constants of the artificial receptive units remain the same) also results in a clear increase in motion blur, as demonstrated in the lower part of figure 3. Inspection of the two wave forms drawn by means of the dashed lines in both upper and lower half of the figure, yields the conclusion, that (apart from rounding errors introduced by the rather small number of artificial filters available), equal amounts of smear will be produced when the product of time constant and pattern velocity is equal. For the upper dashed wave form the velocity was four times smaller but the time constant four times larger than for its equivalent in the lower part of the figure. ADAPTIVE SCHEME In designing a proper image processing procedure our next step is to incorporate the experimentally observed flexibility property of the time constants in the imaging elements of our device. In figure 4a a scheme is shown, which filters the information with fixed time constants, not influenced by the pattern velocity. In figure 4b a network is shown where the time constants also remain fixed no matter what pattern movement is presented, but now at the next level of information processing, a spatially differential network is incorporated in order to enhance blurred contrasts. In the filtering network in figure 4c, first a measurement of the magnitude of the velocity of the moving objects is done by thus far hypothetically introduced movement processing algorithms, modelled here as a set of receptive elements sampling the environment in such a manner that proper estimation of local pattern velocities can be done. Then the time constants of the artificial receptive elements will be tuned according to the estimated velocities and finally the same differential network as in scheme 4b, is used. The actual tuning mechanism used for our simulations is outlined in figure 5: once given the range of velocities for which the model is supposed to be operational, and given a lower limit for the time constant 'f min ('f min can be the smallest value which physically can be realized), the time constant will be tuned to a new value according to the experimentally observed reciprocal relationship, and will, for all velocities within the adaptive range, be larger than the fixed minimum value. As demonstrated in the previous section the corresponding blur in the representation of the moving stimulus will thus always be larger than for the situation in which the filtering is done with fixed and smallest time constants .,. min. More important however is the fact that due to this tuning mechanism the blur will be constant since the product of velocity and time constant is kept constant. So, once the information has been processed by such a system, a velocity independent representation of the image will be the result, which can serve as the input for the spatially differentiating network as outlined in figure 4c. The most elementary form for this differential filtering procedure is the one 668 in which the gradient of two filters K-I and K+l which are the nearest neighbors of filter K, is taken and then added with a constant weighing factor to the central output K as drawn in figure 4b and 4c, where the sign of the gradient depends on the direction of the estimated movement. Essential for our model is that we claim that this weighing factor should be constant throughout the whole set of filters and for the whole high velocity range in which the heterodyne imaging has to be performed. Important to notice is the existence of a so-called settling time, i.e. the minimal time needed for our movement processing device to be able to accurately measure the object velocity. [Note: this time can be set equal to zero in the case that the relative stimulus velocity is known a priori, as demonstrated in figure 3]. Since, without doubt, within this settling period estimated velocity values will come out erroneously and thus no optimal performance of our imaging device can be expected, in all further examples, results after this initial settling procedure will be shown. 2 3 , 5 Fig. 4 A B ; ~ v yV 9' / y ~' 7 .' r~ ;/Y i §. Y C r39 rYO [~l [~l i~J 't"if~'n Pattern movement in this figure is to the right. A: Network consisting of a set of filters with a fixed, pattern velocity independent, time constant in their impulse response. B: Identical network as in figure 4A now followed by a spatially differentiating circuitry which adds the weighed gradients of two neighboring filter outputs K-l and K+I to the central filter output K. C: The time constants of the filtering network are tuned by a hypothetical movement estimating mechanism, visualized here as a number of receptive elements, of which the combined output tunes the filters. A detailed description of this mechanism is shown in figure 5. This tuned network is followed by an identical spatially differentiating circuit as described in figure 4B. Fig. 5 Fig.6 669 increasing velocity • v (<¥s) 1: 1: min ----_ .. decreasing time constant Detailed description of the mechanism used to tune the time constants. The time constant f of a specific neural channel is set by the pattern velocity according to the relationship shown in the insert, which is derived from eq. (2) with cx=- I and 13= I. 6 4 2 =f < 0 w o ::;) -~ a. :J: < 2 o r i', " " r 'I , , , \ , , , , ~ , , ~ - /,----Wi I, " ' \ r;-" ::-:v h 2V 4V , ~be-.--1 I ,,,/' =-.:! I I -~ I , , I , , , , I ... I " I I " : ' \ " V 8V 12 V 16 V I 1: J \ J/---. r------l...- -'1;"- -- --POSITION IN ARTIFICIAL RECEPTOR ARRAY Thick lines: square-wave stimulus pattern with a spatial wavelength overlapping 32 artificial receptive elements. Thick lines: responses for 6 different pattern velocities in a system consisting of paralleling neural filters equipped with time constants, tuned by this velocity, and followed by a spatially differentiating network as described. Dashed lines: responses to the 6 different pattern velocities in a filtering system with fixed time constants, followed by the same spatial differentiating circuitry as before. Note the sharp over- and under shoots for this case. 670 Results obtained with an imaging procedure as drawn in figure 4b and 4c are shown in figure 6. The pattern consists of a square wave, overlapping 32 picture elements. The pattern moves (to the left) with 6 different velocities v, 2v, 4v, 8v, 12v, 16v. At each velocity only one wavelength is shown. Thick lines: square wave pattern. Dashed lines: the outputs of an imaging device as depicted in figure 4b: constant time constants and a constant weighing factor in the spatial processing stage. Note the large differences between the several outputs. Thin continuous lines: the outputs of an imaging device as drawn in figure 4c: tuned time constants according to the reciprocal relationship between pattern velocity and time constant and a constant weighing factor in the spatial processing stage. For further simulation details the reader is referred to Zaagman et al.21 . Now the outputs are almost completely the same and in good agreement with the original stimulus throughout the whole velocity range. Figure 7 shows the effect of the gradient weighing factor on the overall filter performance, estimated as the improvement of the deblurred images as compared with the blurred image, measured in dB. This quantitative measure has been determined for the case of a moving square wave pattern with motion blur Fig. 7 7.-------~------~r-------_r------~ 6 5 IX) "0 4 0) u C ItI 3 E c-o ~ 2 a. cO) ~ 1 ;z: 0 -1 0 1 2 3 4 weighing factor • Effect of the weighing factor on the overall filter performance. Curve measured for the case of a moving square-wave grating. Filter performance is estimated as the improvement in signal to noise ratio: ( I:iI:j«V(i,j)-U(i,j»2) 1=10· 1010g I:iI: j« O(i,j)- u(i,j» 2 where u(i,j) is the original intensity at position (i,j) in the image, v(i,j) is the intensity at the same position (i,j) in the motion blurred image and O(i,j) is the intensity at (i,j) in the image, generated with the adaptive tuning procedure. 671 extents comparable to those used for the simulations to be discussed in section IV. From this curve it is apparent that for this situation there is an optimum value for this weighing factor. Keeping the weight close to this optimum value will result in a constant output of our adaptive scheme, thus enabling an optimal deblurring of the smeared image of the moving object. On the other hand, starting from the point of view that the time constants should remain fixed throughout the filtering process, we should had have to tune the gradient weights to the velocity in order to produce a constant output as demonstrated in figure 6 where the dashed lines show strongly differing outputs of a fixed time constant system with spatial processing with constant weight (figure 4b). In other words, tuning of the time constants as proposed in this section results in: I) the realization of the blur-constancy criterion as formulated previously, and 2) -as a consequence- the possibility to deblur the obtained image oPtimally with one and the same weighing factor of the gradient in the final spatial processing layer over the whole heterodyne velocity range. COMPUTER SIMULATION RESULTS AND CONCLUSIONS The image quality improvement algorithm developed in the present contribution has been implemented on a general purpose DG Eclipse Sjl40 minicomputer for our two dimensional simulations. Figure Sa shows an undisturbed image, consisting of 256 lines of each 256 pixels, with S bit intensity resolution. Figure Sb shows what happens with the original image if the PSF is modelled according to the exponential decay (2). In this case the time constants of all spatial information processing channels have been kept fixed. Again, information content in the higher spatial frequencies has been reduced largely. The implementation of the heterodyne filtering procedure was now done as follows: first the adaptation range was defined by setting the range of velocities. This means that our adaptive heterodyne algorithm is supposed to operate adequately only within the thus defined velocity range and that -in that range- the time constants are tuned according to relationship (2) and will always come out larger than the minimum value 1 min. For demonstration purposes we set Q=I and /3=1 in eq. (2), thus introducing the phenomenon that for any velocity, the two dimensional set of spatial filters with time constants tuned by that velocity, will always produce a constant output, independent of this velocity which introduces the motion blur. Figure Sc shows this representation. It is important to note here that this constant output has far more worse quality than any set of filters with smallest and fixed time constants 1 min would produce for velocities within the operational range. The advantage of a velocity independent output at this level in our simulation model, is that in the next stage a differential scheme can be implemented as discussed in detail in the preceding paragraph. Constancy of the weighing factor which is used in this differential processing scheme is guaranteed by the velocity independency of the obtained image representation. Figure Sd shows the result of the differential operation with an optimized gradient weighing factor. This weighing factor has been optimized based on an almost identical performance curve as described previously in figure 7. A clear and good restoration is apparent from this figure, though close inspection reveals fine structure (especially for areas with high intensities) which is unrelated with the original intensity distribution. These artifacts are caused by the phenomenon that for these high intensity areas possible tuning errors will show up much more pronounced than for low intensities. 672 Fig.8a Fig.8b Fig. 8c Fig.8d a ( b d Original 256x256x8 bit picture. Motion degraded image with a PSF derived from R(t)=c+a -e( -t/r). where T is kept fixed to 12 pixels and the motion blur extent is 32 pixels. Worst case, i.e. the result of motion degradation of the original image with a PSF as in figure 8b, but with tuning of the time constants based on the velocity. Restored version of the degraded image using the heterodyne adaptive processing scheme. In conclusion: a heterodyne adaptive image processing technique, inspired by the fly visual system, has been presented as an imaging device for moving objects. A scalar difference equation model has been used to represent the motion blur degradation process. Based on the experimental results described and on this state space model, we developed an adaptive filtering scheme. which produces at a certain level within the system a constant output, permitting further differential operations in order to produce an optimally deblurred representation of the moving object. ACKNOWLEDGEMENTS The authors wish to thank mT. Eric Bosman for his expert programming 673 assistance, mr. Franco Tommasi for many inspiring discussions and advises during the implementation of the simulation model and dr. Rob de Ruyter van Steveninck for experimental help. This research was partly supported by the Netherlands Organization lor the Advancement 01 Pure Research (Z.W.O.) through the foundation Stichting voor Biolysica. REFERENCES I. K. Hausen, Z. Naturforschung 31c, 629-633 (1976). 2. N. J. Strausfeld, Atlas of an insect brain (Springer Verlag, Berlin, Heidelberg, New York, 1976). 3. K. Hausen, BioI. Cybern. 45, 143-156 (1982). 4. R. Hengstenberg, J. Compo Physiol. 149, 179-193 (1982). 5. W. H. Zaagman, H. A. K. Mastebroek, J. W. Kuiper, BioI. Cybern. 31, 163-168 ( 1978). 6. H. A. K. Mastebroek, W. H. Zaagman, B. P. M. Lenting, Vision Res. 20, 467474 (1980) 7. R. R. de Ruyter van Steveninck, W. H. Zaagman, H. A. K. Mastebroek, BioI. Cybern., 54, 223-236 (1986). 8. W. Reichardt, T. Poggio, Q. Rev. Biophys. 9, 311-377 (1976). 9. W. Reichardt, in Reichardt, W. (Ed.) Processing of optical Data by Organisms and Machines (Academic Press, New York, 1969), pp. 465-493. 10. T. Poggio, W. Reichardt, Q. Rev. Bioph. 9, 377-439 (1976). 11. D. Marr, S. Ullman, Proc. R. Soc. Lond. 211, 151-180 (1981). 12. J. P. van Santen, G. Sperling, J. Opt. Soc. Am. A I, 451-473 (1984). 13. J. L. Harris SR., J. Opt. Soc. Am. 56, 569-574 (1966). 14. A. A. Sawchuk, Proc. IEEE, Vol. 60, No.7, 854-861 (1972). 15. M. M.Sondhi, Proc. IEEE, Vol. 60, No.7, 842-853 (1972). 16. N. E. Nahi, Proc. IEEE, Vol. 60, No.7, 872-877 (1972). 17. A. O. Aboutalib, L. M. Silverman, IEEE Trans. On Circuits And Systems TCAS 75, 278-286 (1975). 18. A. O. Aboutalib, M. S. Murphy, L.M. Silverman, IEEE Trans. Automat. Contr. AC 22, 294-302 (1977). 19. Th. Hildebrand, BioI. Cybern. 36, 229-234 (1980). 20. S. A. Rajala, R. J. P. de Figueiredo, IEEE Trans. On Acoustics, Speech and Signal Processing, Vol. ASSSP-29, No.5, 1033-1042 (1981). 21. W. H. Zaagman, H. A. K. Mastebroek, R. R. de Ruyter van Steveninck, IEEE Trans, Syst. Man Cybern. SMC 13, 900-906 (1983).
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192 PHASE TRANSITIONS IN NEURAL NETWORKS Joshua Chover University of Wisconsin, Madison, WI 53706 ABSTRACT Various simulat.ions of cort.ical subnetworks have evidenced something like phase transitions with respect to key parameters. We demonstrate that. such transi t.ions must. indeed exist. in analogous infinite array models. For related finite array models classical phase transi t.ions (which describe steady-state behavior) may not. exist., but. there can be distinct. quali tative changes in ("metastable") transient behavior as key system parameters pass through crit.ical values . INTRODUCTION Suppose that one st.imulates a neural network - actual or simulated - and in some manner records the subsequent firing activity of cells. Suppose further that. one repeats the experiment. for different. values of some parameter (p) of the system: and that one finds a "cri t.ical value" (p) of the parameter, such that. c (say) for values p > p the act.ivity tends to be much higher than c it. is for values p < p. Then, by analogy with statist.ical c mechanics (where, e.g., p may be temperature, with criUcal values for boiling and freezing) one can say that. the neural network undergoes a "phase transition" at. p. Intracellular phase c transi t.ions, parametrized by membrane potential, are well mown. Here we consider intercellular phase transi t.ions. These have been evidenced in several detailed cort.ical simulations: e.g., of the 1 2 piriform cortex and of the hippocampus In the piriform case, the parameter p represented the frequency of high amplitude spontaneous EPSPs received by a typical pyramidal cell; in the hippocampal case, the parameter was the ratio of inhibitory to excitatory cells in the system. By what. mechanisms could approach to, and retreat. from, a cri t.ical value of some parameter be brought about.? An intriguing conjecture is that. neuromodulators can play such a role in certain 3 networks; temporarily raising or depressing synaptic efficacies What. possible interesting consequences could approach to criticality have for system performance. Good effects could be these: for a network with plasticity, heightened firing response to a stimulus can mean faster changes in synaptic efficacies, which would bring about. faster memory storage. More and longer activi ty could also mean faster access to memory. A bad effect. of © American Institute of Physics 1988 near-criticality - depending on other parameters - can be wild, epileptiform activity. 193 Phase transitions as they might. relate to neural networks have 4 been studied by many authors Here, for clarity, we look at. a particular category of network models - abstracted from the piriform cortex set.ting referred to above - and show the following: a) For "elementary" reasons, phase transition would have to exist if there were infinitely many cells; and the near-subcrit.ical state involves prolonged cellular firing activity in response to an ini t.ial stimulation. b) Such prolonged firing activity takes place for analogous large finite cellular arrays - as evidenced also by computer simulat.ions. What. we shall be examining is space-time patterns which describe the mid-term transient. activity of (Markovian) systems that. tend to silence (with high probability) in the long run. (There is no reference to energy functions, nor to long-run stable firing rates - as such rates would be zero in most. of our cases.) In the following models time will proceed in discrete steps. (In the more complicated set.tings these will be short. in comparison to other time constants, so that. the effect of quant.ization becomes smaller.) The parameter p will be the probability that at. any given t.ime a given cell will experience a certain amount. of exci tatory "spontaneous firing" input.: by itself this amount. will be insufficient. to cause the cell to fire, but. in conjunction wi th sufficiently many exci tatory inputs from other cells it. can assist. in reaching firing threshold. (Other related parameters such as average firing threshold value and average efficacy value give similar results.) In all the models there is a refractory period after a cell fires, during which it cannot fire again; and there may be local (shunt. type) inhibition by a firing cell on near neighbors as well as on itself - but. there is no long-distance inhibi tion. We look first. at. limi ting cases where there are infini tely many cells and - classically - phase transi tion appears in a sharp form. A "SIMPLE" MODEL We consider an infinite linear array of similar cells which obey the following rules, pictured in Fig. lA: (i) If cell k fires at. time n, then it. must. be silent. at. t.ime n+l; (11) if cell k is silent. at. time n but. both of its neighbors k-l and k+l do fire at. time n, then cell k fires at. t.ime n+l; (iii) if cell k is silent at time n and just one of its neighbors (k-l or k+ I) fires at. time n, then ce 11 k wi 11 fire at t.ime n+l with probability p and not. fire with probability l-p, independently of similar decisions at. other cells and at. other times. 194 A TIM~ ~ CELLS"""'> ~ ···0. 0 ~ i 1'\ 000 ~ oollio i DOD Fig. 1. "Simple model". A: firing rules; cells are represented horizontally, time proceeds downwards: filled squares denote firing. B: sample development. Thus, effecUvely, signal propagat ion speed here is one cell per uni t. time, and a cell's firing threshold value is 2 (EPSP units). If we sUmulate ~ cell to fire at time n::O, will its influence necessarily die out or can it. go on forever? (See Fig. lB.) For an answer we note that. in this simple case the firing paUern (if any) at. Ume n must. be an alternat.ing stretch of firing/silent. cells of some length, call it. L. Moreover, n 2 L I = L +2 with probability p (when there are sponteneous n+ n firing assists on both ends of the stretch), or Ln+l = Ln-2 with probability stretch), or 2 (l-p) (when there is no assist at. either. end of the Ln+l = Ln with probability 2p(l-p) (when there is an assist. at. just. one end of the stretch). Start.ing wi th any fini te al ternating stretch La, the successive values L consUtute a "random walk" among the n nonnegat.ive integers. Intui t.ion and simple analysis5 lead to the same conclusion: if the probability for L to decrease «1_p)2) n 2 is greater than that. for it. to increase (p) - i.e. if the average step taken by the random walk is negative then ul t.imately L n will reach a and the firing response dies out. COntrariwise, if 195 2 2 P ) (l-p) then the L can drift. to even higher values wi th n positive probability. In Fig. 2A we sketch the probability for ultimate die-out as a function of p: and in Fig. 2B, the average time until die out. Figs. 2A and B show a classic example of phase transition (p = 1/2) for this infinite array. c A 8 I , \ \ ·1·--- I'~ l<'I) f. Fig. 2. Critical behavior. A: probability of ultimate die out. (or of reaching other traps. in finite array case). B: average time until die-out (or for reaching other traps). Solid curves refer to an infinite array; dashed, to finite arrays. MORE mMPLEX MODELS For an infinite linear array of cells, as sketched in Fig. 3 . we describe now a much more general (and hopefully more realistic) set of rules: (i') A cell cannot fire, nor receive excitatory inputs. at. time n if it has fired at any time during the preceding ~ Hme units (refraction and feedback inhibition). (11 .) Each cell x has a local "inhibitory neighborhood" consisting of a number (j) of cells to its immediate right. and left.. The given cell x cannot. fire or receive excitatory inputs at Hme n if any other cell y in its inhibi tory neighborhood has fired at. any t.ime between t. and t+mI uni ts preceding n, where t. is the t.ime it. would take for a message to travel from y to x at. a speed of VI cells per unit time. (This rule represents local shunt~type inhibition.) (iii') Each cell x has an "excitatory neighborhood" consisting of a number (e) of cells to the immediate right. and left of its inhibitory neighborhood. If a cell y in that. neighborhood fires at a certain time. that firing causes a unit impulse to travel to cell x at a speed of vE cells per uni t. time. The impulse is received at. x subject to rules (i') and (11'). 196 (iv') All cells share a "firing threshold" value 9 and an "integraUon Ume constant." s (s < 9). In addition each cell. at. each t.ime n and independent ly of other times and other cells. can receive a random amount. X of "spontaneous excitatory input.". n The variable Xn can have a general distribution: however. for simplicity we suppose here that. it. assumes only one of two values: b or O. with probabilities p and 1-p respecUvely. (We suppose that. b <. e. so that. the spontaneous "assist." itself is insufficient. for firing.) The above quant.i ties enter into the following firing rule: a cell will fire at. time n if it. is not. prevented by rules (i') and (ii') and if the total number of inputs from other cells. received during the integration "window" last.ing between t.imes n-s+1 and n inclusive. plus the assist. X , n equals or exceeds the threshold 9. (The propagat.ion speeds vI and VE and the neighborhoods are here given left.-right. syrrmetry merely for ease in exposi tion.) o 0 0 0 tl 0 • [J • Jl tl tl [J U' 0 0 0 D • 11 0 0 Jl 0 '" I~h I ~Iit' t '"" I t ~ I • )l,. Fig. 3. Message travel in complex model: see text. rules (i')-(iv'). Wi 11 such a mode 1 d i sp lay phase trans i t i on a t. some cr i t.i cal value of the spontaneous firing frequency p? The dependence of responses upon the ini t.ial condi tions and upon the various parameters is intricate and wi 11 affect. the answer. We briefly discuss here conditions under which the answer is again yes. (1) For a given configuration of parameters and a given ini Ual stimulation (of a stretch of cont.iguous cells) we compare the development. of the model's firing response first. to that. of an auxil iary "more act.ive" system: Suppose that. L now denotes the n distance at. t.ime n between the left:- and right.-most cells which are either firing or in refractory mode. Because no cell can fire wi thout. influence from others and because such influence travels at. a given speed, there is a maximal amount. (D) whereby L 1 can n+ exceed L. n There is also a maximum probability Q(p) - which 197 depends on the spontaneous firing parameter p - that. L 1 ~ L n+ n (whatever n). 'We can compare L with a random walk "A" n n defined so that. An+l = An+D with probability Q(p) and An+l = An-1 with probability l-Q(p). At each transition, An is more likely to increase than L. Hence L is more likely to die n n ou t. than A In the many cases where Q(p) tends to zero as p n does, the average step size of A (viz., DQ(p)+(-I)(I-Q(b») n wi 11 become negat.ive for p below a "cri tical" value p. Thus, a as in the "simple" model above, the probability of ultimate die-out for the A, hence also for the L of the complex model, will be n n 1 when 0 ~ p < p . a (2) There will be a phase transition for the complex model if its probability of die out. - given the same parameters and initial stimulation is in (1) - becomes less than 1 for some p values with p < p < 1. Comparison of the complex process with a simpler a "less act.ive" process is difficul t. in general. However, there are parameter configurat.ions which ul timately can channel all or part. of the firing activity into a (space-t.ime) sublat.t.ice analgous to that. in Fig. 1. Fig. 4 illustrates such a case. For p sufficiently large there is posi tive probabili ty that. the act.ivity will not. die out, just as in the "simple" model. Fig. 4. Activity on a sublattice. (Parameter values: j=2, e=6, MR=2, M1=I, VR=V1=I, 9=3, s=2, and b=I.) Rectangular areas indicate refract.ionlinhibi tion: diagonal lines, excitatory influence. 198 LARGE FINITE ARRAYS Consider now a large finite array of N cells, again as sketched in Fig. 3 ; and operating according to rules similar to (i')-(iv') above, with suitable modifications near the edges. Appropriately encoded, its activity can be described by a (huge) Markov transit.ion matrix, and - depending on the initial st.imulation - must. tend5 to one of a set. of steady-state distribut.ions over firing patterns. For example, (a) if N is odd and the rules are those for Fig. I, then extinct.ion is the unique steady state, for any p (1 (since the L form a random n walk with "reflecUng" upper barrier). But, «(3) if N is even and the cells are arranged in a ring, then, for any P with o < p < 1. both ext.inction and an alternate flip-flop firing pat.tern of period 2 are "traps" for the system - wi th relative long run probabilities determined by the initial state. See the dashed line in Fig. 2A for the extinction probability in the «(3) case, and in Fig. 2B for the expected time until hitting a trap in the 1 (a) case (P(2) and the {(3) case. What quali tat.ive properties related to phase transi tion and critical p values carryover from the infinite to the finite array case? The (a) example above shows that long term activity may now be the same for all 0 ( p (1 but. that parameter intervals can exist. whose key feature is a particularly large expected t.ime before the system hi ts a trap. (Again. the cri tical region can depend upon the ini tial st.imulation.) Prior to being trapped the system spends its time among many states in a kind of "metastable" equilibrium. (We have some preliminary theoretical results on this conditional equilibrium and on its relation to the infinite array case. See also Ref. 6 concerning time scales for which certain corresponding infinite and finite stochastic automata systems display similar behavior . ) Simulat.ion of models satisfying rules (i' )-( iv') does indeed display large changes in length of firing activity corresponding to parameter changes near a critical value. See Fig. 5 for a typical example: As a function of p, the expected time until the system is trapped (for the given parameters) rises approximately linearly in the interval .05<p( .12, wi th most. runs resul ting in extinction - as is the case in Fig. 5A at. time n=115 (for p=.10). But. for p).15 a relatively rigid patterning sets in which leads with high probability to very long runs or to traps other than extinction as is the case in Fig. 5B (p=.20) where the run is arbitrarity truncated at. n=525. (The patterning is highly influenced by the large size of the excitatory neighborhoods.) A _ ........ -...... _ ... . ...• _ .. _ ... _ .. -_'0..... 0' "': ', ' - ' . -,' ., . '0' .. - .: ........ _ .... .. ,.- .-•• _ • ,0 .- .. " - -" ... -.' . .... . ,,- "-" • . : . ,0. ' • • • '. .. . • • .. O' ' 0' ,0 • ---"-' .. ":... . .. . . ... --.. --.' '.' ... . ... _ .... _ ...... . ...... -..... -. . '. .:. .... ". .' ' . " ......... ' . . ' .... 1 TIME ...... .... ' .. . ' .. . ... _ .... --.0 199 B ..... _... ..... : ........ _ .. -...... . .. __ ...... -.. -. . ... ::.: .. : __ ........... ---_0_ o , _. _ ... • • _ . . .. _ • • . : . ..... ..... . . .. '0' • ' . . .. : .... : ... : .---. ,. --...... :. _... ' 0·· ..... ' _• • • ' 0 ' •• _ . . ..... . • .... ,0 '0 • • • .' . . ... '0,. ' 0 ... ' . _ ' _ • ,0 • •• _,' " , _ . • ,'.: •• • • _ .... _ 0° - .... : .... . . , 0 •. •• •••• " --.. -.. ... . ... _. ... . '. "' - ', . • •• ___ •• _ ..... : •• - ....... _ . - 0 • • ..... ... . ,0 , ,-.- ,'.'" ..... -...... _. ,.- ... _.... . .. .0_' ..... .... . -.. • .' ...... . ,0 • __ , ... . _ ,0 .... . . . . _ .... __ ._-.... ... -.. . .. --'- -.. ----, ..... - .. ,. . .._-_ ........... -_. .._-_._ .. :.'. -'. '-----._-.... . -.. , . -----_._-_.. .'--- ----_ ... - ...... __ ._------- ,--"---.-... .. - ...... -. --------.---....•....• '------_.' - ' .. '---------- .-.... _ ..... --_._ .. --.-. --....... --_ . . --------------- '--. ---_.-._---- ".. --- ----------........ ---'.'._-'." .---------' .. .. ---. --------------. --.... _ .. _--.. _-------------.. _-- ... --------'---------....• -.. _-.. _----_._ ..... ... -._ ....... _. ------_. ....... ------'-'- .'-'----------.. _",--. _ .. _----.- ... -' . -.---~ . Fig. 5. Space t.ime firing patterns for one configuration of basic parameters. (There are 200 cells; j=2, e=178, MR=10, M1=9, VR=V1=7, 9=25, s=2, and b=12; 50 are stimulated init.ially.) A: p=.10. B: p=.20. mNa..USION Mechanisms such as neuromodulators, which can (temporarily) bring spontaneous firing levels - or synapt.ic efficacies, or average firing thresholds, or other similar parameters - to near-critical values, can thereby induce large amplification of response act.ivi ty to selected stimul i" The repertoire of such responses is an important. aspect- of the system's function. 200 [Acknowledgement.: Thanks to C. Bezuidenhout. and J. Kane for help wi th simulat.ions.] REFERENCES 1. M. Wilson. J. Bower. J. Chover. L. Haberly. 16th Neurosci. Soc. Mtg. Abstr. 370.11 (1986). 2. R. D. Traub. R. Miles. R.K.S. Wong. 16th Neurosci. Soc. Mtg. Abstr. 196.12 (1986). 3. A. Selverston. this conference. also. Model Neural Networks and Behavior. Plenum (1985); E. Marder. S. Hooper. J. Eisen. SynapUc Function. Wiley (1987) p.305. 4. E.g.: W. Kinzel. Z. Phys. B58, p. 231 (1985); A. Noest .. Phys. Rev. Let .. 57( 1), p. 90 (1986); R. DurreU. (to appear); G. Carpenter, J. Diff. Eqns. 23, p.335 (1977); G. Ermentraut, S. Cohen. BioI. Cyb. 34, p.137 (1979); H. Wilson. S. Cowan. Biophys. J. 12 (1972). 5. W. Feller. An Introd. to Prob. Th·y. and Appl·ns. I. Wiley (1968) Ch. 14. 15. 6. T. Cox and A. Graven (to appear).
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USING NEURAL NETWORKS TO IMPROVE COCHLEAR IMPLANT SPEECH PERCEPTION Manoel F. Tenorio School of Electrical Engineering Purdue University West Lafayette, IN 47907 ABSTRACT 783 An increasing number of profoundly deaf patients suffering from sensorineural deafness are using cochlear implants as prostheses. Mter the implant, sound can be detected through the electrical stimulation of the remaining peripheral auditory nervous system. Although great progress has been achieved in this area, no useful speech recognition has been attained with either single or multiple channel cochlear implants. Coding evidence suggests that it is necessary for any implant which would effectively couple with the natural speech perception system to simulate the temporal dispersion and other phenomena found in the natural receptors, and currently not implemented in any cochlear implants. To this end, it is presented here a computational model using artificial neural networks (ANN) to incorporate the natural phenomena in the artificial cochlear. The ANN model presents a series of advantages to the implementation of such systems. First, the hardware requirements, with constraints on power, size, and processing speeds, can be taken into account together with the development of the underlining software, before the actual neural structures are totally defined. Second, the ANN model, since it is an abstraction of natural neurons, carries the necessary ingredients and is a close mapping for implementing the necessary functions. Third, some of the processing, like sorting and majority functions, could be implemented more efficiently, requiring only local decisions. Fourth, the ANN model allows function modifications through parametric modification (no software recoding), which permits a variety of fine-tuning experiments, with the opinion of the patients, to be conceived. Some of those will permit the user some freedom in system modification at real-time, allowing finer and more subjective adjustments to fit differences on the condition and operation of individual's remaining peripheral auditory system. 1. INTRODUCTION The study of the model of sensory receptors can be carried out either via trying to understand how the natural receptors process incoming signals and build a representation code, or via the construction of artificial replacements. In the second case, we are interested in to what extent those artificial counterparts have the ability to replace the natural receptors. Several groups are now carrying out the design of artificial sensors. Artificial cochleas seem to have a number of different designs and a tradition of experiments. These make them now available for widespread use as prostheses for patients who have sensorineural deafness caused by hair cell damage. © American Institute of Physics 1988 784 Although surgery is required for such implants, their performance has reached a level of maturity to induce patients to seek out these devices voluntarily. Unfortunately, only partial acoustic information is obtained by severely deaf patients with cochlear prosthesis. Useful patterns for speech communication are not yet 'fully recognizable through auditory prostheses. This problem with artificial receptors is true for both single implants, that stimulate large sections of the cochlea with signals that cover a large portion of the spectrum [4,5], and multi channel implants, that stimulate specific regions of the cochlea with specific portions of the auditory spectrum [3,13]. In this paper, we tackle the problem of artificial cochlear implants through the used of neurocomputing tools. The receptor model used here was developed by Gerald Wasserman of the Sensory Coding Laboratory, Department of Psychological Sciences, Purdue University [20], and the implants were performed by Richard Miyamoto of the Department of Otolaryngology, Indiana University Medical School [11]. The idea is to introduce with the cochlear implant, the computation that would be performed otherwise by the natural receptors. It would therefore be possible to experimentally manipulate the properties of the implant and measure the effect of coding variations on behavior. The model was constrained to be portable, simple to implant, fast enough computationally for on-line use, and built with a flexible paradigm, which would allow for modification of the different parts of the model, without having to reconstruct it entirely. In the next section, we review parts of the receptor model, and discuss the block diagram of the implant. Section 3 covers the limitations associated with the technique, and discusses the r.esults obtained with a single neuron and one feedback loop. Section 4 discusses the implementations of these models using feedforward neural networks, and the computational advantages for doing so. 2. COCHLEAR IMPLANTS AND THE NEURON MODEL Although patients cannot reliably recognize randomly chosen spoken words to them (when implanted with either multichannel or single channel devices), this is not to say that no information is extracted from speech. If the vocabulary is reduced to a limited set of words, patients perform significantly better than chance, at associating the word with a member of the set. For these types of experiments, single channel implants correspond to reported performance of 14% to 20% better than chance, with 62% performance being the highest reported. For multiple channels, performances of 95% were reported. So far no one has investigated the differences in performance between the two types of implants. Since the two implants have so many differences, it is difficult to point out the cause for the better performance in the multiple channel case. The results of such experiments are encouraging, and point to the fact that cochlea implants need only minor improvement to be able to mediate ad-lib speech perception successfully. Sensory coding studies have suggested a solution to the implant problem, by showing that the representation code generated by the sensory system is task dependent. This evidence came from comparison of intracellular recordings taken from a single receptor of intact subjects. This coding evidence suggests that the temporal dispersion (time integration) found in natural receptors would be a necessary part of any 785 cochlear implant. Present cochlear implants have no dispersion at all. Figure 2 shows the block diagram for a representative cochlear implant, the House-Urban stimulator. The acoustic signal is picked up by the microphone, which sends it to an AM oscillator. This modulation step is necessary to induce an electro-magnetic coupling between the external and internal coil. The internal coil has been surgically implanted, and it is connected to a pair of wires implanted inside and outside the cochlea. Just incorporating the temporal dispersion model to an existing device would not replicate the fact that in natural receptors, temporal dispersion appears in conjunction to other operations which are strongly non linear. There are operations like selection of a portion of the spectrum, rectification, compression, and time-dispersion to be considered. In figure 3, a modified implant is shown, which takes into consideration some of these operations. It is depicted as a single-channel implant, although the ultimate goal is to make it multichannel. Details of the operation of this device can be found elsewhere [21]. Here, it is important to mention that the implant would also have a compression/rectification function, and it would receive a feedback from the integrator stage in order to control its gain. 3. CHARACTERISTICS AND RESULTS OF THE IMPLANTS The above model has been implemented as an off-line process, and then the patients were exposed to a preprocessed signal which emulated the operation of the device. It is not easy to define the amount of feedback needed in the system or the amount of time dispersion. It could also be that these parameters are variable across different conditions. Another variance in the experiment is the amount of damage (and type) among different individuals. So, these parameters have to be determined clinically. The coupling between the artificial receptor and the natural system also presents problems. If a physical connection is used, it increases the risk of infections. When inductive methods are used, the coupling is never ideal. If portability and limited power is of concern in the implementation, then the limited energy available for coupling has to be used very effectively. The computation of the receptor model has to be made in a way to allow for fast implementation. The signal transformation is to be computed on-line. Also, the results from clinical studies should be able to be incorpora ted fairly easily without having to reengineer the implant. Now we present the results of the implementation of the transfer function of figure 4. Patients, drawn from a population described elsewhere [11,12,14], were given spoken sentences processed off-line, and simultaneously presented with a couple of words related to the context. Only one of them was the correct answer. The patient had two buttons, one for each alternative; he/she was to press the button which corresponded to the correct alternative. The results are shown in the tables below. Patient 1 (Average of the population) Dispersion No disp. 0.1 msec 0.3 msec Percentage of correct alternatives 67% 78% 85% Best performance 786 1 msec 76% 3 msec 72% Table I: Phoneme discrimination in d. two-alternate task. Patient 2 Percentage of correct alternatives D· . lsperSlOn No disp. 50% 1.0 msec 76% Best performance Table II: Sentence comprehension in a two-alternative task. There were quite a lot of variations in the performance of the different patients, some been able to perform better at different dispersion and compression amounts than the average of the population. Since one cannot control the amount of damage in the system of each patient or differences in individuals, it is hard to predict the ideal values for a given patient. Nevertheless, the improvements observed are of undeniable value in improving speech perception. 4. THE NEUROCOMPUTING MODEL In studying the implementation of such a system for on-line use, yet flexible enough to produce a carry-on device, we look at feedforward neurocomputer models as a possible answer. First, we wanted a model that easily produced a parallel implementation, so that the model could be expanded in a multichannel environment without compromising the speed of the system. Figure 5 shows the initial idea for the implementation of the device as a Single Instruction Multiple Data (SIMD) architecture. The implant would be similar to the one described in Figure 4, except that the transfer function of the receptor would be performed by a two layer feed forward network (Figure 6). Since there is no way of finding out the values of compression and dispersion a part from clinical trials, or even if these values do change in certain conditions, we need to create a structure that is flexible enough to modify the program structure by simple manipulation of parameters. This is also the same problem we would face when trying to expand the system to a multichannel implant. Again, neuromorphic models provided a nice paradigm in which the dataflow and the function of the program could be altered by simple parameter (weight) change. For this first implementation we chose to use the no-contact inductive coupling method. The drawback of this method is that all the information has to be compressed in a single channel for reliable transmission and cross talk elimination. Since the inductive coupling of the implant· is critical at every cycle, the most relevant information must be picked out of the processed signal. This information is then given all the available energy, and after all the coupling loss, it should be sufficient to provide for speech pattern discrimination. In a multichannel setting, this corresponds to doing a sorting of all the n signals in the channels, selecting the m highest signals, and adding them up for modulation. In a naive single processor implementation, this could correspond to n 2 comparisons, and in a multiprocessor implementation, log(n) comparisons. Both are dependent on the number of signals to be 787 sorted. We needed a scheme in which the sorting time would be constant with the number of channels, and would be easily implementable in analog circuitry, in case this became a future route. Our scheme is shown in Figure 7. Each channel is connected to a threshold element, whose threshold can be varied externally. A monotonically decreasing function scans the threshold values, from the highest possible value of the output to the lowest. The output of these elements will be high corresponding to the values that are the highest first. These output are summed with a quasi-integrator with threshold set to m. This element, when high, disables the scanning functions; and it corresponds to having found the m highest signals. This sorting is independent of the number of channels. The output of the threshold units are fed into sigma-pi units which gates the signals to be modulated. The output of these units are summed and correspond to the final processed signal (Figure 8). The user has full control of the characteristics of this device. The number of channels can be easily altered; the number of components allowed in the modulation can be changed; the amount of gain, rectificationcompression, and dispersion of each channel can also be individually controlled. The entire system is easily implementable in analog integrated circuits, once the clinical tests have determine the optimum operational characteristics. 6. CONCLUSION We have shown that the study of sensory implants can enhance our understanding of the representation schemes used for natural sensory receptors. In particular, implants can be enhanced significantly if the effects of the sensory processing and transfer functions are incorporated in the model. We have also shown that neuromorphic computing paradigm provides a parallel and easily modifiable framework for signal processing structures, with advantages that perhaps cannot be offered by other technology. We will soon start the use of the first on-line portable model, using a single processor. This model will provide a testbed for more extensive clinical trials of the implant. We will then move to the parallel implementation, and from there, possibly move toward analog circuitry implementation. Another route for the use of neuromorphic computing in this domain is possibly the use of sensory recordings from healthy animals to train selforganizing adaptive learning networks, in order to design the implant transfer functions. [1] [2] REFERENCES Bilger, R.C.; Black, F.O.; Hopkinson, N.T.; and Myers, E.N., "Implanted auditory prosthesis: An evaluation of subjects presently fitted with cochlear implants," Otolaryngology, 1977, Vol. 84, pp. 677682. Bilger, R.C.; Black, F.O.; Hopkinson, N.T.; ~~ers, E.~.; Payne, !.L.; Stenson, N.R.; Vega, A.; and Wolf, R.V., EvaluatiOn of subJects presently fitted with implanted auditory prostheses," Annals of Otology, Rhinology, and Laryngology, 1977, Vol. 86(Supp. 38), pp. 1-176. 788 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] Eddington, D.K.; Dobelle, W.H.; Brackmann, D.E.; Mladejovsky, M.G.; and Parkin, J., "Place and periodicity pitch by stimulation of multiple scala tympani electrodes in deaf volunteers," American Society for Artificial Internal Organs, Transactions, 1978, Vol. 24, pp. 1-5. House, W.F.; Berliner, .K.; Crary, W.; Graham, M.; Luckey, R;; Norton, N.; Selters, W.; Tobm, H.; Urban, J.; and Wexler, M., Cochlear implants," Annals of Otology, Rhinology and Laryngology, 1976, Vol. 85(Supp. 27), pp. 1-93. House, W.F. and Urban, J., "Long term results of electrode implantation and electronic stimulation of the cochlea in man," Annals of Otology, Rhinology and Laryngology, 1973, Vol. 82, No.2, pp. 504-517. Ifukube, T. and White, R.L., "A speech processor with lateral inhibition for an eight channel cochlear implant and its evaluation," IEEE Trans. on Biomedical Engineering, November 1987, Vol. BME-34, No. 11. Kong, K.-L., and Wasserman, G.S., "Changing response measures alters temporal summation in the receptor and spike potentials of the Limulus lateral eye," Sensory Processes, 1978, Vol. 2, pp. 21-31. (a) Kong, K.-L., and Wasserman, G.S., "Temporal summation in the receptor potential of the Limulus lateral eye: Comparison between retinula and eccentric cells," Sensory Processes, 1978, Vol. 2, pp. 9-20. (b) Michelson, R.P., "The results of electrical stimulation of the cochlea in human sensory deafness," Annals of Otology, Rhinology and Laryngology, 1971, Vol. 80, pp. 914-919. Mia dej ovsky, M.G.; Eddington, D.K.; Dobelle, W.H.; and Brackmann, D.E., "Artificial hearing for the deaf by cochlear stimulation: Pitch modulation and some parametric thresholds," American Society for Artificial Internal Organs, Transactions, 1974, Vol. 21, pp. 1-7. Miyamoto, R.T.; Gossett, S.K.; Groom, G.L.; Kienle, M.L.; Pope, M.L.; and Shallop, J.K., "Cochlear implants: An auditory prosthesis for the deaf," Journal of the Indiana State Medical Association, 1982, Vol. 75, pp. 174-177. Miyamoto, R.T.; Myres, W.A.; Pope, M.L.; and Carotta, C.A., "Cochlear implants for deaf children," Laryngoscope, 1986, Vol. 96, pp. 990-996. Pialoux, P.; Chouard, C.H.; Meyer, B.; and Fu,?ain, C., "Indications and results of the multichannel cochlear implant,' Acta Otolaryngology, 1979, Vo .. 87, pp. 185-189. Robbins, A.M.i Osberger, M.J.; Miyamoto, R.T.; Kienle, M.J.; and Myres, W.A., • Speech-tracking performance in single-channel cochlear implant subjects," JourntLl of Speech and Hearing Research, 1985, Vol. 28, pp. 565-578. Russell, I.J. and Sellick, P.M., "The tuning properties of cochlear hair cells," in E.F. Evans and J.P. Wilson (eds.), Psychophysics and Physiology 0 f Hearing, London: Academic Press, 1977. Wasserman, G.S., "Limulus psychophysics: Temporal summation in the ventral eye," Journal of Experimental Psychology: General, 1978, Vol. 107, pp. 276-286. [17] [18] [19] [20] [21] 789 Wasserman, G.S., "Limulus psychophysics: Increment threshold," Perception & Psychophysics, 1981, Vol. 29, pp. 251-260. Wasserman, G.S.; Felsten, G.; and Easland, G.S., "Receptor saturation and the psychophysical function," Investigative Ophthalmology and Visual Science, 1978, Vol. 17, p. 155 (Abstract). Wasserman, G.S.; Felsten, G.; and Easland, G.S., "The psychophysical function: Harmonizing Fechner and Stevens," Science, 1979, Vol. 204, pp. 85-87. Wasserman, G.S., "Cochlear implant codes and speech perception in profoundly deaf," Bulletin of Psychonomic Society, Vol. (18)3, 1987. Wasserman, G.S.; Wang-Bennett, L.T.; and Miyamoto, R.T., "Temporal dispersion in natural receptors and pattern discrimination mediated by artificial receptor," Proc. of the Fechner Centennial Symposium, Hans Buffart (Ed.), Elsevier/North Holland, Amsterdam, 1987. r ~ SENSORY CODING DATA ~ -, 9 7~ I I I 13 I STIMULUS I RECEPTOR SIGNAL 11 CENTRAL ANALYSIS I r----------~ I L ..: PROSTHETIC : ..J I SIGNAL :- I • ...... -r:-. __ . 17 BEHAVIOR 15 Fig. 1. Path of Natural and Prosthetic Signals. Sound Central Nervous System Fig. 2. The House-Urban Cochlear Implant. 790 AMPLIFICATION -----1~ COMPRESSIVE RECTIFIER DISPERSION -----1~ INTEGRATOR Fig. 3. Receptor Model Sound Central Nervous System Fig. 4. Modified Implant Model. PORTABLE PARALLEL NEUROCOMpuTER m EXTERNAL USER CONTROLLED PARAMETERS 16KHz AM MODULATED OUTPUT Fig. 5. Initial Concept for a SIMD Architecture. EXTERNALLY CONTROLLED DISPERSION AMPUFICATION NEURON MODEL NEURON MODEL Fig. 6. Feedforward Neuron Model Implant. SORTER OF N SIGNALS 791 792 SIGNALS INPUTS I j max I . min I SORTER OF n SIGNALS IN 0(1) I--~ RESET SCANNING FUNCTION THRESHOLD CONTROL: THRESHOLD SETOFn, EXTERNALLY CONTROLLED SCANNING FUNCTION FROM I imax TO Ij min Fig. 7. Signal Sorting Circuit. SIGNAL SELECTORS t---~. 0 -leS 1 1 1 J----" OUTPUT SIGNAL In ---+---~.c Fig. 8. Sigma-Pi Units for Signal Composition. 793 USER CONTROLLED PARAMETERS 13121114131 tJ BEST 4 MATCHES DISPERSION 10101312111 ~ ~ ~ SINGLE NEURON GAIN FILTER PROCESSOR PROCESSING BYPASS BYPASS MICROPHONE Fig. 9. Parameter Controls for Clinical Studies.
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SELF-ORGANIZATION OF ASSOCIATIVE DATABASE AND ITS APPLICATIONS Hisashi Suzuki and Suguru Arimoto Osaka University, Toyonaka, Osaka 560, Japan ABSTRACT An efficient method of self-organizing associative databases is proposed together with applications to robot eyesight systems. The proposed databases can associate any input with some output. In the first half part of discussion, an algorithm of self-organization is proposed. From an aspect of hardware, it produces a new style of neural network. In the latter half part, an applicability to handwritten letter recognition and that to an autonomous mobile robot system are demonstrated. INTRODUCTION Let a mapping f : X -+ Y be given. Here, X is a finite or infinite set, and Y is another finite or infinite set. A learning machine observes any set of pairs (x, y) sampled randomly from X x Y. (X x Y means the Cartesian product of X and Y.) And, it computes some estimate j : X -+ Y of f to make small, the estimation error in some measure. Usually we say that: the faster the decrease of estimation error with increase of the number of samples, the better the learning machine. However, such expression on performance is incomplete. Since, it lacks consideration on the candidates of J of j assumed preliminarily. Then, how should we find out good learning machines? To clarify this conception, let us discuss for a while on some types of learning machines. And, let us advance the understanding of the self-organization of associative database . . Parameter Type An ordinary type of learning machine assumes an equation relating x's and y's with parameters being indefinite, namely, a structure of f. It is equivalent to define implicitly a set F of candidates of 1. (F is some subset of mappings from X to Y.) And, it computes values of the parameters based on the observed samples. We call such type a parameter type. For a learning machine defined well, if F 3 f, j approaches f as the number of samples increases. In the alternative case, however, some estimation error remains eternally. Thus, a problem of designing a learning machine returns to find out a proper structure of f in this sense. On the other hand, the assumed structure of f is demanded to be as compact as possible to achieve a fast learning. In other words, the number of parameters should be small. Since, if the parameters are few, some j can be uniquely determined even though the observed samples are few. However, this demand of being proper contradicts to that of being compact. Consequently, in the parameter type, the better the compactness of the assumed structure that is proper, the better the learning machine. This is the most elementary conception when we design learning machines . . Universality and Ordinary Neural Networks Now suppose that a sufficient knowledge on f is given though J itself is unknown. In this case, it is comparatively easy to find out proper and compact structures of J. In the alternative case, however, it is sometimes difficult. A possible solution is to give up the compactness and assume an almighty structure that can cover various 1's. A combination of some orthogonal bases of the infinite dimension is such a structure. Neural networks1,2 are its approximations obtained by truncating finitely the dimension for implementation. © American Institute of Physics 1988 767 768 A main topic in designing neural networks is to establish such desirable structures of 1. This work includes developing practical procedures that compute values of coefficients from the observed samples. Such discussions are :flourishing since 1980 while many efficient methods have been proposed. Recently, even hardware units computing coefficients in parallel for speed-up are sold, e.g., ANZA, Mark III, Odyssey and E-1. Nevertheless, in neural networks, there always exists a danger of some error remaining eternally in estimating /. Precisely speaking, suppose that a combination of the bases of a finite number can define a structure of 1 essentially. In other words, suppose that F 3 /, or 1 is located near F. In such case, the estimation error is none or negligible. However, if 1 is distant from F, the estimation error never becomes negligible. Indeed, many researches report that the following situation appears when 1 is too complex. Once the estimation error converges to some value (> 0) as the number of samples increases, it decreases hardly even though the dimension is heighten. This property sometimes is a considerable defect of neural networks . . Recursi ve Type The recursive type is founded on another methodology of learning that should be as follows. At the initial stage of no sample, the set Fa (instead of notation F) of candidates of I equals to the set of all mappings from X to Y. After observing the first sample (Xl, Yl) E X x Y, Fa is reduced to Fi so that I(xt) = Yl for any I E F. After observing the second sample (X2' Y2) E X x Y, Fl is further reduced to F2 so that i(xt) = Yl and I(X2) = Y2 for any I E F. Thus, the candidate set F becomes gradually small as observation of samples proceeds. The i after observing i-samples, which we write i" is one of the most likelihood estimation of 1 selected in fi;. Hence, contrarily to the parameter type, the recursive type guarantees surely that j approaches to 1 as the number of samples increases. The recursive type, if observes a sample (x" yd, rewrites values 1,-l(X),S to I,(x)'s for some x's correlated to the sample. Hence, this type has an architecture composed of a rule for rewriting and a free memory space. Such architecture forms naturally a kind of database that builds up management systems of data in a self-organizing way. However, this database differs from ordinary ones in the following sense. It does not only record the samples already observed, but computes some estimation of l(x) for any x E X. We call such database an associative database. The first subject in constructing associative databases is how we establish the rule for rewri ting. For this purpose, we adap t a measure called the dissimilari ty. Here, a dissimilari ty means a mapping d : X x X -+ {reals > O} such that for any (x, x) E X x X, d(x, x) > 0 whenever l(x) t /(x). However, it is not necessarily defined with a single formula. It is definable with, for example, a collection of rules written in forms of "if· .. then·· .. " The dissimilarity d defines a structure of 1 locally in X x Y. Hence, even though the knowledge on f is imperfect, we can re:flect it on d in some heuristic way. Hence, contrarily to neural networks, it is possible to accelerate the speed of learning by establishing d well. Especially, we can easily find out simple d's for those l's which process analogically information like a human. (See the applications in this paper.) And, for such /'s, the recursive type shows strongly its effectiveness. We denote a sequence of observed samples by (Xl, Yd, (X2' Y2),···. One of the simplest constructions of associative databases I, after observing i-samples (i = 1,2,.,,) is as follows. Algorithm 1. At the initial stage, let So be the empty set. For every i = 1,2" .. , let i,-l(x) for any x E X equal some y* such that (x*,y*) E S,-l and d(x, x*) = min d(x, x) . (1) (%,y)ES.-t Furthermore, add (x" y,) to S;-l to produce Sa, i.e., S, = S,_l U {(x" y,n. Another version improved to economize the memory is as follows. Algorithm 2, At the initial stage, let So be composed of an arbitrary element in X x Y. For every i = 1,2"", let ii-lex) for any x E X equal some y. such that (x·, y.) E Si-l and d(x, x·) = min d(x, x) . (i,i)ES.-l Furthermore, if ii-l(Xi) # Yi then let Si = Si-l, or add (Xi, Yi) to Si-l to produce Si, i.e., Si = Si-l U {(Xi, Yi)}' In either construction, ii approaches to f as i increases. However, the computation time grows proportionally to the size of Si. The second subject in constructing associative databases is what addressing rule we should employ to economize the computation time. In the subsequent chapters, a construction of associative database for this purpose is proposed. It manages data in a form of binary tree. SELF-ORGANIZATION OF ASSOCIATIVE DATABASE Given a sample sequence (Xl, Yl), (X2' Y2), .. " the algorithm for constructing associative database is as follows. Algorithm 3,' Step I(Initialization): Let (x[root], y[root]) = (Xl, Yd. Here, x[.] and y[.] are variables assigned for respective nodes to memorize data.. Furthermore, let t = 1. Step 2: Increase t by 1, and put x, in. After reset a pointer n to the root, repeat the following until n arrives at some terminal node, i.e., leaf. Notations nand n mean the descendant nodes of n. If d(x" r[n)) ~ d(xt, x[n)), let n = n. Otherwise, let n = n. Step 3: Display yIn] as the related information. Next, put y, in. If yIn] = y" back to step 2. Otherwise, first establish new descendant nodes n and n. Secondly, let (x[n], yIn)) (x[n], yIn)) (x[n], yIn)), (Xt, y,). (2) (3) Finally, back to step 2. Here, the loop of step 2-3 can be stopped at any time and also can be continued. Now, suppose that gate elements, namely, artificial "synapses" that play the role of branching by d are prepared. Then, we obtain a new style of neural network with gate elements being randomly connected by this algorithm. LETTER RECOGNITION Recen tly, the vertical slitting method for recognizing typographic English letters3 , the elastic matching method for recognizing hand written discrete English letters4 , the global training and fuzzy logic search method for recognizing Chinese characters written in square styleS, etc. are published. The self-organization of associative database realizes the recognition of handwritten continuous English letters. 769 770 H 9 /wn" NOV ~ ~ ~ -xk :La.t ~~ ~ ~~~ dw1lo' ~~~~~of~~ ~~~ 4,-¥~~4Fig. 1. Source document. 2~~---------------' lOO~---------------' o o 1000 2000 3000 4000 Number of samples o 1000 2000 3000 4000 NUAlber of sampl es Fig. 2. Windowing. Fig. 3. An experiment result. An image scanner takes a document image (Fig. 1). The letter recognizer uses a parallelogram window that at least can cover the maximal letter (Fig. 2), and processes the sequence of letters while shifting the window. That is, the recognizer scans a word in a slant direction. And, it places the window so that its left vicinity may be on the first black point detected. Then, the window catches a letter and some part of the succeeding letter. If recognition of the head letter is performed, its end position, namely, the boundary line between two letters becomes known. Hence, by starting the scanning from this boundary and repeating the above operations, the recognizer accomplishes recursively the task. Thus the major problem comes to identifying the head letter in the window. Considering it, we define the following. • Regard window images as x's, and define X accordingly. • For a (x, x) E X x X, denote by B a black point in the left area from the boundary on window image X. Project each B onto window image x. Then, measure the Euclidean distance 6 between fj and a black point B on x being the closest to B. Let d(x, x) be the summation of 6's for all black points B's on x divided by the number of B's. • Regard couples of the "reading" and the position of boundary as y's, and define Y accordingly. An operator teaches the recognizer in interaction the relation between window image and reading& boundary with algorithm 3. Precisely, if the recalled reading is incorrect, the operator teaches a correct reading via the console. Moreover, if the boundary position is incorrect, he teaches a correct position via the mouse. Fig. 1 shows partially a document image used in this experiment. Fig. 3 shows the change of the number of nodes and that of the recognition rate defined as the relative frequency of correct answers in the past 1000 trials. Speciiications of the window are height = 20dot, width = 10dot, and slant angular = 68deg. In this example, the levels of tree were distributed in 6-19 at time 4000 and the recognition rate converged to about 74%. Experimentally, the recognition rate converges to about 60-85% in most cases, and to 95% at a rare case. However, it does not attain 100% since, e.g., "c" and "e" are not distinguishable because of excessive lluctuation in writing. If the consistency of the x, y-relation is not assured like this, the number of nodes increases endlessly (d. Fig. 3). Hence, it is clever to stop the learning when the recognition rate attains some upper limit. To improve further the recognition rate, we must consider the spelling of words. It is one of future subjects. OBSTACLE AVOIDING MOVEMENT Various systems of camera type autonomous mobile robot are reported flourishingly6-1O. The system made up by the authors (Fig. 4) also belongs to this category. Now, in mathematical methodologies, we solve usually the problem of obstacle avoiding movement as a cost minimization problem under some cost criterion established artificially. Contrarily, the self-organization of associative database reproduces faithfully the cost criterion of an operator. Therefore, motion of the robot after learning becomes very natural. Now, the length, width and height of the robot are all about O.7m, and the weight is about 30kg. The visual angle of camera is about 55deg. The robot has the following three factors of motion. It turns less than ±30deg, advances less than 1m, and controls speed less than 3km/h. The experiment was done on the passageway of wid th 2.5m inside a building which the authors' laboratories exist in (Fig. 5). Because of an experimental intention, we arrange boxes, smoking stands, gas cylinders, stools, handcarts, etc. on the passage way at random. We let the robot take an image through the camera, recall a similar image, and trace the route preliminarily recorded on it. For this purpose, we define the following. • Let the camera face 28deg downward to take an image, and process it through a low pass filter. Scanning vertically the filtered image from the bottom to the top, search the first point C where the luminance changes excessively. Then, su bstitu te all points from the bottom to C for white, and all points from C to the top for black (Fig. 6). (If no obstacle exists just in front of the robot, the white area shows the ''free'' area where the robot can move around.) Regard binary 32 x 32dot images processed thus as x's, and define X accordingly. • For every (x, x) E X x X, let d(x, x) be the number of black points on the exclusive-or image between x and X. • Regard as y's the images obtained by drawing routes on images x's, and define Y accordingly. The robot superimposes, on the current camera image x, the route recalled for x, and inquires the operator instructions. The operator judges subjectively whether the suggested route is appropriate or not. In the negative answer, he draws a desirable route on x with the mouse to teach a new y to the robot. This opera.tion defines implicitly a sample sequence of (x, y) reflecting the cost criterion of the operator. rmbi Ie unit (robot) Stationary uni t Fig. 4. Configuration of autonomous mobile robot system. IibUBe .::l" ! Roan ~ _. I 22 11 , 23 24 Roan 12 {13 North 14 t y Fig. 5. Experimental environment. 771 772 Wall Preprocessing A : : : !fa • •• . . A Fig. 6. Processing for obstacle avoiding movement. Camera image Preprocessing 0 x Search O Course suggest ion Fig. 1. Processing for position identification. We define the satisfaction rate by the relative frequency of acceptable suggestions of route in the past 100 trials. In a typical experiment, the change of satisfaction rate showed a similar tendency to Fig. 3, and it attains about 95% around time 800. Here, notice that the rest 5% does not mean directly the percentage of collision. (In practice, we prevent the collision by adopting some supplementary measure.) At time 800, the number of nodes was 145, and the levels of tree were distributed in 6-17. The proposed method reflects delicately various characters of operator. For example, a robot trained by an operator 0 moves slowly with enough space against obstacles while one trained by another operator 0' brushes quickly against obstacles. This fact gives us a hint on a method of printing "characters" into machines. POSITION IDENTIFICATION The robot can identify its position by recalling a similar landscape with the position data to a camera image. For this purpose, in principle, it suffices to regard camera images and position data as x's and y's, respectively. However, the memory capacity is finite in actual compu ters. Hence, we cannot but compress the camera images at a slight loss of information. Such compression is admittable as long as the precision of position identification is in an acceptable area. Thus, the major problem comes to find out some suitable compression method. In the experimental environment (Fig. 5), juts are on the passageway at intervals of 3.6m, and each section between adjacent juts has at most one door. The robot identifies roughly from a surrounding landscape which section itself places in. And, it uses temporarily a triangular surveying technique if an exact measure is necessary. To realize the former task, we define the following . • Turn the camera to take a panorama image of 360deg. Scanning horizontally the center line, substitute the points where the luminance excessively changes for black and the other points for white (Fig. 1). Regard binary 360dot line images processed thus as x's, and define X accordingly . • For every (x, x) E X x X, project each black point A on x onto x. And, measure the Euclidean distance 6 between A and a black point A on x being the closest to A. Let the summation of 6 be S. Similarly, calculate S by exchanging the roles of x and X. Denoting the numbers of A's and A's respectively by nand n, define d(x, x) = ~(~ + ~). 2 n n (4) • Regard positive integers labeled on sections as y's (cf. Fig. 5), and define Y accordingly. In the learning mode, the robot checks exactly its position with a counter that is reset periodically by the operator. The robot runs arbitrarily on the passageways within 18m area and learns the relation between landscapes and position data. (Position identification beyond 18m area is achieved by crossing plural databases one another.) This task is automatic excepting the periodic reset of counter, namely, it is a kind of learning without teacher. We define the identification rate by the relative frequency of correct recalls of position data in the past 100 trials. In a typical example, it converged to about 83% around time 400. At time 400, the number of levels was 202, and the levels oftree were distributed in 522. Since the identification failures of 17% can be rejected by considering the trajectory, no pro blem arises in practical use. In order to improve the identification rate, the compression ratio of camera images must be loosened. Such possibility depends on improvement of the hardware in the future. Fig. 8 shows an example of actual motion of the robot based on the database for obstacle avoiding movement and that for position identification. This example corresponds to a case of moving from 14 to 23 in Fig. 5. Here, the time interval per frame is about 40sec. ,~. ~ I' ;~"i.. . ( " . ; i " ~ " t . ..I '.1 I • -: • ,.., 'II Fig. 8. Actual motion of the robot. 773 774 CONCLUSION A method of self-organizing associative databases was proposed with the application to robot eyesight systems. The machine decomposes a global structure unknown into a set of local structures known and learns universally any input-output response. This framework of problem implies a wide application area other than the examples shown in this paper. A defect of the algorithm 3 of self-organization is that the tree is balanced well only for a subclass of structures of f. A subject imposed us is to widen the class. A probable solution is to abolish the addressing rule depending directly on values of d and, instead, to establish another rule depending on the distribution function of values of d. It is now under investigation. REFERENCES 1. Hopfield, J. J. and D. W. Tank, "Computing with Neural Circuit: A Model/' Science 233 (1986), pp. 625-633. 2. Rumelhart, D. E. et al., "Learning Representations by Back-Propagating Errors," Nature 323 (1986), pp. 533-536. 3. Hull, J. J., "Hypothesis Generation in a Computational Model for Visual Word Recognition," IEEE Expert, Fall (1986), pp. 63-70. 4. Kurtzberg, J. M., "Feature Analysis for Symbol Recognition by Elastic Matching," IBM J. Res. Develop. 31-1 (1987), pp. 91-95. 5. Wang, Q. R. and C. Y. Suen, "Large Tree Classifier with Heuristic Search and Global Training," IEEE Trans. Pattern. Anal. & Mach. Intell. PAMI 9-1 (1987) pp. 91-102. 6. Brooks, R. A. et al, "Self Calibration of Motion and Stereo Vision for Mobile Robots," 4th Int. Symp. of Robotics Research (1987), pp. 267-276. 7. Goto, Y. and A. Stentz, "The CMU System for Mobile Robot Navigation," 1987 IEEE Int. Conf. on Robotics & Automation (1987), pp. 99-105. 8. Madarasz, R. et al., "The Design of an Autonomous Vehicle for the Disabled," IEEE Jour. of Robotics & Automation RA 2-3 (1986), pp. 117-125. 9. Triendl, E. and D. J. Kriegman, "Stereo Vision and Navigation within Buildings," 1987 IEEE Int. Conf. on Robotics & Automation (1987), pp. 1725-1730. 10. Turk, M. A. et al., "Video Road-Following for the Autonomous Land Vehicle," 1987 IEEE Int. Conf. on Robotics & Automation (1987), pp. 273-279.
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TEMPORAL PATTERNS OF ACTIVITY IN NEURAL NETWORKS Paolo Gaudiano Dept. of Aerospace Engineering Sciences, University of Colorado, Boulder CO 80309, USA January 5, 1988 Abstract Patterns of activity over real neural structures are known to exhibit timedependent behavior. It would seem that the brain may be capable of utilizing temporal behavior of activity in neural networks as a way of performing functions which cannot otherwise be easily implemented. These might include the origination of sequential behavior and the recognition of time-dependent stimuli. A model is presented here which uses neuronal populations with recurrent feedback connections in an attempt to observe and describe the resulting time-dependent behavior. Shortcomings and problems inherent to this model are discussed. Current models by other researchers are reviewed and their similarities and differences discussed. METHODS / PRELIMINARY RESULTS 297 In previous papers,[2,3] computer models were presented that simulate a net consisting of two spatially organized populations of realistic neurons. The populations are richly interconnected and are shown to exhibit internally sustained activity. It was shown that if the neurons have response times significantly shorter than the typical unit time characteristic of the input patterns (usually 1 msec), the populations will exhibit time-dependent behavior. This will typically result in the net falling into a limit cycle. By a limit cycle, it is meant that the population falls into activity patterns during which all of the active cells fire in a cyclic, periodic fashion. Although the period of firing of the individual cells may be different, after a fixed time the overall population activity will repeat in a cyclic, periodic fashion. For populations organized in 7x7 grids, the limit cycle will usually start 20~200 msec after the input is turned off, and its period will be in the order of 20-100 msec. The point ofinterest is that ifthe net is allowed to undergo synaptic modifications by means of a modified hebbian learning rule while being presented with a specific spatial pattern (i.e., cells at specific spatial locations within the net are externally stimulated), subsequent presentations of the same pattern with different temporal characteristics will cause the population to recall patterns which are spatially identical (the same cells will be active) but which have different temporal qualities. In other words, the net can fall into a different limit cycle. These limit cycles seem to behave as attractors in that similar input patterns will result in the same limit cycle, and hence each distinct limit cycle appears to have a basin of attraction. Hence a net which can only learn a small © American Institute of Physics 1988 298 number of spatially distinct patterns can recall the patterns in a number of temporal modes. If it were possible to quantitatively discriminate between such temporal modes, it would seem reasonable to speculate that different limit cycles could correspond to different memory traces. This would significantly increase estimates on the capacity of memory storage in the net. It has also been shown that a net being presented with a given pattern will fall and stay into a limit cycle until another pattern is presented which will cause the system to fall into a different basin of attraction. If no other patterns are presented, the net will remain in the same limit cycle indefinitely. Furthermore, the net will fall into the same limit cycle independently of the duration of the input stimulus, so long as the input stimulus is presented for a long enough time to raise the population activity level beyond a minimum necessary to achieve self-sustained activity. Hence, if we suppose that the net "recognizes" the input when it falls into the corresponding limit cycle, it follows that the net will recognize a string of input patterns regardless of the duration of each input pattern, so long as each input is presented long enough for the net to fall into the appropriate limit cycle. In particular, our system is capable of falling into a limit cycle within some tens of milliseconds. This can be fast enough to encode, for example, a string of phonemes as would typically be found in continuous speech. It may be possible, for instance, to create a model similar to Rumelhart and McClelland's 1981 model on word recognition by appropriately connecting multiple layers of these networks. If the response time of the cells were increased in higher layers, it may be possible to have the lowest level respond to stimuli quickly enough to distinguish phonemes (or some sub-phonemic basic linguistic unit), then have populations from this first level feed into a slower, word-recognizing population layer, and so On. Such a model may be able to perform word recognition from an input consisting of continuous phoneme strings even when the phonemes may vary in duration of presentation. SHORTCOMINGS Unfortunately, it was noticed a short time ago that a consistent mistake had been made in the process of obtaining the above-mentioned results. Namely, in the process of decreasing the response time of the cells I accidentally reached a response time below the time step used in the numerical approximation that updates the state of each cell during a simulation. The equations that describe the state of each cell depend on the state of the cell at the previous time step as well as on the input at the present time. These equations are of first order in time, and an explicit discrete approximation is used in the model. Unfortunately it is a known fact that care must be taken in selecting the size of the time step in order to obtain reliable results. It is infact the case that by reducing the time step to a level below the response time of the cells the dynamics of the system varied significantly. It is questionable whether it would be possible to adjust some of the population parameters within reson to obtain the same results with a smaller step size, but the following points should be taken into account: 1) other researchers have created similar models that show such cyclic behavior (see for example Silverman, Shaw and Pearson[7]). 2) biological data exists which would indicate the existance of cyclic or periodic bahvior in real neural systems (see for instance Baird[1]). As I just recently completed a series of studies at this university, I will not be able to perform a detailed examination of the system described here, but instead I will more 299 than likely create new models on different research equipment which will be geared more specifically towards the study of temporal behavior in neural networks. OTHER MODELS It should be noted that in the past few years some researchers have begun investigating the possibility of neural networks that can exhibit time-dependent behavior, and I would like to report on some of the available results as they relate to the topic of temporal patterns. Baird[l] reports findings from the rabbit's olfctory bulb which indicate the existance of phase-locked oscillatory states corresponding to olfactory stimuli presented to the subjects. He outlines an elegant model which attributes pattern recognition abilities to competing instabilities in the dynamic activity of neural structures. He further speculates that inhomogeneous connectivity in the bulb can be selectively modified to achieve input-sensitive oscillatory states. Silverman, Shaw and Pearson[7] have developed a model based on a biologically-inspired idealized neural structure, which they call the trion. This unit represents a localized group of neurons with a discrete firing period. It was found that small ensembles of trions with symmetric connections can exhibit quasi-stable periodic firing patterns which do not require pacemakers or external driving. Their results are inspired by existing physiological data and are consistent with other works. Kleinfeld[6], and Sompolinsky and Kanter[8] independently developed neural network models that can generate and recognize sequential or cyclic patterns. Both models rely on what could be summarized as the recirculation of information through time-delayed channels. Very similar results are presented by Jordan[4] who extends a typical connectionist or PDP model to include state and plan units with recurrent connections and feedback from output units through hidden units. He employs supervised learning with fuzzy constraints to induce learning of sequences in the system. From a slightly different approach, Tank and Hopfield[9] make USe of patterned sets of delays which effectively compress information in time. They develop a model which recognizes patterns by falling into local minima of a state-space energy function. They suggest that a systematic selection of delay functions can be done which will allow for time distortions that would be likely to occur in the input. Finally, a somewhat different approach is taken by Homma, Atlas and Marks[5], who generalize a network for spatial pattern recognition to one that performs spatio-temporal patterns by extending classical principles from spatial networks to dynamic networks. In particular, they replace multiplication with convolution, weights with transfer functions, and thresholding with non linear transforms. Hebbian and Delta learning rules are similarly generalized. The resulting models are able to perform temporal pattern recognition. The above is only a partial list of some of the relevant work in this field, and there are probably various other results I am not aware of. DISCUSSION All of the above results indicate the importance of temporal patterns in neural networks. The need is apparent for further formal models which can successfully quantify temporal behavior in neural networks. Several questions must be answered to further 300 clarify the role and meaning of temporal patterns in neural nets. For instance, there is an apparent difference between a model that performs sequential tasks and one that performs recognition of dynamic patterns. It seems that appropriate selection of delay mechanisms will be necessary to account for many types of temporal pattern recognition. The question of scaling must also be explored: mechanism are known to exist in the brain which can cause delays ranging from the millisecond-range (e.g. variations in synaptic cleft size) to the tenth of a second range (e.g. axonal transmission times). On the other hand, the brain is capable of rec"Ignizing sequences of stimuli that can be much longer than the typical neural event, such as for instance being able to remember a song in its entirety. These and other questions could lead to interesting new aspects of brain function which are presently unclear. References [1] Baird, B., "Nonlinear Dynamics of Pattern Formation and Pattern Recognition in the Rabbit Olfactory Bulb". Physica 22D, 150-175. 1986. [2] Gaudiano, P., "Computer Models of Neural Networks". Unpublished Master's Thesis. University of Colorado. 1987. [3] Gaudiano, P., MacGregor, R.J., "Dynamic Activity and Memory Traces in Computer-Simulated Recurrently-Connected Neural Networks". Proceedings of the First International Conference on Neural Networks. 2:177-185. 1987. [4] Jordan, M.I., "Attractor Dynamics and Parallelism in a Connectionist Sequential Machine". Proceedings of the Eighth Annual Conference of the Cognitive Sciences Society. 1986. [5] Homma, T., Atlas, L.E., Marks, R.J.II, "An Artificial Neural Network for SpatioTemporal Bipolar Patterns: Application to Phoneme Classification". To appear in proceedings of Neural Information Processing Systems Conference (AlP). 1987. [6] Kleinfeld, D., "Sequential State Generation by Model Neural Networks". Proc. Natl. Acad. Sci. USA. 83: 9469-9473. 1986. [7] Silverman, D.l., Shaw, G.L., Pearson, l.C. "Associative Recall Properties of the Trion Model of Cortical Organization". Biol. Cybern. 53:259-271. 1986. [8] Sompolinsky, H., Kanter, I. "Temporal Association in Asymmetric Neural Networks". Phys. Rev. Let. 57:2861-2864. 1986. [9] Tank, D.W., Hopfield, l.l. "Neural Computation by Concentrating Information in Time". Proc. Natl. Acad. Sci. USA. 84:1896-1900. 1987.
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Network Generality, Training Required, and PrecisIon Required John S. Denker and Ben S. Wittner 1 AT&T Bell Laboratories Holmdel, New Jersey 07733 219 Keep your hand on your wallet. Leon Cooper, 1987 Abstract We show how to estimate (1) the number of functions that can be implemented by a particular network architecture, (2) how much analog precision is needed in the connections in the network, and (3) the number of training examples the network must see before it can be expected to form reliable generalizations. Generality versus Training Data Required Consider the following objectives: First, the network should be very powerful and versatile, i.e., it should implement any function (truth table) you like, and secondly, it should learn easily, forming meaningful generalizations from a small number of training examples. Well, it is information-theoretically impossible to create such a network. We will present here a simplified argument; a more complete and sophisticated version can be found in Denker et al. (1987). It is customary to regard learning as a dynamical process: adjusting the weights (etc.) in a single network. In order to derive the results of this paper, however, we take a different viewpoint, which we call the ensemble viewpoint. Imagine making a very large number of replicas of the network. Each replica has the same architecture as the original, but the weights are set differently in each case. No further adjustment takes place; the "learning process" consists of winnowing the ensemble of replicas, searching for the one( s) that satisfy our requirements. Training proceeds as follows: We present each item in the training set to every network in the ensemble. That is, we use the abscissa of the training pattern as input to the network, and compare the ordinate of the training pattern to see if it agrees with the actual output of the network. For each network, we keep a score reflecting how many times (and how badly) it disagreed with a training item. Networks with the lowest score are the ones that agree best with the training data. If we had complete confidence in lCurrently at NYNEX Science and Technology, 500 Westchester Ave., White Plains, NY 10604 @) American Institute of Physics 1988 220 the reliability of the training set, we could at each step simply throwaway all networks that disagree. For definiteness, let us consider a typical network architecture, with No input wires and Nt units in each processing layer I, for I E {I·· ·L}. For simplicity we assume NL = 1. We recognize the importance of networks with continuous-valued inputs and outputs, but we will concentrate for now on training (and testing) patterns that are discrete, with N == No bits of abscissa and N L = 1 bit of ordinate. This allows us to classify the networks into bins according to what Boolean input-output relation they implement, and simply consider the ensemble of bins. There are 22N jossible bins. If the network architecture is completely general and powerful, all 22 functions will exist in the ensemble of bins. On average, one expects that each training item will throwaway at most half of the bins. Assuming maximal efficiency, if m training items are used, then when m ~ 2N there will be only one bin remaining, and that must be the unique function that consistently describes all the data. But there are only 2N possible abscissas using N bits. Therefore a truly general network cannot possibly exhibit meaningful generalization 100% of the possible data is needed for training. N ow suppose that the network is not completely general, so that even with all possible settings of the weights we can only create functions in 250 bins, where So < 2N. We call So the initial entropy of the network. A more formal and general definition is given in Denker et al. (1987). Once again, we can use the training data to winnow the ensemble, and when m ~ So, there will be only one remaining bin. That function will presumably generalize correctly to the remaining 2N - m possible patterns. Certainly that function is the best we can do with the network architecture and the training data we were given. The usual problem with automatic learning is this: If the network is too general, So will be large, and an inordinate amount of training data will be required. The required amount of data may be simply unavailable, or it may be so large that training would be prohibitively time-consuming. The shows the critical importance of building a network that is not more general than necessary. Estimating the Entropy In real engineering situations, it is important to be able to estimate the initial entropy of various proposed designs, since that determines the amount of training data that will be required. Calculating So directly from the definition is prohibitively difficult, but we can use the definition to derive useful approximate expressions. (You wouldn't want to calculate the thermodynamic entropy of a bucket of water directly from the definition, either. ) 221 Suppose that the weights in the network at each connection i were not continuously adjustable real numbers, but rather were specified by a discrete code with bi bits. Then the total number of bits required to specify the configuration of the network is (1) Now the total number offunctions that could possibly be implemented by such a network architecture would be at most 2B. The actual number will always be smaller than this, since there are various ways in which different settings of the weights can lead to identical functions (bins). For one thing, for each hidden layer 1 E {1··· L-1}, the numbering of the hidden units can be permuted, and the polarity of the hidden units can be flipped, which means that 250 is less than 2B by a factor (among others) of III Nl! 2N,. In addition, if there is an inordinately large number of bits bi at each connection, there will be many settings where small changes in the connection will be immaterial. This will make 2so smaller by an additional factor. We expect aSO/abi ~ 1 when bi is small, and aSO/abi ~ 0 when bi is large; we must now figure out where the crossover occurs. The number of "useful and significant" bits of precision, which we designate b*, typically scales like the logarithm of number of connections to the unit in question. This can be understood as follows: suppose there are N connections into a given unit, and an input signal to that unit of some size A is observed to be significant (the exact value of A drops out of the present calculation). Then there is no point in having a weight with magnitude much larger than A, nor much smaller than A/N. That is, the dynamic range should be comparable to the number of connections. (This argument is not exact, and it is easy to devise exceptions, but the conclusion remains useful.) If only a fraction 1/ S of the units in the previous layer are active (nonzero) at a time, the needed dynamic range is reduced. This implies b* ~ log(N/S). Note: our calculation does not involve the dynamics of the learning process. Some numerical methods (including versions of back propagation) commonly require a number of temporary "guard bits" on each weight, as pointed out by llichard Durbin (private communication). Another log N bits ought to suffice. These bits are not needed after learning is complete, and do not contribute to So. If we combine these ideas and apply them to a network with N units in each layer, fully connected, we arrive at the following expression for the number of different Boolean functions that can be implemented by such a network: (2) where B ~ LN2 log N (3) These results depend on the fact that we are considering only a very restricted type of processing unit: the output is a monotone function of a weighted sum of inputs. Cover 222 (1965) discussed in considerable depth the capabilities of such units. Valiant (1986) has explored the learning capabilities of various models of computation. Abu-Mustafa has emphasized the principles of information and entropy and applied them to measuring the properties of the training set. At this conference, formulas similar to equation 3 arose in the work of Baum, Psaltis, and Venkatesh, in the context of calculating the number of different training patterns a network should be able to memorize. We originally proposed equation 2 as an estimate of the number of patterns the network would have to memorize before it could form a reliable generalization. The basic idea, which has numerous consequences, is to estimate the number of (bins of) networks that can be realized. References 1. Vasser Abu-Mustafa, these proceedings. 2. Eric Baum, these proceedings. 3. T. M. Cover, "Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition," IEEE Trans. Elec. Comp., EC-14, 326-334, (June 1965) 4. John Denker, Daniel Schwartz, Ben Wittner, Sara Solla, John Hopfield, Richard Howard, and Lawrence Jackel, Complex Systems, in press (1987). 5. Demetri Psaltis, these proceedings. 6. 1. G. Valiant, SIAM J. Comput. 15(2), 531 (1986), and references therein. 7. Santosh Venkatesh, these proceedings.
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HIGH ORDER NEURAL NETWORKS FOR EFFICIENT ASSOCIATIVE MEMORY DESIGN I. GUYON·, L. PERSONNAZ·, J. P. NADAL·· and G. DREYFUS· 233 • Ecole Superieure de Physique et de Chimie Industrielles de la Ville de Paris Laboratoire d'Electronique 10, rue Vauquelin 75005 Paris (France) •• Ecole Normale Superieure Groupe de Physique des Solides 24, rue Lhomond 75005 Paris (France) ABSTRACT We propose learning rules for recurrent neural networks with high-order interactions between some or all neurons. The designed networks exhibit the desired associative memory function: perfect storage and retrieval of pieces of information and/or sequences of information of any complexity. INTRODUCTION In the field of information processing, an important class of potential applications of neural networks arises from their ability to perform as associative memories. Since the publication of J. Hopfield's seminal paper 1, investigations of the storage and retrieval properties of recurrent networks have led to a deep understanding of their properties. The basic limitations of these networks are the following: - their storage capacity is of the order of the number of neurons; - they are unable to handle structured problems; - they are unable to classify non-linearly separable data. © American Institute of Physics 1988 234 In order to circumvent these limitations, one has to introduce additional non-linearities. This can be done either by using "hidden", non-linear units, or by considering multi-neuron interactions2. This paper presents learning rules for networks with multiple interactions, allowing the storage and retrieval, either of static pieces of information (autoassociative memory), or of temporal sequences (associative memory), while preventing an explosive growth of the number of synaptic coefficients. AUTOASSOCIATIVE MEMORY The problem that will be addressed in this paragraph is how to design an autoassociative memory with a recurrent (or feedback) neural network when the number p of prototypes is large as compared to the number n of neurons. We consider a network of n binary neurons, operating in a synchronous mode, with period t. The state of neuron i at time t is denoted by (Ji(t), and the state of the network at time t is represented by a vector ~(t) whose components are the (Ji(t). The dynamics of each neuron is governed by the following relation: (Ji(t+t) = sgn vi(t). (1 ) In networks with two-neuron interactions only, the potential vi(t) is a linear function of the state of the network: For autoassociative memory design, it has been shown3 that any set of correlated patterns, up to a number of patterns p equal to 2n, can be made the stable states of the system, provided the synaptic matrix is computed as the orthogonal projection matrix onto the subspace spanned by the stored vectors. However, as p increases, the rank of the family of prototype vectors will increase, and finally reach the value of n. In such a case, the synaptic matrix reduces to the identity matrix, so that all 2n states are stable and the energy landscape becomes flat. Even if such an extreme case is avoided, the attractivity of the stored states decreases with increasing p, or, in other terms, 235 the number of fixed points which are not the stored patterns increases; this problem can be alleviated to a large extent by making a useful use of these "spurious" fixed points4. Another possible solution consists in "gardening" the state space in order to enlarge the basins of attraction of the fixed points5. Anyway, no dramatic improvements are provided by all these solutions since the storage capacity is always O(n). We now show that the introduction of high-order interactions between neurons, increases the storage capacity proportionally to the number of connections per neuron. The dynamical behaviour of neuron i is still governed by (1). We consider two and three-neuron interactions, extension to higher order are straightforward. The potential vi (t) is now defi ned as It is more convenient, for the derivation of learning rules, to write the potential in the matrix form: ~(t) = C ;t(t), where :¥(t) is an m dimensional vector whose components are taken among the set of the (n2+n)/2 values: a1 , ... , an' a1 a2 , ... , aj al ' ... , an-1 an. As in the case of the two-neuron interactions model, we want to compute the interaction coefficients so that the prototypes are stable and attractor states. A condition to store a set of states Q:k (k=1 to p) is that y'k= Q:k for all k. Among the solutions, the most convenient solution is given by the (n,m) matrix c=I,rl (2) where I, is the (n,p) matrix whose columns are the Q:k and rl is the (p,m) pseudoinverse of the (m,p) matrix r whose columns are the {. This solution satisfies the above requirements, up to a storage capacity which is related to the dimension m of vectors :¥. Thus, in a network with three-neuron 236 interactions, the number of patterns that can be stored is O(n2). Details on these derivations are published in Ref.6. By using only a subset of the products {aj all. the increase in the number of synaptic coefficients can remain within acceptable limits, while the attractivity of the stored patterns is enhanced, even though their number exceeds the number of neurons ; this will be examplified in the simulations presented below. Finally, it can be noticed that, if vector ~contains all the {ai aj}' i=1, ... n, j=1, ... n, only, the computation of the vector potential ~=C~can be performed after the following expression: where ~ stands for the operation which consists in squaring all the matrix coefficients. Hence, the computation of the synaptic coefficients is avoided, memory and computing time are saved if the simulations are performed on a conventional computer. This formulation is also meaningful for optical implementations, the function ell being easily performed in optics 7. In order to illustrate the capabilities of the learning rule, we have performed numerical simulations which show the increase of the size of the basins of attraction when second-order interactions, in addition to the first-order ones, are used. The simulations were carried out as follows. The number of neurons n being fixed, the amount of second-order interactions was chosen ; p prototype patterns were picked randomly, their components being ±1 with probability 0.5 ; the second-order interactions were chosen randomly. The synaptic matrix was computed from relation (2). The neural network was forced into an initial state lying at an initial Hamming distance Hi from one of the prototypes {!k ; it was subsequently left to evolve until it reached a stable state at a distance Hf from {!k. This procedure was repeated many times for each prototype and the Hf were averaged over all the tests and all the prototypes. Figures 1 a. and 1 b. are charts of the mean values of Hf as a function of the number of prototypes, for n = 30 and for various values of m (the dimension of 237 vector ':/.). These curves allowed us to determine the maximum number of prototype states which can be stored for a given quality of recall. Perfect recall implies Hf =0 ; when the number of prototypes increases, the error in recall may reach Hf =H i : the associative memory is degenerate. The results obtained for Hi In =10% are plotted on Figure 1 a. When no high-order interactions were used, Hf reached Hi for pIn = 1, as expected; conversely, virtually no error in recall occured up to pIn = 2 when all second-order interactions were taken into account (m=465). Figure 1 b shows the same quantities for Hi=200/0 ; since the initial states were more distant from the prototypes, the errors in recall were more severe. 1.2 1.2 1.0 1.0 0.8 0.8 :f :f A 0.6 A 0.6 f f v 0.4 (8) v (b) 0.4 0.2 0.2 0.0 0.0 0 2 3 0 2 3 pIn pIn Fig. 1. Improvement of the attractivity by addition of three-neuron interactions to the two-neuron interactions. All prototypes are always stored exactly (all curves go through the origin). Each point corresponds to an average over min(p,10) prototypes and 30 tests for each prototype. [] Projection: m = n = 30; • m = 120 ; • m = 180; 0 m = 465 (all interactions) 1 a: Hi I n =10% ; 1 b : Hi In =20%. TEMPORAL SEQUENCES (ASSOCIATIVE MEMORY) The previous section was devoted to the storage and retrieval of items of information considered as fixed points of the dynamics of the network (autoassociative memory design). However, since fully connected neural networks are basically dynamical systems, they are natural candidates for 238 storing and retrieving information which is dynamical in nature, i.e., temporal sequences of patterns8: In this section, we propose a general solution to the problem of storing and retrieving sequences of arbitrary complexity, in recurrent networks with parallel dynamics. Sequences consist in sets of transitions between states {lk_> Q:k+ 1, k=1, ... , p. A sufficient condition to store these sets of transitions is that y..~ Q:k+ 1 for all k. In the case of a linear potential y"=C Q:, the storage prescription proposed in ref.3 can be used: C=r,+r,I, where r, is a matrix whose columns are the Q:k and r,+ is the matrix whose columns are the successors Q:k+ 1 of Q:k. If P is larger than n, one can use high-order interactions, which leads to introduce a non-linear potential Y..=C ';f. , with ';f. as previously defined. We proposed in ref. 1 0 the following storage prescription : (3) The two above prescriptions are only valid for storing simple sequences, where no patterns occur twice (or more). Suppose that one pattern occurs twice; when the network reaches this bifurcation point, it is unable to make a decision according the deterministic dynamics described in (1), since the knowledge of the present state is not sufficient. Thus, complex sequences require to keep, at each time step of the dynamics, a non-zero memory span. The vector potential Y..=C':J. must involve the states at time t and t-t, which leads to define the vector ';f. as a concatenation of vectors Q:(t), ~(t-t), Q:(t)®Q:(t), Q:(t)®Q:(t-t), or a suitable subset thereof. The subsequent vector Q:(t+t) is still determined by relation (1). In this form, the problem is a generalization of the storage of patterns with high order interactions, as described above. The storage of sequences can be still processed by relation (3). The solution presented above has the following features: i) Sequences with bifurcation points can be stored and retrieved. ii) The dimension of the synaptic matrix is at most (n,2(n2+n)), and at least (n,2n) in the linear case, so that at most 2n(n2+n) and at least 2n2 synapses are required. 239 iii) The storage capacity is O(m), where m is the dimension of the vector ';t . iv) Retrieval of a sequence requires initializing the network with two states in succession. The example of Figure 2 illustrates the retrieval performances of the latter learning rule. We have limited vector ';t to Q:(t}®Q:(t-t). In a network of n=48 neurons, a large number of poems have been stored, with a total of p=424 elementary transitions. Each state is consists in the 6 bit codes of 8 letters. ALOUETTE JETE JE NE PLUMERAI OLVMERAI ALOUETTE AQFUETTE GENTILLE JEHKILLE ALOUETTE SLOUETTE ALOUETTE ALOUETTE JETE JETE PLUMERAI PLUMERAI Fig. 2. One of the stored poems is shown in the first column. The network is initialized with two states (the first two lines of the second column). After a few steps, the network reaches the nearest stored sequence. LOCAL LEARNING Finally, it should be mentioned that all the synaptic matrices introduced in this paper can be computed by iterative, local learning rules. For autoassociative memory, it has been shown analytically9 that the procedure: with Cij(O) = 0, which is a Widrow-Hoff type learning rule, yields the projection matrix, when 240 the number of presentations of the prototypes {~k} goes to infinity, if the latter are linearly independent. A derivation along the same lines shows that, by repeated presentations of the prototype transitions, the learning rules: lead to the exact solutions (relations (2) and (3) respectively), if the vectors }< are Ii nearly independent. GENERALIZATION TASKS Apart from storing and retrieving static pieces of information or sequences, neural networks can be used to solve problems in which there exists a structure or regularity in the sample patterns (for example presence of clumps, parity, symmetry ... ) that the network must discover. Feed-forward networks with multiple layers of first-order neurons can be trained with back-propagation algorithms for these purposes; however, one-layer feed-forward networks with mUlti-neuron interactions provide an interesting alternative. For instance, a proper choice of vector ':I. (second-order terms only) with the above learning rule yields a perfectly straightforward solution to the exclusive-OR problem. Maxwell et al. have shown that a suitable high-order neuron is able to exhibit the "ad hoc network solution" for the contiguity problem 11. CONCLUSION The use of neural networks with high-order interactions has long been advocated as a natural way to overcome the various limitations of the Hopfield model. However, no procedure guaranteed to store any set of information as fixed points or as temporal sequences had been proposed. The purpose of the present paper is to present briefly such storage prescriptions and show 241 some illustrations of the use of these methods. Full derivations and extensions will be published in more detailed papers. REFERENCES 1. J. J. Hopfield, Proc. Natl. Acad. Sci. (USA) la, 2554 (1982). 2. P. Peretto and J. J. Niez, BioI. Cybern. M, 53 (1986). P. Baldi and S. S. Venkatesh, Phys. Rev. Lett . .5.6 , 913 (1987). For more references see ref.6. 3. L. Personnaz, I. Guyon, G. Dreyfus, J. Phys. Lett. ~ , 359 (1985). L. Personnaz, I. Guyon, G. Dreyfus, Phys. Rev. A ~ , 4217 (1986). 4. I. Guyon, L. Personnaz, G. Dreyfus, in "Neural Computers", R. Eckmiller and C. von der Malsburg eds (Springer, 1988). 5. E. Gardner, Europhys. Lett . .1, 481 (1987). G. Poppel and U.Krey, Europhys. Lett.,.1, 979 (1987). 6. L. Personnaz, I. Guyon, G. Dreyfus, Europhys. Lett. .1,863 (1987). 7. D. Psaltis and C. H. Park, in "Neural Networks for Computing", J. S. Denker ed., (A.I.P. Conference Proceedings 151, 1986). 8. P. Peretto, J. J. Niez, in "Disordered Systems and Biological Organization", E. Bienenstock, F. Fogelman, G. Weisbush eds (Springer, Berlin 1986). S. Dehaene, J. P. Changeux, J. P. Nadal, PNAS (USA)~, 2727 (1987). D. Kleinfeld, H. Sompolinsky, preprint 1987. J. Keeler, to appear in J. Cog. Sci. For more references see ref. 9. 9. I. Guyon, L. Personnaz, J.P. Nadal and G. Dreyfus, submitted for publication. 10. S. Diederich, M. Opper, Phys. Rev. Lett . .5.6, 949 (1987). 11. T. Maxwell, C. Lee Giles, Y. C. Lee, Proceedings of ICNN-87, San Diego, 1987.
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642 LEARNING BY ST ATE RECURRENCE DETECfION Bruce E. Rosen, James M. Goodwint, and Jacques J. Vidal University of California, Los Angeles, Ca. 90024 ABSTRACT This research investigates a new technique for unsupervised learning of nonlinear control problems. The approach is applied both to Michie and Chambers BOXES algorithm and to Barto, Sutton and Anderson's extension, the ASE/ACE system, and has significantly improved the convergence rate of stochastically based learning automata. Recurrence learning is a new nonlinear reward-penalty algorithm. It exploits information found during learning trials to reinforce decisions resulting in the recurrence of nonfailing states. Recurrence learning applies positive reinforcement during the exploration of the search space, whereas in the BOXES or ASE algorithms, only negative weight reinforcement is applied, and then only on failure. Simulation results show that the added information from recurrence learning increases the learning rate. Our empirical results show that recurrence learning is faster than both basic failure driven learning and failure prediction methods. Although recurrence learning has only been tested in failure driven experiments, there are goal directed learning applications where detection of recurring oscillations may provide useful information that reduces the learning time by applying negative, instead of positive reinforcement. Detection of cycles provides a heuristic to improve the balance between evidence gathering and goal directed search. INTRODUCflON This research investigates a new technique for unsupervised learning of nonlinear con trol problems with delayed feedback. Our approach is compared to both Michie and Chambers BOXES algorithml, to the extension by Barto, et aI., the ASE (Adaptive Search Element) and to their ASE/ACE (Adaptive Critic Element) system2, and shows an improved learning time for stochastically based learning automata in failure driven tasks. We consider adaptively controlling the behavior of a system which passes through a sequence of states due to its internal dynamics (which are not assumed to be known a priori) and due to choices of actions made in visited states. Such an adaptive controller is often referred to as a learning automaton. The decisions can be deterministic or can be made according to a stochastic rule. A learning automaton has to discover which action is best in each circumstance by producing actions and observing the resulting information. This paper was motivated by the previous work of Barto, et al. to investigate neuronlike adaptive elements that affect and learn from their environment. We were inspired by their current work and the recent attention to neural networks and connectionist systems, and have chosen to use the cart-pole control problem2, to enable a comparison of our results with theirs . ... ! Permanent address: California State University, Stanislaus; Turlock, California. @ American Institute of Physics 1988 643 THE CART ·POLE PROBLEM In their work on the cart-pole problem, Barto, Sutton and Anderson considered a learning system composed of an automaton interacting with an environment. The problem requires the automaton to balance a pole acting as an inverted pendulum hinged on a moveable cart. The cart travels left or right along a bounded one dimensional track; the pole may swing to the left or right about a pivot attached to the cart. The automaton must learn to keep the pole balanced on the cart, and to keep the cart within the bounds of the track. The parameters of the cart/pole system are the cart po~ition and velocity, and the pole angle and angular velocity. The only actions available to the automaton are the applications of a fixed impulsive force to the cart in either right or left direction; one of these actions must be taken. This balancing is an extremely difficult problem if there is no a priori knowledge of the system dynamics, if these dynamics change with time, or if there is no preexisting controller that can be imitated (e.g. Widrow and Smith's3 ADALINE controller). We assumed no a priori knowledge of the dynamics nor any preexisting controller and anticipate that the system will be able to deal with any changing dynamics. Numerical simulations of the cart-pole solution via recurrence learning show substantial improvement over the results of Barto et aI., and of Michie and Chambers, as is shown in figure 1. The algorithms used, and the results shown in figure 1, will be discussed in detail below. 500000 T'unc Until Fai1\R 100000 ~-----------------------------25 so 75 110 Trial No. Figure 1: Perfonnance of the ASE, ASE/ACE, Constant Recurrence (HI) and Shon Recurrence (H2) Algorithms. THE GENERAL PROBLEM: ASSIGNMENT OF CREDIT The cart-pole problem is one of a class of problems known as "credit assignment"4, and in particular temporal credit assignment. The recurrence learning algorithm is an approach to the general temporal credit assignment problem. It is characterized by seeking to improve learning by making decisions about early actions. The goal is to find actions responsible for improved or degraded perfonnance at a much later time. An example is the bucket brigade algorithmS. This is designed to assign credit to rules in the system according to their overall usefulness in attaining their goals. This is done by adjusting the strength value (weight) of each rule. The problem is of modifying these strengths is to permit rules activated early in the sequence to result in successful actions later. 644 Samuels considered the credit assignment problem for his checkers playing program6. He noted that it is easy enough to credit the rules that combine to produce a triple jump at some point in the game; it is much harder to decide which rules active earlier were responsible for changes that made the later jump possible. State recurrence learning assigns a strength to an individual rule or action and modifies that action's strength (while the system accumulates experience) on the basis of the action's overall usefulness in the situations in which it has been invoked. In this it follows the bucket brigade paradigm of Holland. PREVIOUS WORK The problems of learning to control dynamical systems have been studied in the past by Widrow and Smith3, Michie and Chambers!, Barto, Sutton, and Anderson2, and Conne1l7. Although different approaches have been taken and have achieved varying degrees of success, each investigator used the cart/pole problem as the basis for empirically measuring how well their algorithms work. Michie and Chambersl built BOXES, a program that learned to balance a pole on a cart. The BOXES algorithm choose an action that had the highest average time until failure. After 600 trials (a trial is a run ending in eventual failure or by some time limit expiration), the program was able to balance the pole for 72,000 time steps. Figure 2a describes the BOXES learning algorithm. States are penalized (after a system failure) according to recency. Active states immediately preceding a system failure are punished most. Barto, Sutton and Anderson2 used two neuronlike adaptive elements to solve the control problem. Their ASE/ACE algorithm chose the action with the highest probability of keeping the pole balanced in the region, and was able to balance the pole for over 60,000 time steps before completion of the lOOth trial. Figure 2a and 2b: The BOXES and ASE/ACE (Associative Search Elelement Adpative Critic Element) algorithms Figure 2a shows the BOXES (and ASE) learning algorithm paradigm When the automaton enters a failure state (C), all states that it has traversed (shaded rectangles) are punished, although state B is punished more than state A. (Failure states are those at the edges of the diagram.) Figure 2b describes the ASE/ACE learning algorithm. If a system failure occurs before a state's expected failure time, the state is penalized. If a system failure occurs after its expected failure time, the state is rewarded. State A is penalized because a failure occurred at B sooner than expected. State A's expected 645 failure time is the time for the automaton to traverse from state A to failure point C. When leaving state A, the weights are updated if the new state's expected failure time differs from that of state A. Anderson8 used a connectionist system to learn to balance the pole. Unlike the previous experiments, the system did provide well-chosen states a priori. On the average, 10,000 trials were necessary to learn to balance the pole for 7000 time steps. Connell and Utgoff'7 developed an approach that did not depend on partitioning the state space into discrete regions. They used Shepard's function9,l0 to interpolate the degree of desirability of a cart-pole state. The system learned the control task after 16 trials. However, their system used a knowledge representation that had a priori information about the system. O'n-lER RELATED WORK Klopfll proposed a more neurological class of differential learning mechanisms that correlates earlier changes of inputs with later changes of outputs. The adaptation formula used multiplies the change in outputs by the weighted sum of the absolute value of the t previous inputs weights (~Wj)' the t previous differences in inputs (~Xj)' and the t previous time coefficients (c/ Sutton's temporal differences (TD)12 approach is one of a class of adaptive prediction methods. Elements of this class use the sum of previously predicted output values multiplied by the gradient and an exponentially decaying coefficient to modify the weights. Barto and Sutton 13 used temporal differences as the underlying learning procedure for classical conditioning. THERECURRENCELE~G~HOD DEFINITIONS A state is the set of values (or ranges) of parameters sufficient to specify the instantaneous condition of the system. The input decoder groups the environmental states into equivalence classes: elements of one class have identical system responses. Every environmental input is mapped into one of n input states. (All further references to "states" assumes that the input values fall into the discrete ranges determined by the decoder, unless otherwise specified. ) States returned to after visiting one or more alternate states recur. An action causes the modification of system parameters, which may change the system state. However, no change of state need occur, since the altered parameter values may be decoded within the same ranges. A weight, wet), is associated with each action for each state, with the probability of an allowed action dependent on the current value of its weight. A rule determines which of the allowable actions is taken. The rule is not deterministic. It chooses an action stochastically, based on the weights. Weight changes, ~w(t), are made to reduce the likelihood of choosing an action which will cause an eventual failure. These changes are made based on the idea that the previous action of an element, when presented with input x(t), had some influence in causing a similar pattern to occur again. Thus, weight changes are made to increase the likelihood that an element produces the same action f(t) when patterns similar to x(t) occur in the future. 646 For example, consider the classic problem of balancing a pole on a moving cart. The state is specified by the positions and velocities of both the cart and the pole. The allowable actions are fixed velocity increments to the right or to the left, and the rule determines which action to take, based on the current weights. THE ALGORITHM The recurrence learning algorithm presented here is a nonlinear reward-penalty method 14. Empirical results show that it is successful for stationary environments. In contrast to other methods, it also may be applicable to nonstationary environments'. Our efforts have been to develop algorithms that reward decision choices that lead the controller/environment to quasi-stable cycles that avoid failure (such as limit cycles, converging oscillations and absorbing points). Our technique exploits recurrence information obtained during learning trials. The system is rewarded upon return to a previous state, however weight changes are only permitted when a state transition occurs. If the system returns to a state, it has avoided failure. A recurring state is rewarded. A sequence of recurring states can be viewed as evidence for a (possibly unstable) cycle. The algorithm forms temporal "cause and effect" associations. To optimize performance, dynamic search techniques must balance between choosing a search path with known solution costs, and exploring new areas of the search space to find better or cheaper solutions. This is known as the two armed bandit problem l5, i.e. given a two handed slot machine with one arm's observed reward probabilities higher than the other, one should not exclude playing with the arm with the lesser payoff. Like the ASE/ACE system, recurrence learning learns while searching, in contrast to the BOXES and ASE algorithms which learn only upon failure. RANGE DECODING In our work, as in Barto and others, the real valued input parameters are analyzed as members of ranges. This reduces computing resource demands. Only a limited number of ranges are allowed for each parameter. It is possible for these ranges to overlap, although this aspect of range decoding is not discussed in this paper, and the ranges were considered nonoverlapping. When the parameter value falls into one of the ranges that range is active. The specification of a state consists of one of the active ranges for each of the parameters. If the ranges do not overlap, then the set of parameter values specify one unique state; otherwise the set of parameter values may specify several states. Thus, the parameter values at any time determine one or several active states Si from the set of n possible states. The value of each environmental parameter falls into one of a number of ranges, which may be different for different parameters. A state is specified by the active range for each parameter. The set of input parameter values are decoded into one (or more) of n ranges Si' 0<= i <= n. For this problem, boolean values are used to describe the activity level of a state Si. The activity value of a state is 1 if the state is active, or 0 if it is inactive. ACfION DECISIONS Our model is the same as that of the BOXES and ASE/ACE systems, where only one input (and state) is active at any given time. All states were nonoverlapping and mutually exclusive, although there was no reason to preclude them from overlapping 647 other than for consistency with the two previous models. In the ASE/ACE system and in ours as well, the output decision rule for the controller is based on the weighted sum of its inputs plus some stochastic noise. The action (output) decision of the controller is either + 1 or -1, as given by: ( 1) where f( ) = [+ 1 .i f z ~ 0 ] z -llfz<O (2) and noise is a real randomly (Gaussian) distributed value with some mean 11 and variance 0'2. An output, fez), for the car/pole controller is interpreted as a push to the left if fez) = -lor to the right if fez) = + 1. RECURRENCELE~G The goal of the recurrence learning algorithm is to avoid failure by moving toward states that are part of cycles if such states exist, or quasi-stable oscillations if they don't. This concept can be compared to juggling. As long as all the balls are in the air, the juggler is judged a success and rewarded. No consideration is given to whether the balls are thrown high or low, left or right; the controller, like the juggler, tries for the most stable cycles. Optimum performance is not demanded from recurrence learning. Two heuristics have been devised. The fundamental basis of each of them is to reward a state for being repeatedly visited (or repeatedly activated). The first heuristic is to reward a state when it is revisited, as part of a cycle in which no failure had occurred. The second heuristic augments the first by giving more reward to states which panicipate in shorter cycles. These heuristics are discussed below in detail. HEURISTIC HI: If a state has been visited more than once during one trial, reward it by reinforcing its weight. RATIONALE This heuristic assumes that states that are visited more than once have been part of a cycle in which no failure had occurred. The action taken in the previous visit is assumed to have had some influence on the recurrence. By reinforcing a weight upon state revisitation, it is assumed to increase the likelihood that the cycle will occur again. No assumptions are made as to whether other states were responsible for the cycle. RESTRICfION An action may not immediately cause the environment to change to a different state. There may be some delay before a transition, since small changes of parameters may be decoded into the same input ranges, and hence the same state. This inertia is incorporated into the heuristics. When the same state appears twice in succession, its weight is not reinforced, since that would assume that the action, rather than inertia, directly caused the state's immediate recurrence. 648 THE RECURRENCE EQUATIONS The recurrence learning equations stem in part from the weight alteration formula used in the ASE system. The weight of a state is a sum of its previous weight, and the product of the learning rate (a), the reward (r), and the state's eligibility (e). ret) E {-I,O} (3) The eligibility index e/t) is an exponentially decaying trace function. (4) where O<=P<=I, Xi E {0,1}, and Yi E {-I,I}. The output value Yi is the last output decision, and P determines the decay rate. The reward function is: { -1 ret) = ° when the system fails at time t } otherwise REINFORCEMENT OF CYCLES (5) Equations (1) through (5) describe the basic ASE system. Our algorithm extends the weight updating procedure as follows: (6) The term ar(t)ei(t) is the same as in (3), providing failure reinforcement. The term a2r2(t)e2,i(t) provides reinforcement for success. When state i is eligible (by virtue of Xi > 0), there is a weight change by the amount: CXz multiplied by the reward value, r2(t), and the current eligibility e2,i(t). For simplicity, the reward value, r2(t), may be taken to be some positive constant, although it need not be; any environmental feedback, yielding a reinforcement value as a function of time could be used instead. The second eligibility function e2,i(t) yields one of three constant values for HI: -P2' 0, or P2 according to formula (7) below: if t-ti,last = 1 or ti,last = ° } otherwise (7) where ti,last is the last time that state was active. If a state has not previously been active (i.e. xi(t) = ° for all t) then ti,last=O. As the formula shows, e2,i(t) = ° if the state has not been previously visited or if no state transition occurred in the last time step; otherwise, e2,i(t) = P2Xj(t)y(ti,last)· The direction (increase or decrease) of the weight change due to the final term in (6) is that of the last action taken, y(ti,last). 649 Heuristic HI is called constant recurrence learning because the eligibility function is designed to reinforce any cycle. HEURISTIC H2: Reward a short cycle more than a longer one. Heuristic 82 is called short recurrence learning because the eligibility function is designed to reinforce shorter cycle more than longer cycles. REINFORCEMENT OF SHORTER CYCLES The basis of the second heuristic is the conjecture that short cycles converge more easily to absorbing points than long ones, and that long cycles diverge more easily than shorter ones, although any cycle can "grow" or diverge to a larger cycle. The following extension to the our basic heuristic is proposed. The formula for the recurrence eligibility function is: { o if t-ti,last = e2,i(t) = P2 xi(t) y(ti,last) otherwise (P2+t-ti,last) 1 or li,last = 0 } (8) The current eligibility function e2/t) is similar to the previous failure eligibility function in (7); however, e2 i(t) reinforces shorter cycles more, because the eligibility decays with time. The value'returned from e2it) is inversely proportional to the period of the cycle from ti,last to t. H2 reinforces converging oscillations; the term (X.2r2(t)e2/t) in (6) ensures weight reinforcement for returning to an already visited state. Figure 3a and 3b: The Constant Recurrence algorithm and Short Recurrence algorithms Figure 3A shows the Constant Recurrence algorithm (HI). A state is rewarded when it is reactivated by a transition from another state. In the example below. state A is reward by a constant regardless of weather the cycle traversed states B or C. Figure 3b describes the Short Recurrence algorithm (m). A state is rewarded according to the difference between the current time and its last activation time. Small differences are rewarded more than large differences In the example below, state A is rewarded more 650 when the cycle (through state C) traverses the states shown by the dark heavy line rather than when the cycle (through state B) traverses the lighter line, since state A recurs sooner when traversing the darker line. SIMULATION RESULTS We simulated four algorithms: ASE, ASE/ACE and the two recurrence algorithms. Each experiment consisted of ten runs of the cart-pole balancing task, each consisting of 100 trials. Each trial lasted for 500,000 time steps or until the cart-pole system failed (i.e. the pole fell or the cart went beyond the track boundaries). In an effort to conserve cpu time, simulations were also terminated when the system achieved two consecutive trials each lasting for over 500,000 time steps; all remaining trials were assumed to also last 500,000 time steps. This assumption was reasonable: the resulting weight space causes the controller to become deterministic regardless of the influence of stochastic noise. Because of the long time require to run simulations, no attempts were made to optimize parameters of the algorithm. As in Bart02, each trial began with the cart centered, and the pole upright. No assumptions were made as to the state space configuration, the desirability of the initial states, or the continuity of the state space. The first experiment consisted of failure and recurrence reward learning. The ASE failure learning runs averaged 1578 time steps until failure after 100 trials*. Next, the predictive ASE/ACE system was run as a comparative metric, and it was found that this method caused the controller to average 131,297 time steps until failure; the results are comparable to that described by Barto, Sutton and Anderson. In the next experiment, short recurrence learning system was added to the ASE system. Again, ten 100 trial learning session were executed. On the average, the short recurrence learning algorithm ran for over 400,736 time steps after 100th trial, bettering the ASE/ACE system by 205%. In the final experiment, constant recurrence learning with the ASE system was simulated. The constant recurrence learning eliminated failure after only 207,562 time steps. Figure 1 shows the ASE, ASE/ACE, Constant recurrence learning (HI) and Short recurrence learning (H2) failure rates averaged over 10 simulation runs. DISCUSSION Detection of cycles provides a heuristic for the "two armed bandit" problem to decide between evidence gathering, and goal directed search. The algorithm allows the automaton to search outward from the cycle states (states with high probability of revisitation) to the more unexplored search space. The rate of exploration is proportional to the recurrence learning parameter~; as ~ is decreased, the influence of the cycles governing the decision process also decreases and the algorithm explores more of the search space that is not part of any cycle or oscillation path. * However, there was a relatively large degree of variance in the final trials. The last 10 trails (averaged over each of the 10 simulations) ranged from 607 to 15,459 time steps until failure 651 THEFUfURE Our future experiments will study the effects of rewarding predictions of cycle lengths in a manner similar to the prediction of failure used by the ASE/ACE system. The effort will be to minimize the differences of predicted time of cycles in order to predict their period. Results of this experiment will be shown in future reports. We hope to show that this recurrence prediction system is generally superior to either the ASE/ACE predictive system or the short recurrence system operating alone. CONCLUSION This paper presented an extension to the failure driven learning algorithm based on reinforcing decisions that cause an automaton to enter environmental states more than once. The controller learns to synthesize the best values by reinforcing areas of the search space that produce recurring state visitation. Cycle states, which under normal failure driven learning algorithms do not learn, achieve weight alteration from success. Simulations show that recurrence reward algorithms show improved overall learning of the cart-pole task with a substantial decrease in learning time. REFERENCES 1. D. Michie and R. Chambers, Machine Intelligence, E. Dale and D. Michie, Ed.: (Oliver and Boyd, Edinburgh, 1968), p. 137. 2. A. Barto, R. Sutton, and C. Anderson, Coins Tech. Rept., No. 82-20, 1982. 3. B. Widrow and F. Smith, in Computer and Information Sciences, 1. Tou and R. Wilcox Eds., (Clever Hume Press, 1964). 4. M. Minsky, in Proc. IRE, 49, 8, (1961). 5. J. Holland, in Proc. Int. Conj., Genetic Algs. and their Appl., 1985, p. 1. 6. A. Samuel, IBM Journ. Res.and Dev. 3, 211, (1959) 7. M. Connell and P. Utgoff, in Proc. AAAl-87 (Seattle, 1987), p. 456. 8. C. Anderson, Coins Tech. Rept., No. 86-50: Amherst, MA. 1986. 9. R. Barnhill, in Mathematical Software I II, (Academic Press, 1977). 10. L. Schumaker, in Approximation Theory II. (Academic Press, 1976). 11. A. H. Klopf, in IEEE Int. Conf. Neural Networks" June 1987. 12. R. Sutton, GTE Tech. Rept.TR87-509.1, GTE Labs. Inc., Jan. 1987 13. R. Sutton and A. G. Barto, Tech. Rept. TR87-5902.2 March 1987 14. A. Barto and P. Anandan, IEEE Trans. SMC 15, 360 (1985). 15. M. Sato, K. Abe, and H. Takeda, IEEE Trans.SMC 14,528 (1984).
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184 THE CAPACITY OF THE KANERVA ASSOCIATIVE MEMORY IS EXPONENTIAL P. A. Choul Stanford University. Stanford. CA 94305 ABSTRACT The capacity of an associative memory is defined as the maximum number of vords that can be stored and retrieved reliably by an address vithin a given sphere of attraction. It is shown by sphere packing arguments that as the address length increases. the capacity of any associati ve memory is limited to an exponential grovth rate of 1 - h2( 0). vhere h2(0) is the binary entropy function in bits. and 0 is the radius of the sphere of attraction. This exponential grovth in capacity can actually be achieved by the Kanerva associative memory. if its parameters are optimally set . Formulas for these op.timal values are provided. The exponential grovth in capacity for the Kanerva associative memory contrasts sharply vith the sub-linear grovth in capacity for the Hopfield associative memory. ASSOCIATIVE MEMORY AND ITS CAPACITY Our model of an associative memory is the folloving. Let ()(,Y) be an (address. datum) pair. vhere )( is a vector of n ±ls and Y is a vector of m ±ls. and let ()(l),y(I)), ... ,()(M), y(M)). be M (address, datum) pairs stored in an associative memory. If the associative memory is presented at the input vith an address )( that is close to some stored address )(W. then it should produce at the output a vord Y that is close to the corresponding contents y(j). To be specific, let us say that an associative memory can correct fraction 0 errors if an )( vi thin Hamming distance no of )((j) retrieves Y equal to y(j). The Hamming sphere around each )(W vill be called the sphere of attraction, and 0 viII be called the radius of attraction. One notion of the capacity of this associative memory is the maximum number of vords that it can store vhile correcting fraction 0 errors . Unfortunately. this notion of capacity is ill-defined. because it depends on exactly vhich (address. datum) pairs have been stored. Clearly. no associative memory can correct fraction 0 errors for every sequence of stored (address, datum) pairs. Consider. for example, a sequence in vhich several different vords are vritten to the same address . No memory can reliably retrieve the contents of the overvritten vords. At the other extreme. any associative memory ' can store an unlimited number of vords and retrieve them all reliably. if their contents are identical. A useful definition of capacity must lie somevhere betveen these tvo extremes. In this paper. ve are interested in the largest M such that for most sequences of addresses XU), .. . , X(M) and most sequences of data y(l), ... , y(M). the memory can correct fraction 0 errors. We define IThis vork vas supported by the National Science Foundation under NSF grant IST-8509860 and by an IBM Doctoral Fellovship. © American Institute of Physics 1988 185 I most sequences' in a probabilistic sense, as some set of sequences yi th total probability greater than say, .99. When all sequences are equiprobab1e, this reduces to the deterministic version: 991. of all sequences. In practice it is too difficult to compute the capacity of a given associative memory yith inputs of length n and outputs of length Tn. Fortunately, though, it is easier to compute the asymptotic rate at which A1 increases, as n and Tn increase, for a given family of associative memories. This is the approach taken by McEliece et al. [1] toyards the capacity of the Hopfield associative memory. We take the same approach tovards the capacity of the Kanerva associative memory, and tovards the capacities of associative memories in general . In the next section ve provide an upper bound on the rate of grovth of the capacity of any associative memory fitting our general model. It is shown by sphere packing arguments that capacity is limited to an exponential rate of grovth of 1- h2(t5), vhere h2(t5) is the binary entropy function in bits, and 8 is the radius of attraction. In a later section it vill turn out that this exponential grovth in capacity can actually be achieved by the Kanerva associative memory, if its parameters are optimally set. This exponential grovth in capacity for the Kanerva associative memory contrasts sharply yith the sub-linear grovth in capacity for the Hopfield associative memory [1]. A UNIVERSAL UPPER BOUND ON CAPACITY Recall that our definition of the capacity of an associative memory is the largest A1 such that for most sequences of addresses X(1), ... ,X(M) and most sequences of data y(l), ... , y(M), the memory can correct fraction 8 errors. Clearly, an upper bound to this capacity is the largest Af for vhich there exists some sequence of addresses X(1), . . . , X(M) such that for most sequences of data y(l), ... , y(M), the memory can correct fraction 8 errors. We nov derive an expression for this upper bound. Let 8 be the radius of attraction and let DH(X(i) , d) be the sphere of attraction, i.e., the set of all Xs at most Hamming distance d= Ln8J from .y(j). Since by assumption the memory corrects fraction 8 errors, every address X E DH(XU),d) retrieves the vord yW. The size of DH(XU),d) is easily shown to be independent of xU) and equal to vn.d = 2:%=0 (1:), vhere (I:) is the binomial coefficient n!jk!(n - k)!. Thus out of a total of 2n n-bit addresses, at least vn.d addresses retrieve y(l), at least Vn.d addresses retrieve y(2), at least Vn.d addresses retrieve y(~, and so forth. It fol10vs that the total number of distinct yU)s can be at most 2n jvn.d ' Nov, from Stirling's formula it can be shovn that if d:S; nj2, then vn.d = 2nh2(d/n)+O(logn), vhere h2( 8) = -81og2 8 - (1 - 8) log2( 1 - 8) is the binary entropy function in bits, and O(logn) is some function yhose magnitude grovs more slovly than a constant times log n. Thus the total number of distinct y(j)s can be at most 2n(1-h2(S»+O(logn) Since any set containing I most sequences' of Af Tn-bit vords vill contain a large number of distinct vords (if Tn is 186 Figure 1: Neural net representation of the Kanerva associative memory. Signals propagate from the bottom (input) to the top (output). Each arc multiplies the signal by its weight; each node adds the incoming signals and then thresholds. sufficiently large --- see [2] for details), it follovs that M :5 2n(l-h 2(o»+O(logn). (1) In general a function fen) is said to be O(g(n)) if f(n)fg(n) is bounded, i.e. , if there exists a constant a such that If(n)1 :5 a\g(n)1 for all n. Thus (1) says that there exists a constant a such that M :5 2n(l-h2(S»+alogn. It should be emphasized that since a is unknow, this bound has no meaning for fixed n. Hovever, it indicates that asymptotically in n, the maximum exponential rate of grovth of M is 1 - h2( 6). Intui ti vely, only a sequence of addresses X(l), ... , X(M) that optimally pack the address space {-l,+l}n can hope to achieve this upper bound. Remarkably, most such sequences are optimal in this sense, vhen n is large. The Kanerva associative memory can take advantage of this fact. THE KANERVA ASSOCIATIVE MEMORY The Kanerva associative memory [3,4] can be regarded as a tvo-layer neural netvork, as shovn in Figure 1, vhere the first layer is a preprocessor and the second layer is the usual Hopfield style array. The preprocessor essentially encodes each n-bit input address into a very large k-bit internal representation, k ~ n, vhose size will be permitted to grov exponentially in n. It does not seem surprising, then, that the capacity of the Kanerva associative memory can grov exponentially in n, for it is knovn that the capacity of the Hopfield array grovs almost linearly in k, assuming the coordinates of the k-vector are dravn at random by independent flips of a fair coin [1]. 187 Figure 2: Matrix representation of the Kanerva associative memory. Signals propagate from the right (input) to the left (output). Dimensions are shown in the box corners. Circles stand for functional composition; dots stand for matrix multiplication. In this situation, hovever, such an assumption is ridiculous: Since the k-bit internal representation is a function of the n-bit input address, it can contain at most n bits of information, whereas independent flips of a fair coin contain k bits of information. Kanerva's primary contribution is therefore the specification of the preprocessor, that is, the specification of how to map each n-bit input address into a very large k-bit internal representation. The operation of the preprocessor is easily described. Consider the matrix representation shovn in Figure 2. The matrix Z is randomly populated vith ±ls. This randomness assumption is required to ease the analysis. The function fr is 1 in the ith coordinate if the ith row of Z is within Hamming distance r of X, and is Oothervise. This is accomplished by thresholding the ith input against n-2r. The parameters rand k are two essential parameters in the Kanerva associative memory. If rand k are set correctly, then the number of 1s in the representation fr(ZX) vill be very small in comparison to the number of Os. Hence fr(Z~Y) can be considered to be a sparse internal representation of X. The second stage of the memory operates in the usual way, except on the internal representation of X. That is, Y = g(W fr(ZX)), vhere M l-V = LyU)[Jr(ZXU))]t, (2) i=l and 9 is the threshold function whose ith coordinate is +1 if the ith input is greater than 0 and -1 is the ith input is less than O. The ith column of l-V can be regarded as a memory location vhose address is the ith row of Z. Every X vi thin Hamming distance r of the ith rov of Z accesses this location. Hence r is known as the access radius, and k is the number of memory locations. The approach taken in this paper is to fix the linear rate p at which r grovs vith n, and to fix the exponential rate ~ at which k grovs with n. It turns out that the capacity then grovs at a fixed exponential rate Cp,~(t5), depending on p, ~, and 15. These exponential rates are sufficient to overcome the standard loose but simple polynomial bounds on the errors due to combinatorial approximations. 188 THE CAPACITY OF THE KANERVA ASSOCIATIVE MEMORY Fix 0 $ K $1. 0 $ p$ 1/2. and 0 $ 0 $ min{2p,1/2}. Let n be the input address length, and let Tn be the output word length. It is assumed that Tn is at most polynomial in n, i.e., Tn = exp{O(logn)}. Let r = IJmJ be the access radius, let k = 2L"nJ be the number of memory locations, and let d= LonJ be the radius of attraction. Let Afn be the number of stored words. The components of the n-vectors X(l), .. . , X(Mn) , the m-vectors y(l), ... , y(,Yn), and the k X n matrix Z are assumed to be lID equiprobable ±1 random variables. Finally, given an n-vector X, let Y = g(W fr(ZX)) where W = Ef;nl yU)[Jr(ZXW)jf. Define the quantity Cp ,,(0) = { 26 + 2(1- 0)h(P;~~2) + K, - 2h(p) if K, $ K,o(p) (3) 'Cp,ICo(p)(o) if K> K,O(p) , where KO(p) = 2h(p) - 2; - 2(1- ;)h(P~242) + 1- he;) (4) and ; = ~ - J 196 - 2p(1 - p). Theorem: If Af < 2nCp ... (5)+O(logn) n_ then for all f>O, all sufficiently large n, all jE{l, ... ,Afn }. and all X E DH(X(j) , d), P{y -::J y(j)} < f. Proof: See [2]. Interpretation: If the exponential growth rate of the number of stored words Afn is asymptotically less than Cp,,, ( 0), then for every sufficiently large address length n. there is some realization of the nx 2n" preprocessor matrix Z such that the associative memory can correct fraction 0 errors for most sequences of Afn (address, datum) pairs. Thus Cp,IC( 0) is a lover bound on the exponential growth rate of the capacity of the Kanerva associative memory with access radius np and number of memory locations 2nIC • Figure 3 shows Cp,IC(O) as a function of the radius of attraction 0, for K,= K,o(p) and p=O.l, 0.2, 0.3, 0.4 and 0.45. For· any fixed access radius p, Cp,ICO(p)( 0) decreases as 0 increases. This reflects the fact that fewer (address, datum) pairs can be stored if a greater fraction of errors must be corrected. As p increases, Cp,,,o(p)(o) begins at a lower point but falls off less steeply. In a moment we shall see that p can be adjusted to provide the optimal performance for a given O. Not ShOVIl in Figure 3 is the behavior of Cp,,, ( 0) as a function of K,. However, the behavior is simple. For K, > K,o(p), Cp,,,(o) remains unchanged, while for K$ K,o(p), Cp,,,(o) is simply shifted doVIl by the difference KO(p)-K,. This establishes the conditions under which the Kanerva associative memory is robust against random component failures. Although increasing the number of memory locations beyond 2rl11:o(p) does not increase the capacity, it does increase robustness. Random 189 0.8 0.6 '!I.2 ...... --" " • 1 1Il.2 1Il.3 IIl.S Figure 3: Graphs of Cp,lCo(p)(o) as defined by (3). The upper envelope is 1- h2(0). component failures will not affect the capacity until so many components have failed that the number of surviving memory locations is less than 2nlCo(p) . Perhaps the most important curve exhibited in Figure 3 is the sphere packing upper bound 1 - h2( 0). which is achieved for a particular p by b = ~ - J 196 - 2p(1 - p). Equivalently. the upper bound is achieved for a particular 0 by P equal to poCo) = t - Jt - iO(l ~o). Thus (4) and (5) specify the optimal values of the parameters K and P. respectively. These functions are shown in Figure 4. With these optimal values. (3) simplifies to the sphere packing bound. (5) It can also be seen that for 0 = 0 in (3). the exponential growth rate of the capacity is asymptotically equal to K. which is the exponential growth rate of the number of memory locations. k n • That is. Mn = 2n1C+O(logn) = kn . 20 (logn). Kanerva [3] and Keeler [5] have argued that the capacity at 8 =0 is proportional to the number of memory locations, i.e .• Mn = kn . (3. for some constant (3. Thus our results are consistent with those of Kanerva and Keeler. provided the 'polynomial' 20 (logn) can be proved to be a constant. However. the usual statement of their result. M = k·(3. that the capacity is simply proportional to the number of memory locations. is false. since in light of the universal 190 liLS o riJ.S Figure 4: Graphs of KO(p) and co(p), the inverse of Po(<5), as defined by (4) and (5). upper bound, it is impossible for the capacity to grow without bound, with no dependence on the dimension n. In our formulation, this difficulty does not arise because we have explicitly related the number of memory locations to the input dimension: kn =2n~. In fact, our formulation provides explicit, coherent relationships between all of the following variables: the capacity .~, the number of memory locations k, the input and output dimensions n and Tn, the radius of attraction C, and the access radius p. We are therefore able to generalize the results of [3,5] to the case C >0, and provide explicit expressions for the asymptotically optimal values of p and K as well. CONCLUSION We described a fairly general model of associative memory and selected a useful definition of its capacity. A universal upper bound on the growth of the capacity of such an associative memory was shown by a sphere packing argument to be exponential with rate 1 - h2( c), where h2(C) is the binary entropy function and 8 is the radius of attraction. We reviewed the operation of the Kanerva associative memory, and stated a lower bound on the exponential growth rate of its capacity. This lower bound meets the universal upper bound for optimal values of the memory parameters p and K. We provided explicit formulas for these optimal values. Previous results for <5 =0 stating that the capacity of the Kanerva associative memory is proportional to the number of memory locations cannot be strictly true. Our formulation corrects the problem and generalizes those results to the case C > o. REFERENCES 1. R.J. McEliece, E.C. Posner, E.R. Rodemich, and S.S. Venkatesh, "The capacity of the Hopfield associative memory," IEEE Transactions on Information Theory, submi tt ed . 2. P.A. Chou, "The capacity of the Kanerva associative memory," IEEE Transactions on Information Theory, submitted. 3. P. Kanerva, "Self-propagating search: a unified theory of memory," Tech. Rep. CSLI-84-7, Stanford Center for the Study of Language and Information. Stanford. CA, March 1984. 4. P. Kanerva, "Parallel structures in human and computer memory," in Neural Networks for Computing, (J .S. Denker. ed.), Nev York: American Institute of Physics. 1986. 5. J.D. Keeler. "Comparison betveen sparsely distributed memory and Hopfield-type neural netvork models," Tech . Rep. RIACS TR 86 .31, NASA Research Institute for Advanced Computer Science, Mountain Viev. CA, Dec. 1986. 191
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242 THE SIGMOID NONLINEARITY IN PREPYRIFORM CORTEX Frank H. Eeckman University of California, Berkeley, CA 94720 ABSlRACT We report a study ·on the relationship between EEG amplitude values and unit spike output in the prepyriform cortex of awake and motivated rats. This relationship takes the form of a sigmoid curve, that describes normalized pulse-output for normalized wave input. The curve is fitted using nonlinear regression and is described by its slope and maximum value. Measurements were made for both excitatory and inhibitory neurons in the cortex. These neurons are known to form a monosynaptic negative feedback loop. Both classes of cells can be described by the same parameters. The sigmoid curve is asymmetric in that the region of maximal slope is displaced toward the excitatory side. The data are compatible with Freeman's model of prepyriform burst generation. Other analogies with existing neural nets are being discussed, and the implications for signal processing are reviewed. In particular the relationship of sigmoid slope to efficiency of neural computation is examined. INTRODUCTION The olfactory cortex of mammals generates repeated nearly sinusoidal bursts of electrical activity (EEG) in the 30 to 60 Hz. range 1. These bursts ride on top of a slower ( 1 to 2 Hz.), high amplitude wave related to respiration. Each burst begins shortly after inspiration and terminates during expiration. They are generated locally in the cortex. Similar bursts occur in the olfactory bulb (OB) and there is a high degree of correlation between the activity in the two structures!' The two main cell types in the olfactory cortex are the superficial pyramidal cell (type A), an excitatory neuron receiving direct input from the OB, and the cortical granule cell (type B), an inhibitory interneuron. These cell groups are monosynaptically connected in a negative feedback loop2. Superficial pyramidal cells are mutually excitatory3, 4, 5 as well as being excitatory to the granule cells. The granule cells are inhibitory to the pyramidal cells as well as to each other3, 4, 6. In this paper we focus on the analysis of amplitude dependent properties: How is the output of a cellmass (pulses) related to the synaptic potentials (ie. waves)? The concurrent recording of multi-unit spikes and EEG allows us to study these phenomena in the olfactory cortex. The anatomy of the olfactory system has been extensively studied beginning with the work of S. Ramon y Cajal 7. The regular geometry and the simple three-layered architecture makes these structures ideally suitable for EEG recording 4, 8. The EEG generators in the various olfactory regions have been identified and their synaptic connectivities have been extensively studied9, 10,5,4, 11,6. The EEG is the scalar sum of synaptic currents in the underlying cortex. It can be recorded using low impedance « .5 Mohm) cortical or depth electrodes. Multiunit signals are recorded in the appropriate cell layers using high impedance (> .5 Mohm) electrodes and appropriate high pass filtering. Here we derive a function that relates waves (EEG) to pulses in the olfactory cortex of the rat. This function has a sigmoidal shape. The derivative of this curve © American Institute of Physics 1988 243 gives us the gain curve for wave-to-pulse conversion. This is the forward gain for neurons embedded in the cortical cellmass. The product of the forward gain values of both sets of neurons (excitatory and inhibitory) gives us the feedback gain values. These ultimately determine the dynamics of the system under study. MATERIALS AND METI-IODS A total of twenty-nine rats were entered in this study. In each rat a linear array of 6 100 micron stainless steel electrodes was chronically implanted in the prepyriform (olfactory) cortex. The tips of the electrodes were electrolytically sharpened to produce a tip impedance on the order of .5 to 1 megaohm. The electrodes were implanted laterally in the midcortex, using stereotaxic coordinates. Their position was verified electrophysiologically using a stimulating electrode in the olfactory tract. This procedure has been described earlier by Freeman 12. At the end of the recording session a small iron deposit was made to help in histological verification. Every electrode position was verified in this manner. Each rat was recorded from over a two week period following implantation. All animals were awake and attentive. No stimulation (electrical or olfactory) was used. The background environment for recording was the animal's home cage placed in the same room during all sessions. For the present study two channels of data were recorded concurrently. Channel 1 carried the EEG signal, filtered between 10 and 300 Hz. and digitized at 1 ms intervals. Channel 2 carried standard pulses 5 V, 1.2 ms wide, that were obtained by passing the multi-unit signal (filtered between 300 Hz. and 3kHz.) through a window discriminator. These two time-series were stored on disk for off-line processing using a PerkinElmer 3220 computer. All routines were written in FORTRAN. They were tested on data files containing standard sine-wave and pulse signals. DATA PROCESSING The procedures for obtaining a two-dimensional conditional pulse probability table have been described earlier 4. This table gives us the probability of occurrence of a spike conditional on both time and normalized EEG amplitude value. By counting the number of pulses at a fixed time-delay, where the EEG is maximal in amplitude, and plotting them versus the normalized EEG amplitudes, one obtains a sigmoidal function: The Pulse probability Sigmoid Curve (PSC) 13, 14. This function is normalized by dividing it by the average pulse level in the record. It is smoothed by passing it through a digital 1: 1: 1 filter and fitted by nonlinear regression. The equations are: Q = Qmax ( 1- exp [ - ( ev - 1) I Qmax ]) for v> - uO (1 ) Q = -1 for v < - uO where uO is the steady state voltage, and Q = (p-PO)/pO. and Qmax =(Pmax-PO)/pO. PO is the background pulse count, Pmax is the maximal pulse count. These equations rely on one parameter only. The derivation and justification for these equations were discussed in an earlier paper by Freeman 13. 244 RESULTS Data were obtained from all animals. They express normalized pulse counts, a dimensionless value as a function of normalized EEG values, expressed as a Z-score (ie. ranging from - 3 sd. to + 3 sd., with mean of 0.0). The true mean for the EEG after filtering is very close to 0.0 m V and the distribution of amplitude values is very nearly Gaussian. The recording convention was such that high EEG-values (ie. > 0.0 to + 3.0 sd.) corresponded to surface-negative waves. These in turn occur with activity at the apical dendrites of the cells of interest. Low EEG values (ie. from - 3.0 sd. to < 0.0) corresponded to surface-positive voltage values, representing inhibition of the cells. The data were smoothed and fitted with equation (1). This yielded a Qrnax value for every data file. There were on average 5 data files per animal. Of these 5, an average of 3.7 per animal could be fitted succesfully with our technique. In 25 % of the traces, each representing a different electrode pair, no correlations between spikes and the EEG were found. Besides Qmax we also calculated Q' the maximum derivative of the PSC, representing the maximal gain. There were 108 traces in all. In the first 61 cases the Qrnax value described the wave-to-pulse conversion for a class of cells whose maximum firing probability is in phase with the EEG. These cells were labelled type A cells 2. These traces correspond to the excitatory pyramidal cells. The mean for Qmax in that group was 14.6, with a standard deviation of 1.84. The range was 10.5 to 17.8. In the remaining 47 traces the Qmax described the wave-to-pulse conversion for class B cells. Class B is a label for those cells whose maximal firing probability lags the EEG maximum by approximately 1/4 cycle. The mean for Qrnax in that group was 14.3, with a standard deviation of 2.05. The range in this group was 11.0 to 18.8. The overall mean for Qmax was 14.4 with a standard deviation of 1.94. There is no difference in Qmax between both groups as measured by the Student t-test. The nonparametric Wilcoxon rank-sum test also found no difference between the groups ( p = 0.558 for the t-test; p = 0.729 for the Wilcoxon). Assuming that the two groups have Qmax values that are normally distributed (in group A, mean = 14.6, median = 14.6; in group B, mean = 14.3, median = 14.1), and that they have equal variances ( st. deviation group A is 1.84; st. deviation group B is 2.05) but different means, we estimated the power of the t-test to detect that difference in means. A difference of 3 points between the Qmax's of the respective groups was considered to be physiologically significant. Given these assumptions the power of the t-test to detect a 3 point difference was greater than .999 at the alpha .05 level for a two sided test. We thus feel reasonably confident that there is no difference between the Qmax values of both groups. The first derivative of the PSC gives us the gain for wave-to-pulse conversion4. The maximum value for this first derivative was labelled Q'. The location at which the maximum Q' occurs was labelled Vmax. Vmax is expressed in units of standard deviation of EEG amplitudes. The mean for Q' in group A was 5.7, with a standard deviation of .67, in group B it was 5.6 with standard deviation of .73. Since Q' depends on Qmax, the same statistics apply to both: there was no significant difference between the two groups for slope maxima. 245 14 H 12 CII .Q 10 ~ 8 6 4 2 o Figure 1. Distribution of Qmax values group A " , , ~ , " r' r"': , I' ; ; ~ > r-, , ; ,~ . , , " ' , I, , " ' ; r , , , 1', , I~ , ': 1', , , , " , , , " ' ; " " , , ., , 1'; , " v, , " , , " ; , " v, , , , , ; ; .' 1011121314151617181920 Qmax values 14 H 12 CII .Q 10 ~ 8 6 4 2 o group B ~ I" ~" " ~ , , I,~ , , , , " , , ;: , ~ p: , , , ,... , " , , , , , , " , , , , ., , "!l~. , , , , , , " ' , .f71. , , , ; , , , 1011121314151617181920 Qmax values The mean for Vmax was at 2.15 sd. +/- .307. In every case Vmax was on the excitatory side from 0.00, ie. at a positive value of EEG Z-scores. All values were greater than 1.00. A similar phenomenon has been reported in the olfactory bulb 4, 14, 15. Figure 2. Examples of sigmoid fits. A cell B cell 14 14 12 12 10 10 ~ ~ 8 ·rot 8 ~ 6 CII 6 og CII 4 4 11\ ~ 2 2 Po 0 0 -2 -2 -4 -4 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 normalized EEG amplitude Qm = 14.0 Qm = 13.4 3 246 COMPARISON WITH DATA FROM TIIE OB Previously we derived Qrnax values for the mitral cell population in the olfactory bulb14. The mitral cells are the output neurons of the bulb and their axons form the lateral olfactory tract (LOT). The LOT is the main input to the pyramidal cells (type A) in the cortex. For awake and motivated rats (N = 10) the mean Qmax value was 6.34 and the standard deviation was 1.46. The range was 4.41- 9.53. For anesthetized animals (N= 8) the mean was 2.36 and the standard deviation was 0.89. The range was 1.153.62. There was a significant difference between anesthetized and awake animals. Furthermore there is a significant difference between the Qmax value for cortical cells and the Qmaxvalue for bulbar cells (non - overlapping distributions). DISCUSSION An important characteristic of a feedback loop is its feedback gain. There is ample evidence for the existence of feedback at all levels in the nervous system. Moreover specific feedback loops between populations of neurons have been described and analyzed in the olfactory bulb and the prepyriform cortex 3, 9, 4. A monosynaptic negative feedback loop has been shown to exist in the PPC, between the pyramidal cells and inhibitory cells, called granule cells 3, 2, 6, 16. Time series analysis of concurrent pulse and EEG recordings agrees with this idea. The pyramidal cells are in the forward limb of the loop: they excite the granule cells. They are also mutually excitatory 2,4,16. The granule cells are in the feedback limb: they inhibit the pyramidal cells. Evidence for mutual inhibition (granule to granule) in the PPC also exists 17, 6. The analysis of cell firings versus EEG amplitude at selected time-lags allows one to derive a function (the PSC) that relates synaptic potentials to output in a neural feedback system. The first derivative of this curve gives an estimate of the forward gain at that stage of the loop. The procedure has been applied to various structures in the olfactory system 4, 13, 15, 14. The olfactory system lends itself well to this type of analysis due to its geometry, topology and well known anatomy. Examination of the experimental gain curves shows that the maximal gain is displaced to the excitatory side. This means that not only will the cells become activated by excitatory input, but their mutual interaction strength will increase. The result is an oscillatory burst of high frequency ( 30- 60 Hz.) activity. This is the mechanism behind bursting in the olfactory EEG 4, 13. In comparison with the data from the olfactory bulb one notices that there is a significant difference in the slope and the maximum of the PSC. In cortex the values are substantially higher, however the Vmax is similar. C. Gray 15 found a mean value of 2.14 +/- 0.41 for V max in the olfactory bulb of the rabbit (N = 6). Our value in the present study is 2.15 +/- .31. The difference is not statistically significant. There are important aspects of nonlinear coupling of the sigmoid type that are of interest in cortical functioning. A sigmoid interaction between groups of elements ("neurons") is a prominent feature in many artificial neural nets. S. Grossberg has extensively studied the many desirable properties of sigmoids in these networks. Sigmoids can be used to contrast-enhance certain features in the stimulus. Together with a thresholding operation a sigmoid rule can effectively quench noise. Sigmoids can also provide for a built in gain control mechanism 18, 19. 247 Changing sigmoid slopes have been investigated by J. Hopfield. In his network changing the slope of the sigmoid interaction between the elements affects the number of attractors that the system can go to 20. We have previously remarked upon the similarities between this and the change in sigmoid slope between waking and anesthetized animals 14. Here we present a system with a steep slope (the PPC) in series with a system with a shallow slope (the DB). Present investigations into similarities between the olfactory bulb and Hopfield networks have been reported 21, 22. Similarities between the cortex and Hopfieldlike networks have also been proposed 23. Spatial amplitude patterns of EEG that correlate with significant odors exist in the bulb 24. A transmission of "wave-packets" from the bulb to the cortex is known to occur 25. It has been shown through cofrequency and phase analysis that the bulb can drive the cortex 25, 26. It thus seeems likely that spatial patterns may also exist in the cortex. A steeper sigmoid, if the analogy with neural networks is correct, would allow the cortex to further classify input patterns coming from the olfactory bulb. In this view the bulb could form an initial classifier as well as a scratch-pad memory for olfactory events. The cortex could then be the second classifier, as well as the more permanent memory. These are at present speculations that may turn out to be premature. They nevertheless are important in guiding experiments as well as in modelling. Theoretical studies will have to inform us of the likelihood of this kind of processing. REFERENCES 1 S.L. Bressler and W.J. Freeman, Electroencephalogr. Clin. Neurophysiol. ~: 19 (1980). . 2 W.J. Freeman, J. Neurophysiol. ll: 1 (1968). 3 W.J. Freeman, Exptl. Neurol. .lO.: 525 (1964). 4 W.J. Freeman, Mass Action in the Nervous System. (Academic Press, N.Y., 1975), Chapter 3. 5 L.B. Haberly and G.M. Shepherd, Neurophys.~: 789 (1973). 6 L.B. Haberly and J.M. Bower, J. Neurophysiol. ll: 90 (1984). 7 S. Ramon y Cajal, Histologie du Systeme Nerveux de l'Homme et des Vertebres. ( Ed. Maloine, Paris, 1911) . 8 W.J. Freeman, BioI. Cybernetics . .3..5.: 21 (1979). 9 W. Rall and G.M. Shepherd, J. Neurophysiol.ll: 884 (1968). 10 G.M. Shepherd, Physiol. Rev. 5l: 864 (1972). 11 L.B. Haberly and J.L. Price, J. Compo Neurol. .l18.; 711 (1978). 12 W.J. Freeman, Exptl. Neurol. ~: 70 (1962). 13 W.J. Freeman, BioI. Cybernetics.ll: 237 (1979). 14 F.H. Eeckman and W.J. Freeman, AlP Proc. ill: 135 (1986). 15 C.M. Gray, Ph.D. thesis, Baylor College of Medicine (Houston,1986) 16 L.B. Haberly, Chemical Senses, .ll!: 219 (1985). 17 M. Satou et aI., J. Neurophysiol. ~: 1157 (1982). 18 S. Grossberg, Studies in Applied Mathematics, Vol LII, 3 (MIT Press, 1973) p 213. 19 S. Grossberg, SIAM-AMS Proc. U: 107 (1981). 20 J.J Hopfield, Proc. Natl. Acad. Sci. USA 8.1: 3088 (1984). 21 W.A. Baird, Physica 2.m: 150 (1986). 22 W.A. Baird, AlP Proceedings ill: 29 (1986). 23 M. Wilson and J. Bower, Neurosci. Abstr. 387,10 (1987). 248 24 K.A. Grajski and W.J. Freeman, AlP Proc.lS.l: 188 (1986). 25 S.L. Bressler, Brain Res. ~: 285 (1986). 26 S.L. Bressler, Brain Res.~: 294 (1986).
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310 PROBABILISTIC CHARACTERIZATION OF NEURAL MODEL COMPUTATIONS Richard M. Golden t University of Pittsburgh, Pittsburgh, Pa. 15260 ABSTRACT Information retrieval in a neural network is viewed as a procedure in which the network computes a "most probable" or MAP estimate of the unknown information. This viewpoint allows the class of probability distributions, P, the neural network can acquire to be explicitly specified. Learning algorithms for the neural network which search for the "most probable" member of P can then be designed. Statistical tests which decide if the "true" or environmental probability distribution is in P can also be developed. Example applications of the theory to the highly nonlinear back-propagation learning algorithm, and the networks of Hopfield and Anderson are discussed. INTRODUCTION A connectionist system is a network of simple neuron-like computing elements which can store and retrieve information, and most importantly make generalizations. Using terminology suggested by Rumelhart & McClelland 1, the computing elements of a connectionist system are called units, and each unit is associated with a real number indicating its activity level. The activity level of a given unit in the system can also influence the activity level of another unit. The degree of influence between two such units is often characterized by a parameter of the system known as a connection strength. During the information retrieval process some subset of the units in the system are activated, and these units in turn activate neighboring units via the inter-unit connection strengths. The activation levels of the neighboring units are then interpreted as t Correspondence should be addressed to the author at the Department of Psychology, Stanford University, Stanford, California, 94305, USA. © American Institute of Physics 1988 the retrieved information. During the learning process, the values of the interunit connection strengths in the system are slightly modified each time the units in the system become activated by incoming information. DERIV ATION OF TIIE SUBJECITVE PF Smolensky 2 demonstrated how the class of possible probability distributions that could be represented by a Hannony theory neural network model can be derived from basic principles. Using a simple variation of the arguments made by Smolen sky , a procedure for deriving the class of probability distributions associated with any connectionist system whose information retrieval dynamics can be summarized by an additive energy function is briefly sketched. A rigorous presentation of this proof may be found in Golden 3. Let a sample space, Sp, be a subset of the activation pattern state space, Sd, for a particular neural network model. For notational convenience, define the term probability function (pf) to indicate a function that assigns numbers between zero and one to the elements of Sp. For discrete random variables, the pf is a probability mass function. For continuous random variables, the pf is a probability density function. Let a particular stationary stochastic environment be represented by the scalar-valued pf, Pe(X)' where X is a particular activation pattern. The pf, Pe(X), indicates the relative frequency of occurrence of activation pattern X in the network model's environment. A second pf defined with respect to sample space Sp also must be introduced. This probability function, ps(X), is called the network's subjective pf. The pf Ps(X) is interpreted as the network's belief that X will occur in the network's environment. The subjective pf may be derived by making the assumption that the information retrieval dynamical system, D s' is optimal. That is, it is assumed that D s is an algorithm designed to transform a less probable state X into a more probable state X* where the probability of a state is defined by the subjective pf ps(X;A), and where the elements of A are the connection strengths among the units. Or in traditional engineering terminology, it is assumed that D s is a MAP (maximum a posteriori) estimation algorithm. The second assumption is that an energy function, V(X), that is minimized by the system during the information retrieval process can be found with an additivity property. The additivity property says that if the neural network were partitioned into two physically 311 312 unconnected subnetworks, then Vex) can be rewritten as VI (Xl) + V 2(X2) where VIis the energy function minimized by the first subnetwork and V 2 is the energy function minimized by the second subnetwork. The third assumption is that Vex) provides a sufficient amount of information to specify the probability of activation pattern X. That is, p (X) = G(V(X» where G is some continuous function. And the final assumpti;n (following Smolen sky 2) is that statistical and physical independence are equivalent. To derive ps(X), it is necessary to characterize G more specifically. Note that if probabilities are assigned to activation patterns such that physically independent substates of the system are also statistically independent, then the additivity property of V(X) forces G to be an exponential function since the onz continuous function that maps addition into multiplication is the exponential . After normalization and the assignment of unity to an irrelevant free parameter 2, the unique subjective pf for a network model that minimizes V(X) during the information retrieval process is: p s(X;A) = Z -1 exp [ - V (X;A)] (1) Z = Jexp[ - V (X;A)]dX (2) provided that Z < C < 00. Note that the integral in (2) is taken over sp. Also note that the pf, Ps' and samfle space, Sp, specify a Markov Random Field since (1) is a Gibbs distribution . Example 1: Subjective pfs for associative back-propagation networks The information retrieval equation for an associative back-propagation 6 network can be written in the form ~[I;A] where the elements of the vector 0 are the activity levels for the output units and the elements of the vector I are the activity levels for the input units. The parameter vector A specifies the values of the "connection strengths" among the units in the system. The function cl> specifies the architecture of the network. A natural additive energy function for the information retrieval dynamics of the least squares associative back-propagation algorithm is: V(O) = I ()-.4>(I;A) 12, (3) If Sp is defined to be a real vector space such that 0 esp, then direct substitution of V(O) for V iX;A) into (1) and (2) yields a multivariate Gaussian density function with mean cl>(I;A) and covariance matrix equal to the identity matrix multiplied by 1!2. This multivariate Gaussian density function is ps(OII;A). That is, with respect to ps(OII;A), information retrieval in an associative backpropagation network involves retrieving the "most probable" output vector, 0, for a given input vector, I. Example 2: Subjective pis/or Hopfield and BSB networks. The Hopfield 7 and BSB model 8,9 neural network models minimize the following energy function during information retrieval: T Vex) =-X MX (4) where the elements of X are the activation levels of the units in the system. and the elements of M are the connection strengths among the units. Thus, the subjective pf for these networks is: 313 314 X Z-l T P s< ) = exp [X M X] where Z = l:exp [XT M X] (5) where the summation is taken over Sp. APPLICATIONS OF TIlE TIIEORY If the subjective pf for a given connectionist system is known, then traditional analyses from the theory of statistical inference are immediately applicable. In this section some examples of how these analyses can aid in the design and analysis of neural networks are provided. Evaluating Learning Algorithms Learning in a neural network model involves searching for a set of connection strengths or parameters that obtain a global minimum of a learning energy function. The theory proposed here explicitly shows how an optimal learning energy function can be constructed using the model's subjective pf and the environmental pf. In particular, optimal learning is defined as searching for the most probable connection strengths, given some set of observations (samples) drawn from the environmental pf. Given some mild restrictions upon the fonn of the a priori pf associated with the connection strengths, and for a sufficiently large set of observations, estimating the most probable connection strengths (MAP estimation) is equivalent to maximum likelihood estimation 10 A well-known result 11 is that if the parameters of the subjective pf are represented by the parameter vector A, then the maximum likelihood estimate of A is obtained by finding the A * that minimizes the function: E(A) =- <.LOG [p s(X;A)]> (6) where < > is the expectation operator taken with respect to the environmental pf. Also note that (6) is the Kullback-Leibler 12 distance measure plus an irrelevant constant. Asymptotically, E(A) is the logarithm of the probability of A given some set of observations drawn from the environmental pf. Equation (6) is an important equation since it can aid in the evaluation and design of optimal learning algorithms. Substitution of the multivariate Gaussian associated with (3) into (6) shows that the back-propagation algorithm is doing gradient descent upon the function in (6). On the other hand, substitution of (5) into (6) shows that the Hebbian and Widrow-Hoff learning rules proposed for the Hopfield and BSB model networks are not doing gradient descent upon (6). Evaluating Network Architectures The global minimum of ~6) occurs if and only if the subjective and environmental pfs are equivalent 2. Thus, one crucial issue is whether any set of connection strengths exists such that the neural network's subjective pf can be made equivalent to a given environmental pf. If no such set of connection strengths exists, the subjective pf, p s' is defined to be misspecified. White 11 and Lancaster 13 have introduced a statistical test designed to re~ct the null hypothesis that the subjective pf, Ps' is not misspecified. Golden suggests a version of this test that is suitable for subjective pfs with many parameters. REFERENCES 1. D. E. Rumelhart, J. L. McClelland, and the PDP Research Group, Parallel distributed processing: Explorations in the microstructure of cognition, 1, (MIT Press, Cambridge, 1986). 2. P. Smolensky, In D. E. Rumelhart, J. L. McClelland and the PDP Research Group (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition, 1, (MIT Press, Cambridge, 1986), pp. 194-281. 315 316 3. R. M. Golden, A unified framework for connectionist systems. Unpublished manuscript. 4. C. Goffman, Introduction to real analysis. (Harper and Row, N. Y., 1966), p. 65. 5. J. L. Marroquin, Probabilistic solution of inverse problems. A.I. Memo 860, MIT Press (1985). 6. D. E. Rumelhart, G. E. Hinton, & R. J. Williams, In D. E. Rumelhart, 1. L. McClelland, and the PDP Research Group (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition, 1, (MIT Press, Cambridge, 1986), pp. 318-362. 7. J. 1. Hopfield, Proceedings of the National Academy of Sciences, USA, 79, 2554-2558 (1982). 8. J. A. Anderson, R. M. Golden, & G. L. Murphy, In H. Szu (Ed.), Optical and Hybrid Computing, SPIE, 634,260-276 (1986). 9. R. M. Golden, Journal of Mathematical Psychology, 30,73-80 (1986). 10. H. L. Van Trees, Detection, estimation, and modulation theory. (Wiley, N. Y.,1968). 11. H. White, Econometrica, 50, 1-25 (1982). 12. S. Kullback & R. A. Leibler, Annals of Mathematical Statistics, 22, 79-86 (1951). 13. T. Lancaster, Econometrica, 52, 1051-1053 (1984). ACKNOWLEDGEMENTS This research was supported in part by the Mellon foundation while the author was an Andrew Mellon Fellow in the Psychology Department at the University of Pittsburgh, and partly by the Office of Naval Research under Contract No. N-OOI4-86-K-OI07 to Walter Schneider. This manuscript was revised while the author was an NIH postdoctoral scholar at Stanford University. This research was also supported in part by grants from the Office of Naval Research (Contract No. NOOOI4-87-K-0671), and the System Development Foundation to David Rumelhart. I am very grateful to Dean C. Mumme for comments, criticisms, and helpful discussions concerning an earlier version of this manuscript. I would also like to thank David B. Cooper of Brown University for his suggestion that many neural network models might be viewed within a unified statistical framework.
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840 LEARNING IN NETWORKS OF NONDETERMINISTIC ADAPTIVE LOGIC ELEMENTS Richard C. Windecker* AT&T Bell Laboratories, Middletown, NJ 07748 ABSTRACT This paper presents a model of nondeterministic adaptive automata that are constructed from simpler nondeterministic adaptive information processing elements. The first half of the paper describes the model. The second half discusses some of its significant adaptive properties using computer simulation examples. Chief among these properties is that network aggregates of the model elements can adapt appropriately when a single reinforcement channel provides the same positive or negative reinforcement signal to all adaptive elements of the network at the same time. This holds for multiple-input, multiple-output, multiple-layered, combinational and sequential networks. It also holds when some network elements are "hidden" in that their outputs are not directly seen by the external environment. INTRODUCTION There are two primary motivations for studying models of adaptive automata constructed from simple parts. First, they let us learn things about real biological systems whose properties are difficult to study directly: We form a hypothesis about such systems, embody it in a model, and then see if the model has reasonable learning and behavioral properties. In the present work, the hypothesis being tested is: that much of an animal's behavior as determined by its nervous system is intrinsically nondeterministic; that learning consists of incremental changes in the probabilities governing the animal's behavior; and that this is a consequence of the animal's nervous system consisting of an aggregate of information processing elements some of which are individually nondeterministic and adaptive. The second motivation for studying models of this type is to find ways of building machines that can learn to do (artificially) intelligent and practical things. This approach has the potential of complementing the currently more developed approach of programming intelligence into machines. We do not assert that there is necessarily a one-to-one correspondence between real physiological neurons and the postulated model information processing elements. Thus, the model may be loosely termed a "neural network model," but is more accurately described as a model of adaptive automata constructed from simple adaptive parts. * The main ideas in this paper were conceived and initially developed while the author was at the University of Chiang Mai, Thailand (1972-73). The ideas were developed further and put in a form consistent with existing switching and automata theory during the next four years. For two of those years, the author was at the University of Guelph, Ontario, supported of National Research Council of Canada Grant #A6983. © American Institute of Physics 1988 841 It almost certainly has to be a property of any acceptable model of animal learning that a single reinforcement channel providing reinforcement to all the adaptive elements in a network (or subnetwork) can effectively cause that network to adapt appropriately. Otherwise, methods of providing separate, specific reinforcement to all adaptive elements in the network must be postulated. Clearly, the environment reinforces an animal as a whole and the same reinforcement mechanism can cause the animal to adapt to many types of situation. Thus, the reinforcement system is non-specific to particular adaptive elements and particular behaviors. The model presented here has this property. The model described here is a close cousin to the family of models recently described by Barto and coworkers 1-4. The most significant difference are: 1) In the present model, we define the timing discipline for networks of elements more explicitly and completely. This particular timing discipline makes the present model consistent with a nondeterministic extension of switching and automata theory previously described 0. 2) In the present model, the reinforcement algorithm that adjusts the weights is kept very simple. With this algorithm, positive and negative reinforcement have symmetric and opposite effects on the weights. This ensures that the logical signals are symmetric opposites of each other. (Even small differences in the reinforcement algorithm can make both subtle as well as profound differences in the behavior of the model.) We also allow, null, or zero, reinforcemen t. As in the family of models described by Barto, networks constructed within the present model can get "stuck" at a sUboptimal behavior during learning and therefore not arrive at the optimal adapted state. The complexity of the Barto reinforcement algorithm is designed partly to overcome this tendency. In the present work, we emphasize the use of training strategies when we wish to ensure that the network arrives at an optimal state. (In nature, it seems likely that getting "stuck" at suboptimal behavior is common.) In all networks studied so far, it has been easy to find strategies that prevent the network from getting stuck. The chief contributions of the present work are: 1) The establishment of a close connection between these types of models and ordinary, nonadaptive, switching and automata theory 0. This makes the wealth of knowledge in this area, especially network synthesis and analysis methods, readily applicable to the study of adaptive networks. 2) The experimental demonstration that sequential ("recurrent") nondeterministic adaptive networks can adapt appropriately. Such networks can learn to produce outputs that depend on the recent sequence of past inputs, not just the current inputs. 3) The demonstration that the use of training strategies can not only prevent a network from getting stuck, but may also result in more rapid learning. Thus, such strategies may be able to compensate, or even more than compensate, for reduced complexity in the model itself. References 2-4 and 6 provide a comprehensive background and guide to the literature on both deterministic and nondeterministic adaptive automata including those constructed from simple parts and those not. THE MODEL ADAPTIVE ELEMENT The model adaptive element postulated in this work is a nondeterministic, adaptive generalization of threshold logic 7. Thus, we call these elements Nondeterministic Adaptive Threshold-logic gates (NATs). The output chosen by a NAT at any given time is not a function of its inputs. Rather, it is chosen by a stochastic process according to certain probabilities. It is these probabilities that are a function of the inputs. A NAT is like an ordinary logic gate in that it accepts logical inputs that are two-valued and produces a logical output that is two-valued. We let these values be 842 + 1 and -1. A NAT also has a timing input channel and a reinforcement input channel. The NAT operates on a three-part cycle: 1) Logical input signals are changed and remain constant. 2) A timing signal is received and the NAT selects a new output based on the inputs at that moment. The new output remains constant. 3) A reinforcement signal is received and the weights are incremented according to certain rules. Let N be the number of logical input channels, let Xi represent the ith input signal, and let z be the output. The NAT has within it N+ 1 "weights," wo, WI! ... , WN. The weights are confined to integer values. For a given set of inputs, the gate calculates the quantity W: Then the probability that output z = + 1 is chosen is: P(z = +1) w _--=-=W/v2q 1 J e 2q2 dx = _1_ J e-(l d~ .j2; u - 00 ..;; - 00 (2) where ~ = xjV2u. (An equivalent formulation is to let the NAT generate a random number, Wq, according to the normal distribution with mean zero and variance u2 . Then if W > - Wq, the gate selects the output z = + 1. If W < - Wq, the gate selects output z = -1. If W = Wq, the gate selects output -1 or + 1 with equal probability.) Reinforcement signals, R, may have one of three values: + 1, -1, and 0 representing positive, negative, and no reinforcement, respectively. If + 1 reinforcement is received, each weight is incremented by one in the direction that makes the current output, z, more likely to occur in the future when the same inputs are applied; if -1 reinforcement is received, each weight is incremented in the direction that makes the current output less likely; if 0 reinforcement is received, the weights are not changed. These rules may be summarized: ~wo = zR and ~Wj = xjzR for i > o. NATs operate in discrete time because if the NAT can choose output + 1 or -1, depending on a stochastic process, it has to be told when to select a new output. It cannot "run freely," or it could be constantly changing output. Nor can it change output only when its inputs change because it may need to select a new output even when they do not change. The normal distribution is used for heuristic reasons. If a real neuron (or an aggregate of neurons) uses a stochastic process to produce nondeterministic behavior, it is likely that process can be described by the normal distribution. In any case, the exact relationship between P{z = + 1) and W is not critical. What is important is that P(z = + 1) be monotonically increasing in W, go to 0 and 1 asymptotically as W goes to - 00 and + 00, respectively, and equal 0.5 at W = O. The parameter u is adjustable. We use 10 in the computer simulation experiments described below. Experimentally, values near 10 work reasonably well for networks of NATs having few inputs. Note that as u goes to zero, the behavior of a NAT approximates that of an ordinary deterministic ada pt,ive threshold logic gate with the difference that the output for the case W = 0 is not arbitrary: The NAT will select output +1 or -1 with equal probability. Note that for all values of W, the probabilit,ies are greater than zero that either + 1 or -1 will be chosen, although for large values of W (relative to u) for all 843 practical purposes, the behavior is deterministic. There are many values of the weights that cause the NAT to approximate the behavior of a deterministic threshold logic gate. ~or the same reasons that deterministic threshold logic gates cannot realize all 22 functions of N variables 7, so a NAT cannot learn to approximate any deterministic function; only the threshold logic functions. Note also that when the weights are near zero, a NAT adapts most rapidly when both positive and negative reinforcement are used in approximately equal amounts. As the NAT becomes more likely to produce the appropriate behavior, the opportunity to use negative reinforcement decreases while the opportunity to use positive reinforcement increases. This means that a NAT cannot learn to (nearly) always select a certain output if negative reinforcement alone is used. Thus, positive reinforcement has an important role in this model. (In most deterministic models, positive reinforcement is not useful.) Note further that there is no hysteresis in NAT learning. For a given configuration of inputs, a + 1 output followed by a + 1 reinforcement has exactly the same effect on all the weights as a -1 output followed by a -1 reinforcement. So the order of such events has no effect on the final values of the weights. Finally, if only negative reinforcement is applied to a NAT, independent of output, for a particular combination of inputs, the weights will change in the direction that makes W tend toward zero and once there, follow a random walk centered on zero. (The further W is from zero, the more likely its next step will be toward zero.) If all possible input combinations are applied with more or less equal probability, all the weights will tend toward zero and then follow random walks centered on zero. In this case, the NAT will select + 1 or -1 with more or less equal probability without regard to its inputs. NETWORKS NATs may be connected together in networks (NAT-nets). The inputs to a NAT in such a network can be selected from among: 1) the set of inputs to the entire network, 2) the set of outputs from other NATs in the network, and 3) its own output. The outputs of the network may be chosen from among: 1) the inputs to the network as a whole, and 2) the outputs of the various NATs in the network. Following Ref. 5, we impose a timing discipline on a NAT-net. The network is organized into layers such that each NAT belongs to one layer. Letting L be the number of layers, the network operates as follows: 1) All NATs in a given layer receive timing signals at the same time and select a new output at the same time. 2) Timing signals are received by the different layers, in sequence, from 1 to L. 3) Inputs to the network as a whole are levels that may change only before Layer 1 receives its timing signal. Similarly, outputs from the network as a whole are available to the environment only after Layer L has received its timing signal. Reinforcement to the network as a whole is accepted only after outputs are made available to the environment. The same reinforcement signal is distributed to all NATs in the network at the same time. With these rules, NAT-nets operate through a sequence of timing cycles. In each cycle: 1) Network inputs are changed. 2) Layers 1 through L select new outputs, in sequence. 3) Network outputs are made available to the environment. 4) Reinforcement is received from the environment. We call each such cycle a "trial" and a sequence of such trials is a "session." This model is very general. If, for each gate, inputs are selected only from among the inputs to the network as a whole and from the outputs of gates in layers preceding it in the timing cycle, then the network is combinational. In this case, the probability of the network producing a given output configuration is a function of the inputs at the start of the timing cycle. If at least one NAT has one input from a 844 NAT in the same layer or from a subsequent layer in the timing cycle, then the network is sequential. In this case, the network may have "internal states" that allow it to remember information from one cycle to the next. Thus, the probabilities governing its choice of outputs may depend on inputs in previous cycles. So sequential NAT-nets may have short-term memory embodied in internal states and long-term memory embodied in the weights. In Ref. 5, we showed that sequential networks can be constructed by adding feedback paths to combinational networks and any sequential network can be put in this standard form. In information-theoretic terms: 1) A NAT-net with no inputs and some outputs is an "information source." 2) A NAT-net with both inputs and outputs is an information "channel." 3) A combinational NAT-net is "memory-less" while a sequential NAT-net has memory. In this context, note that a NAT-net may operate in an environment that is either deterministic or nondeterministic. Both the logical and the reinforcement inputs can be selected by stochastic processes. Note also that nondeterministic and deterministic elements as well as adaptive and nonadaptive elements can be combined in one network. (It may be that the decision-making parts of an animal's nervous system are nondeterministic and adaptive while the information transmitting parts (sensory data-gathering and the motor output parts) are deterministic and nonadaptive.) One capability that combinational NAT-nets possess is that of "pattern recognizers." A network having many inputs and one or a few outputs can "recognize" a small subset of the potential input patterns by producing a particular output pattern with high probability when a member of the recognized subset appears and a different output pattern otherwise. In practice, the number of possible input patterns may be so large that we cannot present them all for training purposes and must be content to train the network to recognize one subset by distinguishing it (with different output pattern) from another subset. In this case, if a pattern is subsequently presented to the network that has not been in one of the training sets, the probabilities governing its output may approach one or zero, but may well be closer to 0.5. The exact values will depend on the details of the training period. If the new pattern is similar to those in one of the training sets, the NAT-net will often have a high probability of producing the same output as for that set. This associative property is the analog of the well known associative property in deterministic models. If the network lacks sufficient complexity for the separation we wish to make, then it cannot be trained. For example, a single Ninput NAT cannot be trained to recognize any arbitrary set of input patterns by selecting the + 1 output when one of them is presented and -1 otherwise. It can only be trained to make separations that correspond to threshold functions. A combinational NAT-net can also produce patterns. By analogy with a pattern recognizer, a NAT-net with none or a few inputs and a larger number of outputs can learn for each input pattern to produce a particular subset of the possible output patterns. Since the mapping may be few-to-many, instead of many-to-few, the goal of training in this case mayor may not be to have the network approximate deterministic behavior. Clearly, the distinction between pattern recognizers and pattern prod ucers is somewhat arbitrary: in general, NATnets are pattern transducers that map subsets of input patterns into subsets of output patterns. A sequential network can "recognize" patterns in the timesequence of network inputs and produce patterns in the time-sequence of outputs. SIMULATION EXPERIMENTS In this Section, we discuss computer simulation results for three types of multiple-element networks. For two of these types, certain strategies are used to train the networks. In general, these strategies have two parts that alternate, as 845 needed. The first part is a general scheme for providing network inputs and reinforcement that tends to train all elements in the network in the desired direction. The second part is substituted temporarily when it becomes apparent that the network is getting stuck in some suboptimal behavior. It is focussed on getting the network unstuck. The strategies used here are intuitive. In general, there appear to be many strategies that will lead the network to the desired behavior. While we have made some attempt to find strategies that are reasonably efficient, it is very unlikely that the ones used are optimal. Finally, these strategies have been tested in hundreds of training sessions. Although they worked in all such sessions, there may be some (depending on the sequence of random numbers generated) in which they would not work. In describing the networks simulated, Figs. 1-3, we use the diagramatic conventions defined in Ref. 5: We put all NATs in the same layer in a vertical line, with the various layers arranged from left to right in their order in the timing cycle. Inputs to the entire network corne in from the left; outputs go out to the right. Because the timing cycle is fixed, we omit the timing inputs in these figures. For similar reasons, we also omit the reinforcement inputs. In the simulations described here, the weights in the NATs start at zero making the network outputs completely random in the sense that on any given trial, all outputs are equally likely to occur, independent of past or present inputs. As learning proceeds, some or all the weights become large, so that the NAT-net's selection of outputs is strongly influenced by some or all of its inputs and internal connections. (Note that if the weights do not start at zero, they can be driven close to zero by using negative reinforcement.) In general, the optimum behavior toward which the network adapts is deterministic. However, because the probabilities are never identically equal to zero or one, we apply an arbitrary criterion and say that a NAT-net has learned the appropriate behavior when that criterion is satisfied. In real biological systems, we cannot know the weights or the exact probabilities governing the behavior of the individual adaptive elements. Therefore, it is appropriate to use a criterion based on observable behavior. For example, the criterion might be that the network selects the correct response (and continues to receive appropriate reinforcement) 25 times in a row. Note that NAT-nets can adapt appropriately when the environment is not deliberately trying to make the them behave in a particular way. For example, the environment may provide inputs according to some (not necessarily deterministic) pattern and there may be some independent mechanism that determines whether the NAT-net is responding appropriately or not and provides the reinforcement accordingly. One paradigm for this situation is a game in which the NAT-net and the environment are players. The reinforcement scheme is simple: if, according to the rules of the game, the NAT-net wins a play (= trial) of the game, reinforcement is + 1 , if it loses, -1. For a NAT-net to adapt appropriately in this situation, the game must consist of a series of similar plays. If the game is competitive, the best strategy a given player has depends on how much information he has about the opponent and vice versa. If a player assumes that his opponent is all-knowing, then his best strategy is to minimize his maximum loss and this often means playing at random, or a least according to certain probabilities. If a player knows a lot about how his opponent plays, his best strategy may be to maximize gain. This often means playing according to some deterministic strategy. The example networks described here are special cases of three types: pattern producing (combinational multiple-output) networks, pattern recogmzmg (combinational multiple-input, multiple-layered, few-output) networks, and game playing (sequential) networks. The associative properties of NATs and NAT-nets 846 are not emphasized here because they are analogous to the well known associative properties of other related models. A Class of Simple Pattern Producing Networks A simple class of pattern producing networks consists of the single-layer type shown in Fig. 1. Each of M NATs in such a network has no inputs, only an output. As a consequence, each has only one weight, Woo The network is a simple, adaptive, information source. Consider first the case in which the network contains only one NAT and we wish to train it to always produce a simple "pattern," + 1. We give positive reinforcement when it selects + 1 and negative reinforcement otherwise. If Wo starts at 0, it will quickly gr.ow large making the probability of selecting + 1 approach unity. The criterion we use for deciding that the network is trained is that it produce a string of 25 correct outputs. Table I o~-..... z, 0--.' Z2 O~-..... ~3 · · · · · · · • • • • • · 0--.. Z18 Fig. 1. A Simple Pattern Producing Network shows that in 100 sessions, this one-NAT network selected + 1 output for the next 25 trials starting, on average, at trial 13. Next consider a network with two NATs. They can produce four different output patterns. If both weights are 0, they will produce each of the patterns with equal probability. But they can be trained to produce one pattern (nearly) all the time. If we wish to train this subnetwork to produce the pattern (in vector notation) [+1 +1], one strategy is to give no M 1 2 4 8 16 Min 1 8 18 44 49 Ave 13 25 35 70 115 Max 26 43 60 109 215 reinforcement if it produces patterns [-1 +1] or [+1 -1), give it positive reinforcement if it produces [+1 +1] and negative reinforcement if it produces [-1 -1]. Table I shows that in 100 sessions, this network learned to produce the desired pattern (by producing a string of 25 correct outputs) in about 25 trials. Because we initially gave reinforcement only about 50% of the time, it took longer to train two NATS Table I. Training Times For than one. Networks Per Fig. 1. Next, consider the 16-NAT network in Fig. 1. Now there are 216 possible patterns the network can produce. When all the weights are zero, each has probability 2- 16 of being produced. An ineffective strategy for training this network is to provide positive reinforcement when the desired pattern is produced, negative reinforcement when its opposite is produced, and zero reinforcement otherwise. A better strategy is to focus on one output of the network at a time, training each NAT separately (as above) to have a high probability of producing the desired output. Once all are trained to a relatively high level, the network as a whole has a reasonable chance of producing exactly the correct output. Now we can provide positive reinforcement when it does and no reinforcement otherwise. With this two-stage hybrid strategy, the network will soon meet the training criterion. The time it takes to train a network of M elements with a strategy of this type is roughly proportional to M, not 2(M 1), as for the first strategy. 847 A still more efficient strategy is to alternate between a general substrategy and a substrategy focussed on keeping the network from getting "stuck." One effective general substrategy is to give positive reinforcement when more than half of the NATs select the desired output, negative reinforcement when less than half select the desired output, and no reinforcement when exactly half select the desired output. This substrategy starts out with approximately equal amounts of positive and negative reinforcement being applied. Soon, the network selects more than half of the outputs correctly more and more of the time. Unfortunately, there will tend to be a minority subset with low probability of selecting the correct output. At this stage, we must recognize this subset and switch to a substrategy that focuses on the elements of this subset following the strategy for one or two elements, above. When all NATs have a sufficiently high probability of selecting the desired output, training can conclude with the first substrategy. The strategies used to obtain the results for M = 4,8, and 16 in Table I were slightly more complicated variants of this two-part strategy. In all of them, a running average was kept of the number of right responses given by each NAT. Letting OJ be the "correct" output for Zj, the running average after the tt" trial, Aj( t), is: Aj(t) = BAj(t 1) + CjZj(t) (3) where B is a fraction generally in the range 0.75 to 0.9. If Aj(t) for a particular i gets too far below the combined average for all i, then training focuses on the it" element until its average improves. The significance of the results given in Table I is not the details of the strategies used, nor how close the training times may be to the optimum. Rather, it is the demonstration that training strategies exist such that the training time grows significantly more slowly than in proportion to M. A Simple Pattern Recognizing Network As mentioned above, there are fewer threshold logic functions of N variables (for N > 1) than the total possible functions. For N = 2, there are 14. The remining two are the "exclusive or" (XOR) and its complement. Multi-layered networks are x, -~))~---......p)oo--•. Z X2 _-0lil_1:;,.__ __ needed to realize these functions, and an Fig. 2. A Two-Element Network important test of any adaptive network That Learns XOR model is its ability to learn XOR. The network in Fig. 2 is one of the simplest networks capable of learning this function. Table II gives the results of 100 training sessions with this network. The strategy used to obtain these results again had two parts. The general part consisted of supplying each of the four possible input patterns to the network in rotation, glvmg appropriate Network Function Min Ave Fig. 2 Fig. 2 Ref. 2 Ref. 8 OR 18 57 XOR 218 681 XOR -700 -3500 XOR 2232 Table II. Training Times For The Network In Fig. 2. Max 106 1992 -14,300 reinforcement each trial. The second part involved keeping a running average (similar to Eq. (3)) of the responses of the network by input combination. When the average for one combination fell significantly behind 848 the average for all, training was focused on just that combination until performance improved. The criterion used for deciding when training was complete was a sequence of 50 correct responses (for all input patterns together). For comparison, Table II shows results for the same network trained to realize the normal OR function. Also shown for comparison are numbers taken from Refs. 2 and 8 for the equivalent network in those different models. These are nondeterministic and deterministic models, respectively. The numbers from Ref. 2 are not exactly comparable with the present results for several reasons. These include: 1) The criterion for judging when the task was learned was not the same; 2) In Ref. 2, the "wrong" reinforcement was deliberately applied 10% of the time to test learning in this situation; 3) Neither model was optimized for the particular task at hand. Nonetheless, if these (and other) differences were taken into account, it is likely that the NAT-net would have learned the XOR function significantly faster. The significance of the present results is that they suggest that the use of a training strategy can not only prevent a network from getting stuck, but may also facilitate more rapid learning. Thus, such strategies can compensate, or more than compensate, for reduced complexity in the reinforcement algorithm. A Simple Game-Playing Network Here, we consider NAT-nets in the context of the game of "matching pennies." In this game, each player has a stack of pennies. At each play of the game, each player places one of his pennies, heads up or heads down, but covered, in front of him. Each player uncovers his penny at the same time. If they match, player A adds both to his stack, otherwise, player B takes both. Game theory says that the strategy of each player that minimizes his maximum loss is to play heads and tails at random. Then A cannot predict B's behavior and at best can win 50% of the time and likewise for B with respect to A. This is a conservative strategy on the part of each player because each assumes that the other has (or can derive through a sequence of plays), and can use, information about the other player's strategy. Here, we make the different assumption that: 1) Player B does not play at random, 2) Player B has no information about A's strategy, and 3) Player B is incapable of inferring any information about A through a sequence of plays and in any event is incapable of changing its strategy. Then, if A has no information about B's pattern of playing at the start of the game, A's best course of action is to try to infer a non-random pattern in B's playing through a sequence of plays and subsequently take advantage of that knowledge to win more often than 50% of the time. An adaptive NAT-net, as A, can adapt appropriately in situations of this type. For example, suppose a single NAT of the type in Fig. 1 plays A, where + 1 output means heads, -1 output means tails. A third agent supplies reinforcement + 1 if the NAT wins a play, -1 otherwise. Suppose B plays heads with 0.55 probability and tails with 0.45 probability. Then A will learn over time to play heads 100% of the time and thereby maximize its total winnings by winning 55% of the time. A more complicated situation is the following. Suppose B repeats its own move two plays ago 80% of the time, and plays the opposite 20% of the time. A NAT-net with the potential to adapt to this strategy and win 80% of the time is shown in Fig. 3. This is a sequential network shown in the standard form of a combinational network (in the dotted rectangle) plus a feedback path. The input to the network at time tis B's play at t 1. The output is A's move. The top NAT selects its output at time t based partly on the bottom NAT's output at time t 1. The bottom NAT selects its output at t 1 based on its input at that time which is B's output at t 2. Thus, the network as a whole can learn to select its 849 output based on B's play two time increments past. Simulation of 100 sessions resulted in the network learning to do this 98 times. On average, it took 468 plays (Min 20, max 4137) to reach the point at which the network repeated B's move two times past on the next 50 plays. For two sessions the network got stuck (for an unknown number of plays greater than 25,000) playing the opposite of B's last move or always playing tails. {The first two-part strategy found that trains the network to repeat B's output two time increments past without getting stuck (not in the game-playing context) took an average of 260 trials (Min 25, Max 1943) to meet the training criterion.) x----~ Hi----.... Z Fig. 3. A Sequential GamePlaying Network The significance of these results is that a sequential NAT-net can learn to produce appropriate behavior. Note that hidden NATs contributed to appropriate behavior for both this network and the one that learned XOR, above. CONCLUDING REMARKS The examples above have been kept simple in order to make them readily understandable. They are not exhaustive in the sense of covering all possible types of situations in which NAT-nets can adapt appropriately. Nor are they definitive in the sense of proving generally and in what situations NAT-nets can adapt appropriately. Rather, they are illustrative in the sense of demonstrating a variety of significant adaptive abilities. They provide an existence proof that NAT-nets can adapt appropriately and relatively easily in a wide variety of situations. The fact that nondeterministic models can learn when the same reinforcement is applied to all adaptive elements, while deterministic models generally cannot, supports the hypothesis that animal nervous systems may be (partly) nondeterministic. Experimental characterization of how animal learning does, or does not get "stuck," as a function of learning environment or training strategy, would be a useful test of the ideas presented here. REFERENCES 1. Barto, A. G., "Game-Theoretic Cooperativity in Networks of Self-Interested Units," pp. 41-46 in Neural Networks for Computing, J. S. Denker, Ed., AlP Conference Proceedings 151, American Institute of Physics, New York, 1986. 2. Barto, A. G., Human Neurobiology, 4, 229-256, 1985. 3. Barto, A. G., R. S. Sutton, and C. W. Anderson, IEEE Transactions on Systems, Man, and Cybernetics, SMC-13, No.5, 834-846, 1983. 4. Barto, A. G., and P. Anandan, IEEE Transactions on Systems, Man, and Cybernetics, SMC-15, No.3, 360-375, 1985. 5. Windecker, R. C., Information Sciences, 16, 185-234 (1978). 6. Rumelhart, D. E., and J. L. McClelland, Parallel Distributed Processing, MIT Press, Cambridge, 1986. 7. Muroga, S., Threshold Logic And Its Applications, Wiley-Interscience, New York, 1971. 8. Rumelhart, D. E., G. E. Hinton, and R. J. Williams, Chapter 8 in Ref. 6.
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HIGH DENSITY ASSOCIATIVE MEMORIES! A"'ir Dembo Information Systems Laboratory, Stanford University Stanford, CA 94305 Ofer Zeitouni Laboratory for Information and Decision Systems MIT, Cambridge, MA 02139 ABSTRACT 211 A class of high dens ity assoc iat ive memories is constructed, starting from a description of desired properties those should exhib it. These propert ies include high capac ity, controllable bas ins of attraction and fast speed of convergence. Fortunately enough, the resulting memory is implementable by an artificial Neural Net. I NfRODUCTION Most of the work on assoc iat ive memories has been structure oriented, i.e.. given a Neural architecture, efforts were directed towards the analysis of the resulting network. Issues like capacity, basins of attractions, etc. were the main objects to be analyzed cf., e.g. [1], [2], [3], [4] and references there, among others. In this paper, we take a different approach, we start by explicitly stating the desired properties of the network, in terms of capacity, etc. Those requirements are given in terms of axioms (c.f. below). Then, we bring a synthesis method which enables one to design an architecture which will yield the desired performance. Surprisingly enough, it turns out that one gets rather easily the following properties: (a) High capacity (unlimited in the continuous state-space case, bounded only by sphere-packing bounds in the discrete state case). (b) Guaranteed basins of attractions in terms of the natural metric of the state space. (c) High speed of convergence in the guaranteed basins of attraction. Moreover, it turns out that the architecture suggested below is the only one which satisfies all our axioms (-desired properties-)I Our approach is based on defining a potential and following a descent algorithm (e.g., a gradient algorithm). The main design task is to construct such a potential (and, to a lesser extent, an implementat ion of the descent algorithm via a Neural network). In doing so, it turns out that, for reasons described below, it is useful to regard each des ired memory locat ion as a -part iclein the state space. It is natural to require now the following requirement from a IAn expanded version of this work has been submitted to Phys. Rev. A. This work was carried out at the Center for Neural Sc ience, Brown University. © American Institute of Physics 1988 212 .eJlOry: (Pl) The potential should be linear w.r.t. adding partic les in the sense that the potential of two particles should be the sum of the potentials induced by the individual particles (i.e •• we do not allow interparticles interaction). (P2) Part icle locat ions are the only poss ible sites of stable .emory locations. (P3) The system should be invariant to translations and rotations of the coordinates. We note that the last requirement is made only for the sake of simplicity. It is not essential and may be dropped without affecting the results. In the sequel. we construct a potential which satisfies the above requirements. We refer the reader to [5] for details of the proofs. etc. Acknowledgements. We would like to thank Prof. L.N. Cooper and C.M. Bachmann for many fruitful discussions. In particular. section 2 is part of a joint work with them ([6]). 2. HIGH DENSIlY STORAGE MODEL In what follows we present a particular case of a method for the construct ion of a high storage dens ity neural memory. We define a function with an arbitrary number of minima that lie at preassigned points and define an appropriate relaxat ion procedure. The general case in presented in [5]. Let i 1 ..... i m be a set of m arb itrary d ist inct memories in RN. The ·energy· function we will use is: m ~ = - i 2 Qi Iii - ii I-L (1) i=l where we assume throughout that N ~ 3. L ~ (N - 2). and Qi > 0 and use 1 ••• 1 to denote the Euclidean distance. Note that for L = 1. NF3. ~ is the electrostat ic potent ial induced by negat ive fixed part ic les with charges -Qi. This ·energy· funct ion possesses global minima at i 1 ••••• i m (where ~(ii) .. -) and has no local minima except at these points. A rigorous proof is presented in [5] together with the complete characterization of functions having this property. As a relaxation procedure. we can choose any dynamical system for which ~ is strictly decreasing. uniformly in compacts. In this instance. the theory of dynamical systems guarantees that for almost any initial data. the trajectory of the system converges to one of the desired points i 1 ••••• i m• However. to give concrete results and to further exploit the resemblance to electrostatic. consider the relaxation: 213 . .. -= EII Il -= (2) i=1 where for N=3. L=1. equation (2) describes the motion of a positive t~st particle in the electrost!tic f~eld ~ generated by the negative f1xed charges -Q1 •••• L -~ at xl ••••• xm• Since the field E;i is just minus the gradient of e. it is clear that along trajectories of (2). de/dt ~ O. with equality only at the fbed points of (2). which are exactly the stat ionary po ints of e. Therefore. using (2) as the relaxation procedure. we can conclude that entering at any ~(O). the system converges to a stationary point of e. The space of inputs is partitioned into m domains of attraction. each one corresponding to a different memory. and the boundaries (a set of measure zero). on which p(O) will converge to a saddle point of e. We can now explain why e~ has no spurious local minima. at least for L=1. N=3. using elementary physical arguments. Suppose e has a spurious local minima at y ~ xl ••••• xm• then in a s!!all neighborhood of y which does not include any of the xi. the field ~ points towards y. Thus. on any closed surface in that neighborhood. the integral of the normal inward component of ~ is positive. However. this integral is just the total charge included inside the surface. which is zero. Thus we arrive at a contradiction. so y can not be a local minimum. We now have a relaxation procedure. such that almost any ~(O) is attracted by one of the xi. but we have not yet spec ified the shapes of the basins of attraction. By varying the charges Qi. we can enlarge one basin of attraction at the expense of the others (and vice versa). Even when all of the Qi are eqmal. the position of the xi might cause ~(O) not to converge to the closest memory. as emphasized in the example in fig. 1. However. let r = min1~i~j~mlxi i j 1 be the minimal distance between any two memoriesJ then if I~(O) - ii I~ ,[ • .,lIk) it can be shown that ~(O) will converge to xi. (provided that k L +! =-N+i 11). Thus. if thamemories are densely packed in a hypersphere. by choosing k large enough (i.e. enlarging the parameter L). convergence to the closest memory for any -interestinginput. that is an input i;:(O) with a distinct closest memory. is guaranteed. The detailed proof of the above property is given in [5]. It is based on bound ing the number of xj • j~i. in a hypersphere of radius R(Rlr) around xi. by [2R/r + 1]N. tlien bounding the magnitude of the field induced 'I. any Xj. j~i. on the boundar, of such a hypersphere by (R-li;:(O)-xiP+1). and finally integrat ing to show that for I~(O)-ii 15. (i~~I/~ ,with e<1. the convergence of ~(O) to xi is within finite time T. which behaves like e L+2 for L » 1 and e < 1 and fixed. Intuitively the reason for 214 this behaviour equat ion (2) • convergence rate '. ,I I " Figure 1 R » I and 0 « 1 is the short-range nature of the fields used in Because of this. we also expect extremely low for inputs ~(O) far away from all of the xi. The radial nature of these fields suggests a way to overcome this difficulty. that is to increase the convergence rate from points very far away. without disturbing all of the aforementioned desirable properties of the model. Assume that we know in advance that all of the xi lie inside some large hypersphere S around the origin. Then. at any point ~ outside S. the field ~ has a positive projection radially into S. By adding a longrange force to B-. effective only outside of S. we can hasten the mgvement towards S. from points far away, without creating additional minima inside of S. As an example the force (-~ for ji , S, 0 for ji 8 S) will pull any test input ji(O) to the boundary of S within the small finite time T ~ 1/1SI. and from then on the system wil} behave inside S according to the original field Bu. Up to this point. our derivations have been for a continuous system. but from it we can deduce a discrete system. We shall do this mainly for a clearer comparison between our high density memory model and the discrete version of Hopfield's model. Before continuing in that direction. note that our continuous system has unl imited storage capacity unlike Hopfield's continuous system. which like his discrete model, has limited capac ity. For the discrete system, assume that the Xi are composed of elements ±1 and replace the Euclid\an dJstance in (1) with the normal ized Hamming 4 istance lii1 ~21 = 1; I '=1111~ 11~ I. This places the vec tors :i i on the un it hypersphere. J J J The relaxation process for the discrete system will be of the type defined in Hopfield's model in [11 Choose at random a component to be updated (that is, a neighbor ~' of ii such that Iii' - iii = 2/N). calculate the "energy" difference. r.e = ~(ii~-~(ii). and only if r.e < O. change this component, that is: 11· ~f.l. sign(~(~~ ~(ji», 1 1 (3) where e(ii) is the potent ial energN in (1). Since there is a finite number of possible ~ vectors (2), convergence in finite time is guaranteed. This relaxation procedure is rigid since the movement is limited to points with components +1. Therefore. although the local minima of ~(ii) defined in (2) are only at the desired points Xi' the relaxation may get stuck at some ii which is not a stationary point of ~(ii). However, the short range behaviour of the potential e(~), unlike the long-range behavior of the quadratic potential used by Hopfield, gives 215 rise to results similar to those we have quoted for the continuous ll10del (equation (1». Specifically. let the stored me~ories i 1 ••••• i m be separated from one another by having at least pN different components (0 < p i 1/2 and p fixed), and let ~(O) agree up to at least one ii with at most epN errors between them (0 i e < 1/2. with e fixed), then jHO) converges monotonically to i i by the relaxat ion procedure given in equat ion (3). This result holds independently of m. provided that N is large enough (typically. Np In(1~e) L 1) and L is chosen so that f i In(!~e) The proof is constructed by bounding the cummulative effect of terms I~ - ii rL. j;&i. to t~e energy difference Se and showing that it is dominafed by I~ - ii 1 L. For details. we refer the reader again to [5]. Note the importance of this property: unlike the Hopfield model which is limited to miN. the suggested system is optimal in the sense of Information Theory. since for every set of memories i 1 ••••• i m separated from each other by a Hamming distance pN. up to 1/2 pN errors in the input can be corrected. provided that N is large and L properly chosen. As for the complexity of the system. we note that the nonlinear operat ion a -L. for a}O and L integer (which is at (the heart of our system computationally)' is equivalent to e-Lln a) and can be implemented. therefore. by a simple electrical circuit composed of diodes. which have exponential input-output characteristics. and resistors. which can carry out the necessary multiplications (cf. the implementation of section 3). Further. since both liil and I~I are held fixed in the discrete system. where all states are on the unit hypersphere. I~ - ii 12 is equivalent to the inner product of ~ and ii' up to a constant. To conclude. the suggested model involves about m'N multiplications. followed by m nonlinear operations. and then m'N additions. The original model of Hopfield involves Nf multiplications and additions. and then N nonlinear operations. but is limited to miN. Therefore. whenever the Hopfield model is applicable the complexity of both ll10dels is comparable. 3. IMPLEMENI'ATION We propose below one possible network which implements the discrete time and space version of the model described above. An implementation for the ocntinuous time case. which is even simpler. is also hinted. We point out that the implementation described below is by no means unique. (and maybe even not the simplest one). Moreover. the -neurons· used are artificial neurons which perform various tasks. as follows: There are (N+1) neurons which are delay elements. and \'l'\. pOintwise non-linear functions (which may be interpreted as delayless. intermediate neurons). There are ~N synaptic connections between those two layers of neurons. In addition. as in the Hopfield 216 model, we have at each iteration to specify (either deterministically or stochastically) which coordinate are we updating. To do that, we use an N dimensional ·control register· whose content is always a unit vector of {O, l}N (and the location of the '1' will denote the next coordiante to be changed). This vector may be varied from instant n to n + 1 either by shift (·sequential coordinate update·) or at random. Let Ai' UUN be the i-th output of the ·co1!,trol· register, xi' l~UN and V be the (N+1) I!eurons inputs and xi = xi (l-2Ai ) the corresponding outputs (where xi' xi8{+1,-1), Ai 8{0,1}, but V is a real number), _j' l~j~ be the input of the j-th inte;medi~te neuron (-1~_ ~1), ~j = -(1-_ )-L be its output, and 'ji = uij IN be the synaptiC weight of thJ ij - th synapsis, where u~j) refers here to the i-th element of the j-th memory. The system's equations are: The V = + CD. we made As sphere) 1 < i ~ N (4a) 1 ~ j < m (4b) ~ "" -(1 __ )-L j j (4c) (4d) 1 S = i"(l-sign(V - V» (4e) 1 < i ~ N (4f) V ~V + SV (4g) system is initialized by xi = xi (0) (the probe vector), and A block diagram of this sytem appears in Fig. 2. Note that use of N + m + 1 neurons and O(Nm) connections. for the continuous time case (with memories on the unit we will get the equations: 217 m LN 2 Xi + 2m VX i = "jil1j. 1 ~ i ~ N (Sa) j=l N N N 2 " . i X .• 6 2 2 -j = = x .• J 1 1 1 ~ j ~ m (Sb) i=l i .. l _(L + 1) l1j = (1 + 6 2_ j) 'I" • 1 < j ~ m (Sc) ~ - 2 V = l1j (Sd) j=l with similar interpretation (here there is no 'control' register as all components are updated continuously). s Legend @] Deloy Unit (Neuron) i _~o Synoptic Switch (0 =Zi, c =0) fc t c = I ( { , c=O) Synoptic Switch 0= .. '2 C = I Computation UnIt (0= 1/2(1-sign(i2-i, Il) Figure 2 Neural Network Implementotion 218 REFERENCES 1. 1.1. Bopfield. -Neural Networks and Physical Systems with Emergent Collective Computational Abilities-. Proc. Nat. Acad. Sci. U.S.A •• Vol. 79 (1982). pp. 2554-2558. 2. R.I. McEliece. et al •• -The Capacity of the Hopfield Associative Memory-. IEEE" Trans. on Inf. Theory. Vol. IT-33 (1987). pp. 461482. 3. A. Dembo. -On the Capac ity of the Hopfield Memory-. submitted. IEEE Trans. on Inf. Theory. 4. Kohonen. T •• Self Organization and Associative Memory. Springer. Berlin. 1984. 5". Dembo. A. and Ze itouni. 0 •• General Potent ial Surfaces and Neural Networks. submitted. Phys. Rev. A. 6. Bachmann. C.M.. Cooper. L.N., Dembo. A. and Zeitouni. 0.. A relazation Model for Memory with high storage density. to appear. Proc. Natl. Ac. Science.
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A MEAN FIELD THEORY OF LAYER IV OF VISUAL CORTEX AND ITS APPLICATION TO ARTIFICIAL NEURAL NETWORKS* Christopher L. Scofield Center for Neural Science and Physics Department Brown University Providence, Rhode Island 02912 and Nestor, Inc., 1 Richmond Square, Providence, Rhode Island, 02906. ABSTRACT 683 A single cell theory for the development of selectivity and ocular dominance in visual cortex has been presented previously by Bienenstock, Cooper and Munrol. This has been extended to a network applicable to layer IV of visual cortex2 . In this paper we present a mean field approximation that captures in a fairly transparent manner the qualitative, and many of the quantitative, results of the network theory. Finally, we consider the application of this theory to artificial neural networks and show that a significant reduction in architectural complexity is possible. A SINGLE LAYER NETWORK AND THE MEAN FIELD APPROXIMATION We consider a receive signals from the layer (Figure 1). single layer network of ideal neurons which outside of the layer and from cells within The activity of the ith cell in the network is c' - m' d + ~ T .. c' 1 1 ""' ~J J' J (1) d is a vector of afferent signals to the network. Each cell receives input from n fibers through the matrix of synapses is then transmitted through synapses L. outside of the cortical network mi' Intra-layer input to each cell the matrix of cortico-cortical © American Institute of Physics 1988 684 ... . . Afferent > Signals d m1 m2 mn ~ r;. ",.... : L 1 2 ... .. ~ ,~ , ... .. , ~ c Figure 1: The general single layer recurrent network. Light circles are the LGN -cortical synapses. Dark circles are the (nonmodifiable) cortico-cortical synapses. We now expand the response of the ith cell into individual terms describing the number of cortical synapses traversed by the signal d before arriving through synapses Lij at cell i. Expanding Cj in (1), the response of cell i becomes ci = mi d + l: ~j mj d + l: ~jL Ljk mk d + 2: ~j 2Ljk L Lkn mn d + ... (2) J J K J K' n Note that each term contains a factor of the form This factor describes the first order effect, on cell q, of the cortical transformation of the signal d. The mean field approximation consists of estimating this factor to be a constant, independant of cell location (3) 685 This assumption does not imply that each cell in the network is selective to the same pattern, (and thus that mi = mj). Rather, the assumption is that the vector sum is a constant This amounts to assuming that each cell in the network is surrounded by a population of cells which represent, on average, all possible pattern preferences. Thus the vector sum of the afferent synaptic states describing these pattern preferences is a constant independent of location. Finally, if we assume that the lateral connection strengths are a function only of i-j then Lij becomes a circular matrix so that r. Lij ::: ~ Lji = Lo = constan t. 1 J Then the response of the cell i becomes (4) for I ~ I < 1 where we define the spatial average of cortical cell activity C = in d, and N is the average number of intracortical synapses. Here, in a manner similar to that in the theory of magnetism, we have replaced the effect of individual cortical cells by their average effect (as though all other cortical cells can be replaced by an 'effective' cell, figure 2). Note that we have retained all orders of synaptic traversal of the signal d. Thus, we now focus on the activity of the layer after 'relaxation' to equilibrium. In the mean field approximation we can therefore write (5) where the mean field a =am with 686 and we asume that inhibitory). Lo < 0 (the network is, on average, Afferent > Signals d Figure 2: The single layer mean field network. Detailed connectivity between all cells of the network is replaced with a single (nonmodifiable) synapse from an 'effective' cell. LEARNING IN THE CORTICAL NETWORK We will first consider evolution of the network according to a synaptic modification rule that has been studied in detail, for single cells, elsewhere!· 3. We consider the LGN - cortical synapses to be the site of plasticity and assume for maximum simplicity that there is no modification of cortico-cortical synapses. Then (6) . Lij = O. In what follows c denotes the spatial average over cortical cells, while Cj denotes the time averaged activity of the ith cortical cell. The function cj> has been discussed extensively elsewhere. Here we note that cj> describes a function of the cell response that has both hebbian and anti-hebbian regions. 687 This leads to a very complex set of non-linear stochastic equations that have been analyzed partially elsewhere2 . In general, the afferent synaptic state has fixed points that are stable and selective and unstable fixed points that are nonselective!, 2. These arguments may now be generalized for the network. In the mean field approximation (7) The mean field, a has a time dependent component m. This varies as the average over all of the network modifiable synapses and, in most environmental situations, should change slowly compared to the change of the modifiable synapses to a single cell. Then in this approximation we can write • (mi(a)-a) = cj>[mi(a) - a] d. (8) We see that there is a mapping mi' <-> mica) - a (9) such that for every mj(a) there exists a corresponding (mapped) point mj' which satisfies the original equation for the mean field zero theory. It can be shown 2, 4 that for every fixed point of mj( a = 0), there exists a corresponding fixed point mj( a) with the same selectivity and stability properties. The fixed points are available to the neurons if there is sufficient inhibition in the network (ILo I is sufficiently large). APPLICATION OF THE MEAN FIELD NETWORK TO LAYER IV OF VISUAL CORTEX Neurons in the primary visual cortex of normal adult cats are sharply tuned for the orientation of an elongated slit of light and most are activated by stimulation of either eye. Both of these properties--orientation selectivity and binocularity--depend on the type of visual environment experienced during a critical 688 period of early postnatal development. For example, deprivation of patterned input during this critical period leads to loss of orientation selectivity while monocular deprivation (MD) results in a dramatic shift in the ocular dominance of cortical neurons such that most will be responsive exclusively to the open eye. The ocular dominance shift after MD is the best known and most intensively studied type of visual cortical plasticity. The behavior of visual cortical cells in various rearing conditions suggests that some cells respond more rapidly to environmental changes than others. In monocular deprivation, for example, some cells remain responsive to the closed eye in spite of the very large shift of most cells to the open eyeSinger et. al.5 found, using intracellular recording, that geniculo-cortical synapses on inhibitory interneurons are more resistant to monocular deprivation than are synapses on pyramidal cell dendrites. Recent work suggests that the density of inhibitory GABAergic synapses in kitten striate cortex is also unaffected by MD during the cortical period 6, 7. These results suggest that some LGN -cortical synapses modify rapidly, while others modify relatively slowly, with slow modification of some cortico-cortical synapses. Excitatory LGNcortical synapses into excitatory cells may be those that modify primarily. To embody these facts we introduce two types of LGN -cortical synapses: those (mj) that modify and those (Zk) that remain relatively constant. In a simple limit we have and (10) We assume for simplicity and consistent with the above physiological interpretation that these two types of synapses are confined to two different classes of cells and that both left and right eye have similar synapses (both m i or both Zk) on a given cell. Then, for binocular cells, in the mean field approximation (where binocular terms are in italics) 689 where dl(r) are the explicit left (right) eye time averaged signals arriving form the LGN. Note that a1(r) contain terms from modifiable and non-modifiable synapses: al(r) = a (ml(r) + zl(r»). Under conditions of monocular deprivation, the animal is reared with one eye closed. For the sake of analysis assume that the right eye is closed and that only noise-like signals arrive at cortex from the right eye. Then the environment of the cortical cells is: d = (di, n) (12) Further, assume that the left eye synapses have reached their 1 r selective fixed point, selective to pattern d 1• Then (mi' mi) = (m:*, xi) with IXil «lm!*1. Following the methods of BCM, a local linear analysis of the <I> function is employed to show that for the closed eye Xi = a (1 - }..a)-li.r. (13) where A. = NmIN is the ratio of the number modifiable cells to the total number of cells in the network. That is, the asymptotic state of the closed eye synapses is a scaled function of the meanfield due to non-modifiable (inhibitory) cortical cells. The scale of this state is set not only by the proportion of non-modifiable cells, but in addition, by the averaged intracortical synaptic strength Lo. Thus contrasted with the mean field zero theory the deprived eye LGN-cortical synapses do not go to zero. Rather they approach the constant value dependent on the average inhibition produced by the non-modifiable cells in such a way that the asymptotic output of the cortical cell is zero (it cannot be driven by the deprived eye). However lessening the effect of inhibitory synapses (e.g. by application of an inhibitory blocking agent such as bicuculine) reduces the magnitude of a so that one could once more obtain a response from the deprived eye. 690 We find, consistent with previous theory and experiment, that most learning can occur in the LGN-cortical synapse, for inhibitory (cortico-cortical) synapses need not modify. Some non-modifiable LGN-cortical synapses are required. THE MEAN FIELD APPROXIMATION AND ARTIFICIAL NEURAL NETWORKS The mean field approximation may be applied to networks in which the cortico-cortical feedback is a general function of cell activity. In particular, the feedback may measure the difference between the network activity and memories of network activity. In this way, a network may be used as a content addressable memory. We have been discussing the properties of a mean field network after equilibrium has been reached. We now focus on the detailed time dependence of the relaxation of the cell activity to a state of equilibrium. Hopfield8 introduced a simple formalism for the analysis of the time dependence of network activity. In this model, network activity is mapped onto a physical system in which the state of neuron activity is considered as a 'particle' on a potential energy surface. Identification of the pattern occurs when the activity 'relaxes' to a nearby minima of the energy. Thus mlmma are employed as the sites of memories. For a Hopfield network of N neurons, the intra-layer connectivity required is of order N2. This connectivity is a significant constraint on the practical implementation of such systems for large scale problems. Further, the Hopfield model allows a storage capacity which is limited to m < N memories8, 9. This is a result of the proliferation of unwanted local minima in the 'energy' surface. Recently, Bachmann et al. l 0, have proposed a model for the relaxation of network activity in which memories of activity patterns are the sites of negative 'charges', and the activity caused by a test pattern is a positive test 'charge'. Then in this model, the energy function is the electrostatic energy of the (unit) test charge with the collection of charges at the memory sites E = -IlL ~ Qj I J-l- Xj I - L, J (14) 691 where Jl (0) is a vector describing the initial network activity caused by a test pattern, and Xj' the site of the jth memory. L is a parameter related to the network size. This model has the advantage that storage density is not restricted by the the network size as it is in the Hopfield model, and in addition, the architecture employs a connectivity of order m x N. Note that at each stage in the settling of Jl (t) to a memory (of network activity) Xj' the only feedback from the network to each cell is the scalar ~ Q. I Jl- X· I - L J J J (15) This quantity is an integrated measure of the distance of the current network state from stored memories. Importantly, this measure is the same for all cells; it is as if a single virtual cell was computing the distance in activity space between the current state and stored states. The result of the computation is then broadcast to all of the cells in the network. This is a generalization of the idea that the detailed activity of each cell in the network need not be fed back to each cell. Rather some global measure, performed by a single 'effective' cell is all that is sufficient in the feedback. DISCUSSION We have been discussing a formalism for the analysis of networks of ideal neurons based on a mean field approximation of the detailed activity of the cells in the network. We find that a simple assumption concerning the spatial distribution of the pattern preferences of the cells allows a great simplification of the analysis. In particular, the detailed activity of the cells of the network may be replaced with a mean field that in effect is computed by a single 'effective' cell. Further, the application of this formalism to the cortical layer IV of visual cortex allows the prediction that much of learning in cortex may be localized to the LGN-cortical synaptic states, and that cortico-cortical plasticity is relatively unimportant. We find, in agreement with experiment, that monocular deprivation of the cortical cells will drive closed-eye responses to zero, but chemical blockage of the cortical inhibitory pathways would reveal non-zero closed-eye synaptic states. 692 Finally, the mean field approximation allows the development of single layer models of memory storage that are unrestricted in storage density, but require a connectivity of order mxN. This is significant for the fabrication of practical content addressable memories. ACKNOWLEOOEMENTS I would like to thank Leon Cooper for many helpful discussions and the contributions he made to this work. *This work was supported by the Office of Naval Research and the Army Research Office under contracts #NOOOI4-86-K-0041 and #DAAG-29-84-K-0202. REFERENCES [1] Bienenstock, E. L., Cooper, L. N & Munro, P. W. (1982) 1. Neuroscience 2, 32-48. [2] Scofield, C. L. (I984) Unpublished Dissertation. [3] Cooper, L. N, Munro, P. W. & Scofield, C. L. (1985) in Synaptic Modification, Neuron Selectivity and Nervous System Organization, ed. C. Levy, J. A. Anderson & S. Lehmkuhle, (Erlbaum Assoc., N. J.). [4] Cooper, L. N & Scofield, C. L. (to be published) Proc. Natl. Acad. Sci. USA .. [5] Singer, W. (1977) Brain Res. 134, 508-000. [6] Bear, M. F., Schmechel D. M., & Ebner, F. F. (1985) 1. Neurosci. 5, 1262-0000. [7] Mower, G. D., White, W. F., & Rustad, R. (1986) Brain Res. 380, 253-000. [8] Hopfield, J. J. (1982) Proc. Natl. A cad. Sci. USA 79, 2554-2558. [9] Hopfield, J. J., Feinstein, D. 1., & Palmer, R. O. (1983) Nature 304, 158-159. [10] Bachmann, C. M., Cooper, L. N, Dembo, A. & Zeitouni, O. (to be published) Proc. Natl. Acad. Sci. USA.
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NEURAL NETWORKS FOR TEMPLATE MATCHING: APPLICATION TO REAL-TIME CLASSIFICATION OF THE ACTION POTENTIALS OF REAL NEURONS Yiu-fai Wongt, Jashojiban Banikt and James M. Bower! tDivision of Engineering and Applied Science !Division of Biology California Institute of Technology Pasadena, CA 91125 ABSTRACT 103 Much experimental study of real neural networks relies on the proper classification of extracellulary sampled neural signals (i.e. action potentials) recorded from the brains of experimental animals. In most neurophysiology laboratories this classification task is simplified by limiting investigations to single, electrically well-isolated neurons recorded one at a time. However, for those interested in sampling the activities of many single neurons simultaneously, waveform classification becomes a serious concern. In this paper we describe and constrast three approaches to this problem each designed not only to recognize isolated neural events, but also to separately classify temporally overlapping events in real time. First we present two formulations of waveform classification using a neural network template matching approach. These two formulations are then compared to a simple template matching implementation. Analysis with real neural signals reveals that simple template matching is a better solution to this problem than either neural network approach. INTRODUCTION For many years, neurobiologists have been studying the nervous system by using single electrodes to serially sample the electrical activity of single neurons in the brain. However, as physiologists and theorists have become more aware of the complex, nonlinear dynamics of these networks, it has become apparent that serial sampling strategies may not provide all the information necessary to understand functional organization. In addition, it will likely be necessary to develop new techniques which sample the activities of multiple neurons simultaneouslyl. Over the last several years, we have developed two different methods to acquire multineuron data. Our initial design involved the placement of many tiny micro electrodes individually in a tightly packed pseudo-floating configuration within the brain2. More recently we have been developing a more sophisticated approach which utilizes recent advances in silicon technology to fabricate multi-ported silicon based electrodes (Fig. 1). Using these electrodes we expect to be able to readily record the activity patterns of larger number of neurons. As research in multi-single neuron recording techniques continue, it has become very clear that whatever technique is used to acquire neural signals from many brain locations, the technical difficulties associated with sampling, data compressing, storing, analyzing and interpreting these signals largely dwarf the development of the sampling device itself. In this report we specifically consider the need to assure that neural action potentials (also known as "spikes") on each of many parallel recording channels are correctly classified, which is just one aspect of the problem of post-processing multi-single neuron data. With more traditional single electrode/single neuron recordings, this task usually in© American Institute or Physics 1988 104 volves passing analog signals through a Schmidt trigger whose output indicates the occurence of an event to a computer, at the same time as it triggers an oscilloscope sweep of the analog data. The experimenter visually monitors the oscilloscope to verify the accuracy of the discrimination as a well-discriminated signal from a single neuron will overlap on successive oscilloscope traces (Fig. Ic). Obviously this approach is impractical when large numbers of channels are recorded at the same time. Instead, it is necessary to automate this classification procedure. In this paper we will describe and contrast three approaches we have developed to do this. Traces on upper layer ~ 'a. E IV 1~ 0 2 4 Traces Ume (msec) on lower layer C., &. Recording s~e b. 75sq.jllT1 Fig. 1. Silicon probe being developed in our lababoratory for multi-single unit recording in cerebellar cortex. a) a complete probe; b) surface view of one recording tip; c) several superimposed neuronal action potentials recorded from such a silicon electrode ill cerebellar cortex. While our principal design objective is the assurance that neural waveforms are adequately discriminated on multiple channels, technically the overall objective of this research project is to sample from as many single neurons as possible. Therefore, it is a natural extention of our effort to develop a neural waveform classification scheme robust enough to allow us to distinguish activities arising from more than one neuron per recording site. To do this, however, we now not only have to determine that a particular signal is neural in origin, but also from which of several possible neurons it arose (see Fig. 2a). While in general signals from different neurons have different waveforms aiding in the classification, neurons recorded on the same channel firing simultaneously or nearly simultaneously will produce novel combination waveforms (Fig. 2b) which also need to be classified. It is this last complication which particularly 105 bedevils previous efforts to classify neural signals (For review see 5, also see 3-4). In summary, then, our objective was to design a circuit that would: 1. distinguish different waveforms even though neuronal discharges tend to be quite similar in shape (Fig. 2a); 2. recognize the same waveform even though unavoidable movements such as animal respiration often result in periodic changes in the amplitude of a recorded signal by moving the brain relative to the tip of the electrode; 3. be considerably robust to recording noise which variably corrupts all neural recordings (Fig. 2); 4. resolve overlapping waveforms, which are likely to be particularly interesting events from a neurobiological point of view; 5. provide real-time performance allowing the experimenter to detect problems with discrimination and monitor the progress of the experiment; 6. be implementable in hardware due to the need to classify neural signals on many channels simultaneously. Simply duplicating a software-based algorithm for each channel will not work, but rather, multiple, small, independent, and programmable hardware devices need to be constructed. I 50 Jl.V b. signal recorded c. electrode a. Fig. 2. a) Schematic diagram of an electrode recording from two neuronal cell bodies b) An actual multi-neuron recording. Note the similarities in the two waveforms and the overlapping event. c) and d) Synthesized data with different noise levels for testing classificat.ion algorithms (c: 0.3 NSR ; d: 1.1 NSR) . 106 METHODS The problem of detecting and classifying multiple neural signals on single voltage records involves two steps. First, the waveforms that are present in a particular signal must be identified and the templates be generated; second, these waveforms must be detected and classified in ongoing data records. To accomplish the first step we have modified the principal component analysis procedure described by Abeles and Goldstein3 to automatically extract templates of the distinct waveforms found in an initial sample of the digitized analog data. This will not be discussed further as it is the means of accomplishing the second step which concerns us here. Specifically, in this paper we compare three new approaches to ongoing waveform classification which deal explicitly with overlapping spikes and variably meet other design criteria outlined above. These approaches consist of a modified template matching scheme, and two applied neural network implementations. We will first consider the neural network approaches. On a point of nomenclature, to avoid confusion in what follows, the real neurons whose signals we want to classify will be referred to as "neurons" while computing elements in the applied neural networks will be called "Hopons." Neural Network Approach Overall, the problem of classifying neural waveforms can best be seen as an optimization problem in the presence of noise. Much recent work on neural-type network algorithms has demonstrated that these networks work quite well on problems of this sort6-8. In particular, in a recent paper Hopfield and Tank describe an A/D converter network and suggest how to map the problem of template matching into a similar context8. The energy functional for the network they propose has the form: - 1 E = '" '" T. v..v. - '" VI 2 ~~ I] I] ~ II (1) 1 ] 1 where Tij = connectivity between Hopon i and Hopon y', V; = voltage output of Hopon i, Ii = input current to Hopon i and each Hopon has a sigmoid input-output characteristic V = g(u) = 1/(1 + exp( -au)). If the equation of motion is set to be: du;fdt = -oE/oV = L T;jVj + Ii j (la) then we see that dE/dt = -(I:iTijVj + Ii)dV/dt = - (du/dt)(dV/dt) = -g'{u)(du/dt)2 :s: O. Hence E will go to to a minimum which, in a network constructed as described below, will correspond to a proposed solution to a particular waveform classification problem. Template Matching using a Hopfield-type Neural Net We have taken the following approach to template matching using a neural network. For simplicity, we initially restricted the classification problem to one involving two waveforms and have accordingly constructed a neural network made up of two groups of Hopons, each concerned with discriminating one or the other waveform. The classification procedure works as follows: first, a Schmidt trigger 107 is used to detect the presence of a voltage on the signal channel above a set threshold. When this threshold is crossed, implying the presence of a possible neural signal, 2 msecs of data around the crossing are stored in a buffer (40 samples at 20 KHz). Note that biophysical limitations assure that a single real neuron cannot discharge more than once in this time period, so only one waveform of a particular type can occur in this data sample. Also, action potentials are of the order of 1 msec in duration, so the 2 msec window will include the full signal for single or overlapped waveforms. In the next step (explained later) the data values are correlated and passed into a Hopfield network designed to minimize the mean-square error between the actual data and the linear combination of different delays of the templates. Each Hopon in the set of Hopons concerned with one waveform represents a particular temporal delay in the occurrence of that waveform in the buffer. To express the network in terms of an energy function formulation: Let x(t) = input waveform amplitude in the tth time bin, Sj(t) = amplitude of the ph template, Vjk denote if Sj(t - k)(J·th template delayed by k time bins)is present in the input waveform. Then the appropriate energy function is: (2) The first term is designed to minimize the mean-square error and specifies the best match. Since V E [0,1]' the second term is minimized only when each Vjk assumes values 0 or 1. It also sets the diagonal elements Tij to o. The third term creates mutual inhibition among the processing nodes evaluating the same neuronal signal, which as described above can only occur once per sample. Expanding and simplifying expression (2), the connection matrix is: (3a) and the input current (3b) As it can be seen, the inputs are the correlations between the actual data and the various delays of the templates subtracting a constant term. Modified Hopfield Network As documented in more detail in Fig. 3-4, the above full Hopfield-type network works well for temporally isolated spikes at moderate noise levels, but for overlapping spikes it has a local minima problem. This is more severe with more than two waveforms in the network. 108 Further, we need to build our network in hardware and the full Hopfield network is difficult to implement with current technology (see below) . For these reasons, we developed a modified neural network approach which significantly reduces the necessary hardware complexity and also has improved performance. To understand how this works, let us look at the information contained in the quantities Tij and Iij (eq. 3a and 3b ) and make some use of them. These quantities have to be calculated at a pre-processing stage before being loaded into the Hopfield network. If after calculating these quantities, we can quickly rule out a large number of possible template combinations, then we can significantly reduce the size of the problem and thus use a much smaller (and hence more efficient) neural network to find the optimal solution. To make the derivation simple, we define slightly modified versions of 1';j and Iij (eq. 4a and 4b) for two-template case. Iij = L x(t) [~SI(t - i) + ~S2(t - j)] - ~ L si(t - i) - ~ L s~(t - j) (4b) t t t In the case of overlaping spikes the 1';j'S are the cross-correlations between SI (t) and S2(t) with different delays and Ii;'s are the cross-correlations between input x(t) and weighted combination of SI(t) and S2(t). Now if x(t) = SI(t - i) + S2(t - J') (i.e. the overlap of the first template with i time bin delay and the second template with j time bin delay), then I:::.ij = l1';j - Iijl = O. However in the presence of noise, I:::.ij will not be identically zero, but will equal to the noise, and if I:::.ij > l:::.1';j (where l:::.1';j = l1';j - 1';'j.1 for i =f: i' and j =f: l) this simple algorithm may make unacceptable errors. A solution to this problem for overlapping spikes will be described below, but now let us consider the problem of classifying non-overlapping spikes. In this case, we can compare the input cross-correlation with the auto-correlations (eq. 4c and 4d). T! = Lsi(t - i); T!, = Ls~(t - i) ( 4c) t (4d) So for non-overlapping cases, if x(t) = SI(t - i), then I:::.~ = IT: - 1:1 = O. If x(t) = S2(t - i), then 1:::.:' = IT:' - 1:'1 = o. In the absence of noise, then the minimum of I:::.ij , 1:::.: and I:::.? represents the correct classification. However, in the presence of noise, none of these quantities will be identically zero, but will equal the noise in the input x(t) which will give rise to unacceptible errors. Our solution to this noise related. problem is to choose a few minima (three have chosen in our case) instead of one. For each minimum there is either a known corresponding linear combination of templates for overlapping cases or a simple template for non-overlapping cases. A three neuron Hopfield-type network is then programmed so that each neuron corresponds to each of the cases. The input x(t) is fed to this tiny network to resolve whatever confusion remains after the first step of "cross-correlation" comparisons. (Note: Simple template matching as described below can also be used in the place of the tiny Hopfield type network.) 109 Simple Template Matching ~ To evaluate the performances of these neural network approaches, we decided to implement a simple template matching scheme, which we will now describe. However, as documented below, this approach turned out to be the most accurate and require the least complex hardware of any of the three approaches. The first step is, again, to fill a buffer with data based on the detection of a possible neural signal. Then we calculate the difference between the recorded waveform and all possible combinations of the two previously identified templates. Formally, this consists of calculating the distances between the input x(m) and all possible cases generated by all the combinations of the two templates. d,j = L Ix(t) - {Sl(t - i) + S2(t - Jonl t d~ = L Ix(t) - Sl(t - i)l; d~' = L Ix(t) - S2(t - i)1 t t dmin = min(dij,d~,dn dm,n gives the best fit of all possible combinations of templates to the actual voltage signal. TESTING PROCEDURES To compare the performance of each of the three approaches, we devised a common set of test data using the following procedures. First, we used the principal component method of Abeles and Goldstein3 to generate two templates from a digitized analog record of neural activity recorded in the cerebellum of the rat. The two actual spike waveform templates we decided to use had a peak-to-peak ratio of 1.375. From a second set of analog recordings made from a site in the cerebellum in which no action potential events were evident, we determined the spectral characteristics of the recording noise. These two components derived from real neural recordings were then digitally combined, the objective being to construct realistic records, while also knowing absolutely what the correct solution to the template matching problem was for each occurring spike. As shown in Fig. 2c and 2d, data sets corresponding to different noise to signal ratios were constructed. We also carried out simulations with the amplitudes of the templates themselves varied in the synthesized records to simulate waveform changes due to brain movements often seen in real recordings. In addition to two waveform test sets, we also constructed three waveform sets by generating a third template that was the average of the first two templates. To further quantify the comparisons of the three diffferent approaches described above we considered non-overlapping and overlapping spikes separately. To quantify the performance of the three different approaches, two standards for classification were devised. In the first and hardest case, to be judged a correct classification, the precise order and timing of two waveforms had to be reconstructed. In the second and looser scheme, classification was judged correct if the order of two waveforms was correct but timing was allowed to vary by ±lOO Jlsecs(i.e. ±2 time bins) which for most neurobiological applications is probably sufficient resolution. Figs. 3-4 compare the performance results for the three approaches to waveform classification implemented as digital simulations. 110 PERFORMANCE COMPARISON Two templates - non-overlapping waveforms: As shown in Fig. 3a, at low noise-to-signal ratios (NSRs below .2) each of the three approaches were comparable in performance reaching close to 100% accuracy for each criterion. As the ratio was increased, however the neural network implementations did less and less well with respect to the simple template matching algorithm with the full Hopfield type network doing considerably worse than the modified network. In the range of NSR most often found in real data (.2 - .4) simple template matching performed considerably better than either of the neural network approaches. Also it is to be noted that simple template matching gives an estimate of the goodness of fit betwwen the waveform and the closest template which could be used to identify events that should not be classified (e.g. signals due to noise). a. , . .. c. ,. • .. .. .. .. .. noise level: 3a/peak amplitude -14 -12 -tli -I 1.1 b. . , , // \, / , , , , , , , ,: -2 degrees of overlap light line absolute criteria heavy line less stringent criteria .. .. . . 1.1 noise level: 3a/peak amplitude , . . .-----------,.-.-. I I I :' I I ,I \,' 12 simple template matching Hopfield network modified Hopfield network Fig. 3. Comparisons of the three approaches detecting two non-overlapping (a), and overlapping (b) waveforms, c) compares the performances of the neural network approaches for different degrees of waveform overlap. Two' templates - overlapping waveforms: Fig. 3b and 3c compare performances when waveforms overlapped. In Fig. 3b the serious local minima problem encountered in the full neural network is demonstrated as is the improved performance of the modified network. Again, overall performance in physi111 ological ranges of noise is clearly best for simple template matching. When the noise level is low, the modified approach is the bet ter of the two neural networks due to the reliability of the correlation number which reflects the resemblence between the input data and the template. When the noise level is high, errors in the correlation numbers may exclude the right combination from the smaller network. In this case its performance is actually a little worse than the larger Hopfield network. Fig. 3c documents in detail which degrees of overlap produce the most trouble for the neural network approaches at average NSR levels found in real neural data. It can be seen that for the neural networks, the most serious problem is encountered when the delays between the two waveforms are small enough that the resulting waveform looks like the larger waveform with some perturbation. Three templates - overlapping and non-overlapping: In Fig. 4 are shown the comparisons between the full Hopfield network approach and the simple template matching approach. For nonoverlapping waveforms, the performance of these two approaches is much more comparable than for the two waveform case (Fig. 4a), although simple template matching is still the optimal method. In the overlapping waveform condition, however, the neural network approach fails badly (Fig. 4b and 4c). For this particular application and implementation, the neural network approach does not scale well. a. ~ . !:! ... o .. v ~ .. 28 c. ~ .. '" ... ... o 50 V ~ .. 2. b. .2 .. .S 1. 1 noise level: 3a /peak amplitude .2 .6 . 8 1. • noise level: 3a /peak amplitude .. .2 .4 .. .S .. I noise level: 3a /peak amplitude Hopfield network simple template matching light line absolute criteria heavy line less stringent criteria a = variance of the noise Fig. 4. Comparisons of performance for three waveforms. a) nonoverlapping waveforms; b) two waveforms overlapping; c) three waveforms overlapping. HARDWARE COMPARISONS As described earlier, an important design requi~ement for this work was the ability to <letect neural signals in analog records in real-time originating from 112 many simultaneously active sampling electrodes. Because it is not feasible to run the algorithms in a computer in real time for all the channels simultaneously, it is necessary to design and build dedicated hardware for each channel. To do this, we have decided to design VLSI implementations of our circuitry. In this regard, it is well recognized that large modifiable neural networks need very elaborate hardware implementations. Let us consider, for example, implementing hard wares for a two-template case for comparisons. Let n = no. of neurons per template (one neuron for each delay of the template), m = no. of iterations to reach the stable state (in simulating the discretized differential equation, with step size = 0.05), [ = no. of samples in a template tj(m). Then, the number of connections in the full Hopfield network will be 4n2 • The total no. of synaptic calculations = 4mn2• So, for two templates and n = 16, m = 100,4mn2 = 102,400. Thus building the full Hopfield-type network digitally requires a system too large to be put in a single VLSI chip which will work in real time. If we want to build an analog system, we need to have many (O{ 4n2)) easily modifiable synapses. As yet this technology is not available for nets of this size. The modified Hopfield-type network on the other hand is less technically demanding. To do the preprocessing to obtain the minimum values we have to do about n 2 = 256 additions to find all possible Iijs and require 256 subtractions and comparisons to find three minima. The costs associated with doing input cross-correlations are the same as for the full neural network (i.e. 2nl = 768(l = 24) mUltiplications). The saving with the modified approach is that the network used is small and fast (120 multiplications and 120 additions to construct the modifiable synapses, no. of synaptic calculations = 90 with m = 10, n = 3). In contrast to the neural networks, simple temrlate matching is simple indeed. For example, it must perform about n 2[ + n = 10,496 additions and n 2 = 256 comparisons to find the minimum dij . Additions are considerably less costly in time and hardware than multiplications. In fact, because this method needs only addition operations, our preliminary design work suggests it can be built on a single chip and will be able to do the two-template classification in as little as 20 microseconds. This actually raises the possibility that with switching and buffering one chip might be able to service more than one channel in essentially real time. CONCLUSIONS Template matching using a full Hopfield-type neural network is found to be robust to noise and changes in signal waveform for the two neural waveform classification problem. However, for a three-waveform case, the network does not perform well. Further, the network requires many modifiable connections and therefore results in an elaborate hardware implementation. The overall performance of the modified neural network approach is better than the full ~Iopfield network approach. The computation has been reduced largly and the hardware requirements are considerably less demanding demonstrating the value of designing a specific network to a specified problem. However, even the modified neural network performs less well than a simple template-matching algorithm which also has the simplest hardware implementation. Using the simple template matching algorithm, our simulations suggest it will be possible to build a two or three waveform classifier on a single VLSI chip using CMOS technology that works in real time with excellent error characteristics. Further, such a chip will be able to accurately classify variably overlapping 113 neural signals. REFERENCES [1] G. L. Gerstein, M. J. Bloom, 1. E. Espinosa, S. Evanczuk & M. R. Turner, IEEE Trans. Sys. Cyb. Man., SMC-13, 668(1983). 2 J. M. Bower & R. Llinas, Soc. Neurosci. Abst.,~, 607(1983). 3 M. Abeles & M. H. Goldstein, Proc. IEEE, 65, 762(1977). 4 W. M. Roberts & D. K. Hartline, Brain Res., 94, 141(1976). 5 E. M. Schmidt, J. of Neurosci. Methods, 12, 95(1984). 6 J. J. Hopfield, Proc. Natl. Acad. Sci. (USA), 81, 3088(1984). 7 J. J. Hopfield & D. W. Tank, BioI. Cybern., 52, 141(1985). 8 D. W. Tank & J. J. Hopfield, IEEE Trans. Circuits Syst., CAS-33, 533(1986). ACKNOWLEDGEMENTS We would like to acknowledge the contribution of Dr. Mark Nelson to the intellectual development of these projects and the able assistance of Herb Adams, Mike Walshe and John Powers in designing and constructing support equipment. This work was supported by NIH grant NS22205, the Whitaker Foundation and the Joseph Drown Foundation.
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412 CAPACITY FOR PATTERNS AND SEQUENCES IN KANERVA'S SDM AS COMPARED TO OTHER ASSOCIATIVE MEMORY MODELS James O. Keeler Chemistry Department, Stanford University, Stanford, CA 94305 and RIACS, NASA-AMES 230-5 Moffett Field, CA 94035. e-mail: [email protected] ABSTRACT The information capacity of Kanerva's Sparse, Distributed Memory (SDM) and Hopfield-type neural networks is investigated. Under the approximations used here, it is shown that the total information stored in these systems is proportional to the number connections in the network. The proportionality constant is the same for the SDM and HopJreld-type models independent of the particular model, or the order of the model. The approximations are checked numerically. This same analysis can be used to show that the SDM can store sequences of spatiotemporal patterns, and the addition of time-delayed connections allows the retrieval of context dependent temporal patterns. A minor modification of the SDM can be used to store correlated patterns. INTRODUCTION Many different models of memory and thought have been proposed by scientists over the years. In (1943) McCulloch and Pitts prorosed a simple model neuron with two states of activity (on and off) and a large number of inputs. Hebb (1949) considered a network of such neurons and postulated mechanisms for changing synaptic strengths 2 to learn memories. The learning rule considered here uses the outer-product of patterns of +Is and -Is. Anderson (1977) discussed the effect of iterative feedback in such a system.) Hopfield (1982) showed that for symmetric connections,4 the dynamics of such a network is governed by an energy function that is analogous to the energy function of a spin glass . .5 Numerous investigations have been carried out on similar models. 6-8 Several limitations of these binary interaction, outer-product models have been pointed out. For example, the number of patterns that can be stored in the system (its capacity) is limited to a fraction of the length of the pattern vectors. Also, these models are not very successful at storing correlated patterns or temporal sequences. Other models have been proposed to overcome these limitations. For example, one can allow higher-order interactions among the neurons.9•10 In the following, I focus on a model developed by Kanerva (1984) called the Sparse, Distributed Memory (SOM) model. J1 The SOM can be viewed as a three layer network that uses an outer-product learning between the second and third layer. As discussed below, the SDM is more versatile than the above mentioned networks because the number of stored patterns can increased independent of the length of the pattern. and the SDM can be used to store spatiotemporal patterns with context retrieval. and store correlated patterns. The capacity limitations of outer-product models can be alleviated by using higher-order interaction models or the SOM, but a price must be paid for this added capacity in tenns of an increase in the number of connections. How much information is gained per connection? It is shown in the following that the total infonnation stored in each system IS proportional to the number of connections in the network. and that the proportionality constant is independent of the particular model or the order of the model. This result also holds if the connections are limited to one bit of precision (clipped weights). The analysis presented here requires certain simplifying assumptions. The approximate results are compared numerically to an exact calculation developed by Chou.12 SIMPLE OUTER·PRODUCT NEURAL NETWORK MODEL As an example or a simple first-order neural network model, I consider in detail the model developed by Hopfield.4 This model will be used to introduce the mathematics and the concepts that will be generalized for the analysis of the SOM. The "neurons" are simple two-state © American Institute of Phv!lir.s 19M 413 threshold devices: The state of the i'" neuron. Uj. is either either +1 (on). or -1 (off). Consider a set of n such neurons with net input (local field). hj • to the i'" neuron given by /I hj = Vjj Uj. (1) j where Tjj represents the interaction strength between the i'" neuron and the j"'. The state of each neuron is updated asynchronously (at random) according to the rule Uj +-g (h j ). (2) where the function g is a simple threshold function g (x) = sign (x). Suppose we are given M randomly chosen patterns (strings of length n of ±ls) which we wish to store in this system. Denote these M memory patterns as pattern vectors: pQ = (p f,pf .... ,p"Q). ex = 1.2.3, ... ,M. For example. pI might look like (+1,-1.+1,-1.-1 •...• +1). One method of storing these patterns is the outer-product (Hebbian) learning rule: Start with T=O, and accumulate the outer-products of the pattern vectors. The resulting connection matrix is given by M Tjj = LPjQpt, Tjj = o. =1 (3) The system described above is a dynamical system with attracting fixed points. To obtain an approximate upper bound on the total information stored in this network. we sidestep the issue of the basins of attraction, and we check to see if each of the patterns stored by Eq. (3) is actually a fixed point of (2). Suppose we are given one of the patterns, p~, say, as the initial configuration of the neurons. I will show that p~ is expected to be a fixed point of Eq. (2). After inserting (3) for T into (1), the net input to the i'" neuron becomes M " h j = LPt[ L ptpl]· (4) =1 j The important term in the sum on ex is the one for which ex =~ . This term represents the • 'signal" between the input p~ and the desired output. The rest of the sum represents "noise" resulting from crosstalk with all of the other stored patterns. The expression for the net input becomes hj = signal; + noisej where " signalj = Pj~[L pI pI], (5) j M " noisej = L pjQ[ 2. pt pI]· (6) Q;t~ Summing on all of the h, in (6) yields signalj = (n-l)pj~. Since n is positive, the sign of the signal term and pj~ will be the same. Thus. if the noise term were exactly zero, the signal would give the same sign as pj~ with a magnitude of:: nd • and p~ would be a fixed point of (2). Moreover, patterns close to pI' would give nearly the same signal, so that p~ should be an attracting fixed point. For randomly chosen patterns. <noise> = 0, where < > indicates statistical expectation. and its variance will be a'- = (n-l)d (M -1). The probability that there will be an error on recall of pj~ is given by the probability that the noise is greater than the signal. For n large, the noise distribution is approximately gaussian, and the probability that there is an error in the i'" bit is (7) INFORMATION CAPACITY The number of patterns that can be stored in the network is known as its capacityP·14 However, for a fair comparison between all of the models discussed here. it is more relevant to compare the total number of bits (total information) stored in each model rather than the number of 414 patterns. This allows comparison of information storage in models with different lengths of the pattern vectors. If we view the memory model as a black box which receives input bit strings and outputs them with some small probability of error in each bit, then the definition of bit-capacity used here is exactly the definition of channel capacity used by Shannon.1S Define the bit-capacity as the number of bits that can be stored in a network with fixed probability of ~etting an error in a recalled bit, i.e. Pe = constant in (10). Explicitly, the bit-capacity is given by 6 B = bit capacity = nMll, (8) where 11 = (I + Pelog2fJe + (l-Pe )log2(I-Pe)). Note that 11=1 for Pe =0. Setting Pe to a constant is tantamount to keeping the signal-to-noise ratio (fidelity) constant, where the fidelity, R, is given by R = I signail/a. Explicitly, the relation between (constant) Pe and R, is just R = ~-I(l - Pe ), where R ~R) = (1I2Jt)'h J e-t212dt. (9) Hence, the bit-capacity of these networks can be investigated by examining the fidelity of the models as a function of n, M, and R. From (8) and (9) the fidelity of the Hopfield model is is R2 = nl(n(M-l»Y.. (n>I). Solving for M in terms of (fixed) R and 11, the bit-capacity becomes B = l1[(n 2IR 2)+n]. The results above can be generalized to models with d th order interactions.17,18 The resulting expression for the bit-capacity for d,h order interaction models is just nd+1 B = 11[-2 +n]. (10) R Hence, we see that the number of bits stored in the system increases with the order d. However, to store these bits, one must pay a price by including more connections in the connection tensor. To demonstrate the relationship between the number of connections and the information stored, define the information capacity, y, to be the total information stored in the network divided by the number of bits in the connection tensor (note that this is different than the definition used by AbuMostafa et al.).19 Thus y is just the bit-capacity divided by the number of bits in the tensor T { and represents the efficiency with which information is stored in the network. Since T has n d+ elements, the information capacity is found to be Y -....!L (11) R 2b' where b is the number of bits of precision used per tensor element (b ~ log2M for no clipping of the weights). For large n, the information stored per neuronal coimection is y = lllR 2b, independent of the order of the model (compare this result to that of Peretto, et al.).11J To illustrate this point, suppose one decides that the maximum allowed probability of getting an error in a recalled bit is Pe = 111000, then this would fix the minimum value of R at 3.1. Thus, to store 10,000 bits with a probability of getting an error of a recalled bit of 0.001, equation (15) states that it would take =96,OOOb bits, independent of the order of the model, or =O.ln patterns can be stored with probability 111000 of getting an error in a recalled bit. KANERVA'S SDM Now we focus our attention on Ka.nerva's Sparse, Distributed Memory model (SDM).l1 The SDM can be viewed as a 3-layer network with the middle layer playing the role of hidden units. To get an autoassociative network, the output layer can be fed back into the input layer, effectively making this a two layer network. The first layer of the SDM is a layer of n, ±l input units (the input address, a), the middle layer is a layer of m, hidden units, S, and the third layer consists of the n ±l output units (the data, d). The connections between the input units and the hidden units are random weights of ±I and are given by the m xn matrix A. The connections between the hidden units and the output units are given by the n xm connection matrix C, and these matrix elements are modified by an outer-product learning rule (C ii analogous to the matrix T of the Hopfield model). Given an input pattern a, the hidden unit activations are determined by s = Or (A a), 415 (12) where Or is the Hamming-distance threshold function: The k'" element is 1 if the input a is at most r Hamming units away from the k,1t row in A, and 0 if it is further than r units away, i.e., { I if Y2(n -Xj)~ °r(x)j = 0 if Y2(n-x;»r . (13) The hidden-units vector, or select vector, s, is mostly Os with an average of Sm 1s, where S is some small number dependent on r; S<l. Hence, s represents a large. sparsely coded vector of Os and SIs representing the input address. The net input, h. to the final layer can be simply expressed as the product of C with s: h= Cs. (14) Finally, the output data is given by d = g(h), where gj (hj ) = sign (hj ). To store the M patterns, pl,p2, ... pM, form the outer-product of these pattern vectors and their corresponding select vectors, (15) where T denotes the trampose of the vector, and where each select vector is formed by the corresponding address, so. = Or (A pa). The storage algorithm (15) is an outer-product learning rule similar to (3). Suppose that the M patterns (pl,p2, ... pM) have been stored according to (15). Following the analysis presented for the Hopfield model, I show that if the ~stem is presented with p~ as input, the output will be p~. (i.e. p~ is a fixed point). Setting a = p in (16) and separating terms as before. the net input (18) becomes M h = dl3(s~'s~) + L pa(sa·s~). (16) a.t~ where the first term represents the signal and the second is the noise. Recall that the select vectors have an average of Sm Is and the remainder Os, so that the expected value of the signal is &n srJ. Assuming that the addresses and data are randomly chosen, the expected value of the noise is zero. To evaluate the fidelity, I make certain approximations. First, I assume that the select vectors are independent of each other. Second, I assume that the variance of the signal alone is zero or small compared to the variance of noise term alone. The first assumption will be valid for m S2<1, and the second assumption will be valid for M S>l. With these assumptions, we can easily calculate the variance of the noise term, because each of the select vectors are i.i.d. vectors of length m with mostly Os and::&n Is. With these assumptions, the fidelity is given by (17) In the limit of large m , with Sm :: constant, the number of stored bits scales as mn B - n[ + n] (18) - '. R2(1+S2m) . If we divide this by the number of elements in C, we find the information capacity, y = 1l1R2b, just as before, so the information capacity is the same for the two models. (If we divide the bit capacity by the number of elements in C and A then we get y = 1lIR2(b+1), which is about the same for large M .) A few comments before we continue. FIrst, it should be pointed out that the assumption made by Kanerva11 and Keeler17,18 that the variance of the signal term is much less than that of the noise is not valid over the entire range. If we took this infO account, then the magnitude of the denominator would be increased by the variance of the signal term. Further, if we read at a distance I away from the write address, then it is easy to see that the signal changes to be m S(l), where &(1) the overlap of two spheres of radius r length I apart in the binomial space n 416 (8 = ~O». The fidelity for reading at a distance I away from the write address is m 282(l) R2-----------------~~----~~----(19) m~IXl~l) + (M-l)m82+(M-l)84m2(I-lIm) ' Compare this to the formula derived by ChOU,12 for the exact signal-to-noise ratio: m 282(l) R2-----------------~~~ ____ ~ ______ __ m~IXI_8(I» + (M-l)mJ.l .. ".+(M-l)0;".m2(I-lIm» ' (20) where J.l .. " is the average overlap of the spheres of radius r binomially distributed with parameters (n ,112) and cr is the square of this overlap. The difference in these two formulas lies in the denominator in the terms 82 verses J.l .. ". and 84 vs. 0;".. The difference comes from the fact that Chou correctly calculates the overlap of the spheres without using the independence assumption. How do these formula's differ? First of all, it is found numerically that 82 is identical with J.l .. ".. Hence, the only difference comes from 84 verses 0;".. For m82 < 1, the 84 term is negligible compared to the other terms in the denominator. In addition, 84 and 0 2 are approximately equal for large n and r=n 12. Hence, in the limit n ~oo the two fonnulas agree over most of the range if M=O.lm, m<2". However, for finite n, the two fonnulas can disagree when m82=1 (see Figure 1). 30 20 10 o o Signal-to-Noise Ratios + Eq. (17) o Eq. (19) * Eq. (20) 20 40 60 Hamming Radius 80 Figure 1: A ~omparison of the fidelity calculations of the SDM for typical n, M, andm values. Eq~atlon (17) .was derived assuming no variance of the signal term, and is shown ~y the + line. Equauon (19) ~ses the approximation that all of the select vectors are indePf2ndeot denoted by the 0 line. EquatIon (20) (·'s) is the exact derivation done by Chou . The values used here were n = 150, m = 2000, M = 100. 417 Equation (20) suggests that the5 is a best read-write Hamming radius for the SOM. By setting I = 0 in (19) and by setting ~ = 0, we get an approximate expression for the best Hamming radius: 8"..., =(2Mntll3. This ttend is qualitatively shown in Figure 2. II) Q) u c: o L. :> I.) I.) o ...... o o Figure 2: Numerical investigation of the capacity of the SOM. The vertical axis is the percent of recovered patterns with no errors. The x-axis (left to right) is the Hamming distance used for reading and writing. The y-axis (back to forward) is the number of patterns that were written into the memory. For this investigation, n = 128, m = 1024, and M ranges from 1 to 501. Note the similarity of a cross-section of this graph at constant M with Figure 1. This calculation was performed by Oavid Cohn at RIACS, NASA-Ames. Figure 1 indicates that the fonnula (17) that neglected the variance of the signal term is incorrect over much of the range. However, a variant of the SOM is to constrain the number of selected locations to be constant; circuitry for doing this is easily built.21 The variance of the signal term would be zero in that case, and the approximate expression for the fidelity is given by Eq. (17). There are certain problems where it would be better to keep 8 = constant, as in the case of correlated patterns (see below). The above analysis was done assuming that the elements (weights) in the outer-product matrix are not clipped i.e. that there are enough bits to store the largest value of any matrix element It is interestmg to consider what happens if we allow these values to be represented by only a few bits. If we consider the case case b = 1, i.e. the weights are clipped at one bit, it is easy to showl7 that r-2llf1tR:Z for the dth older models and for the SOM, which yields y = 0.07 for reasonable R, (this is substantially less than Willsbaw's 0.69). 418 SEQUENCES In an autoassociative memory, the system relaxes to one of the stored patterns and stays fixed in time until a new input is presented. However, there are many problems where the recalled patterns must change sequentially in time. For example, a song can be remembered as a string of notes played in the correct sequence; cyclic patterns of muscle contractions are essential for walking, nding a bicycle, or dribbling a basketball. As a first step we consider the very simplistic sequence production as put forth by Hopfield (1982) and Kanerva (1984). Suppose that we wished to store a sequence of patterns in the SOM. Let the pattern vectors be given by (p 1,p2, ...• pM). This sequence of patterns could be stored by having each pattern point to the next pattern in the se~uence. Thus, for the SOM, the patterns would be stored as mput-output pairs (aIX,dIX), where a = pIX and dlX = pa+l for a = 1,2.3, ... ,M -1. Convergence to this sequence works as follows: If the SOM is presented with an address that is close to p i the read data will be close to p2. Iterating the system with p2 as the new input address, the read data will be even closer to p3. As this iterative process continues, the read data will converge to the stored sequence, with the next pattern in the sequence being presented at each time step. The convergence statistics are essentially the same for sequential patterns as that shown above for autoassociative patterns. Presented with pIX as an input address, the signal for the stored sequence is found as before <signal> = Om pIX+l. (21) Thus, given pIX, the read data is expected to be pCX+l. Assuming that the patterns in the sequence are randomly chosen, the mean value of the noise is zero, with variance <a2> = (M-l)82m(I+82(m-l». (22) Hence, the length of a sequence that can be stored in the SOM increases linearly with m for large m. Attempting to store sequences like this in the Horfield model is not very successful due to the asynchronous updating use in the Hopfield mode. A synchronously updated outer-product model (for example [6]) would work just as described for the SOM, but it would still be limited to storing fraction of the word size as the maximum sequence length. Another method for storing sequences in HOp'field-like networks has been proposed independently by Kleinfeld22 and Sompolinsky and Kanter.23 These models relieve the problem created by asynchronous updating by using a time-delayed sequential term. This time-delay storage algorithm has different dynamics than the synchronous SOM model. In the time-delay algorithm, the system allows time for the units to relax to the first pattern before proceeding on to the next pattern. whereas in the synchronous algorithms, the sequence is recalled imprecisely from imprecise input for the first few iterations and then correctly after that. In other words. convergence to the sequence takes place "on the fly" in the synchronous models the system does not wait to zero in on the fitst pattern before proceeding on to recover the following patterns. 1bis allows the synchronous algorithms to proceed k times as fast as the asynchronous time-delay algorithms with half as many (variable) matrix elements. This difference should be able to be detected in biological systems. TIME DELAYS AND HYSTERESIS: FOLDS The above scenario for storing sequences is inadequate to explain speech recognition or pattern generation. For example, the above algorithm cannot store sequences of the form ABAC • or overlapping sequences. In Kanerva's original work, he included the concept of time delays as a general way of storing sequences with hysteresis. The problem addressed by this is the following: Suppose we wish to store two sequences of patterns that overlap. For example, the two pattern sequences (a,b,c,d,e,f .... ) and (x,y,z,d,w,v, ... ) overlap at the pattern d. If the system only has knowledge of the present state, then when given the input d, it cannot decide whether to output w or e. To store two such sequences, the system must have some knowledge of the immediate past. Kanerva incoIporates this idea into the SOM by using .. folds." A system with F + 1 folds has a time history of F past states. These F states may be over the past F time steps or they may go even further back in time, skipping some time steps. The algorithm for reading from the SOM with folds becomes d(t+l) = g(Co's(t) + C l'S(t-'tl) + ... + C F 'S(t-'tF», (23) 419 where s(t-'t~= 9r(Aa(t-'t~». Tg store the Q pattern sequences (pl.Pf ..... p~l). (Pi.pi ..... P2 2) •... (P~.pJ .... ,PQCl), construct the matrix of the W" fold as follows: QMtJ ~ C~ = w~IL.p~+lxsa tJ. (24) a.1~1 where any vector with a superscript less than 1 is taken to be zero. s~~" = 9r(Ap~-~"), and w~ is a weighting factor that would normally decrease with increasing ~. Why do 3tese folds work? Suppose that the system is presented with the pattern sequence (pl,pr ..... PI I). with each pattern presented sequentially as input until the 'tF time step. For simplicity. assume that w~ = 1 for all~. Each tenn in Eq. (39) will contribute a signal similar to the signal for the single-fOld system. Thus. on the i!" time step. the signal term coming from Eq. (39) is <signal(t+l» = F&nptl. The signal will have this value until the end of the pattern sequence is reached. 'The mean of the noise tenns is zero. with variance <noiseZ:> = F (M -1 )82m (1 +82(m -1». Hence. the signal-to-noise ratio is ff times as strong as it is for the SDM without folds. Suppose further that the second stored pattern sequence happens to match the first stored sequence at t = 'to The signal term would then be signal(t+l) = F8mp(+"1 + amprl. (25) With no history of the past (F = 1) the signal is split between p~+l and JJ21+I. and the output is ambiguous. However. for F>I. the signal for the first pattern sequence dominates and allows retrieval of the remainder of the correct sequence. lbis formulation allows context to aid in the retrieval of stored sequences. and can differentiate between overlapping sequences by using time delays. The above formulation is still too simplistic in terms of being able to do real recognition problems such as speech recognition. First. the above algorithm can only recall sequences at a fixed time rate, whereas speech recognition occurs at widely varying rates. Second. the above algorithm does not allow for deletions in the incoming data. For example "seqnce" can be recot nized as "sequence" even though some letters are missing. Third, as pointed out by Lashley speech processing relies on hierarchical structures. Although Kanerva's original algorithm is too simplistic. a straightforward modification allows retrieval at different rates with deletions. To achieve this, we can add on the time-delay terms with weights which are smeared out in time. Kanerva's (1984) formulation can thus be viewed as a discrete-time formulation of that put forth by Hopfield and Tank, (1987).15 Explicitly we could write F ~ h = L. I W~C~s(t-'t~), (26) ~= I A:=~F where the coefficients W ~ are a discrete approximation to a smooth function which spreads the delayed signal out over tlme. As a further step, we could modify these weights dynamically to optimize the signal coming out. The time-delay patterns could also be placed in a hierarchical structure as in the matched filter avalanche structure put forth by Grossberg et al. (1986).26 CORRELATED PA ITERNS In the above associative memories. all of the patterns were taken to be randomly chosen. unifonnly distributed binary vectors of length n. However, there are many applications where the set of input patterns is not uniformly distributed; the input patterns are correlated. In mathematical terms, the set K of input patterns would not be uniformly distributed over the entire space of 2/1 possible patterns. Let the probabi1i!)' distribution function for the Hamming distance between two randomly chosen vectors pQ and p~ from the distribution K be given by the function p(d(pQ-p~», where d(x-y) is the Hamming distance between x and y. 'The SDM can be generalized from Kanerva's original fonnulation so that correlated input patterns can be associated with output patterns. For the moment, assume that the distribution set K and the probability density function p(x) are known a priori. Instead of constructing the rows of the matrix A from the entire space of 2" patterns, construct the rows of A from the distribution 1C. Adjust the Hamming distance r so that ~ = am = constant number of locations are selected. 420 In other words, adjust r so that the value of a is the same as given above, where a is detennined by r [P(X )dx a=--2/1 (27) This implies that r would have to be adjusted dynamical!-y. This could be done, for example, by a feedback loop. Circuitry for doing this is easily built, and a similar structure appears in the Golgi cells in the Cerebellum.27. Using the same distribution for the rows of A as the distribution of the patterns in 1C. and using (27) to specify the choice of r, all of the above analysis is applicable (assuming randomly chosen output patterns). If the outputs do not have equal Is and -Is the mean of the noise is not O. However, if the distribution of outputs is also known, the system can still be made to work by storing IIp+ and IIp_ for Is and -Is respectively, where p± is the probability of getting a 1 or a-I respectively. Using this storage algorithm, all of the above formulas hold, (as long as the distribution is smooth enough and not extremely dense). The SOM will be able to recover data stored with correlated inputs with a fidelity given by Equation (17). What if the distribution function K is not known a priori? In that case, we would need to have the matrix A learn the distribution p(x). There are many ways to build A to mimic p. One such way is to start with a random A matrix and modify the entries of a randomly chosen rows of A at each step accordinft!o the statistics of the most recent input patterns. Another method is to use competitive learning 30 to achieve the proper distribution of At. The competitive learning algorithm is a method for adjusting the wei~hts A;j between the first and second layer to match this probability density function, p(x). The i row of the address matrix A can be viewd as a vector A,. The competitive learning algorithm holds a competition between these vectors, and a few vectors that are the closest (within the Hamming sphere r) to the input pattern x are the winners. Each of these winners are then modified slightly in the direction of x. For large eno~ m, this algorithm almost always converges to a distribution of the Aj that is the same as p(x). The updating equation for the selected addresses is just A;'''"''' = Arid - 'A.{Arld - x) (28) Note for A. = I, this reduces to the so-called unary representation of Baum et al. 31 Which gives the maximum efficiency in terms of capacity. DISCUSSION The above analysis said nothing about the basins of attraction of these memory states. A measure of the perfonnance of a content addressable memory shoUld also say something about the avera~e radius of convergence of the basin of attraction. The basins are in general quite complicated and have been investigated numerically for the unclipped models and values of n and m ranging in the 100S.21 The basins of attraction for the SOM and the d=1 model are very similar in their characteristics and their average radius of convergence. However, the above results give an upper bound on the capacity by looking at the fixed points of the system (if there is no fixed point, there is no basin). In summary, the above arguments show that the total information stored in outer-product neural networks is a constant times the number of connections between the neurons. This constant is independent of the order of the model and is the same (1lIR2b) for the SOM as well as higherorder Hopfield-type networks. The advantage of going to an architecture like the SOM is that the number of patterns that can be stored in the network is independent of the size of the pattern, whereas the number of stored patterns is limited to a fraction of the word size for the Wills haw or Hopfield architecture. The point of the above analysis is that the efficiency of the SOM in terms of information stored per bit is the same as for Hopfield-type models. It was also demonstrated how sequences of patterns can be stored in the SOM, and how time delays can be used to recover contextual information. A minor modification of the SOM could be used to recover time sequences at slightly different rates of presentation. Moreover, another minor modification allows the storage of correlated patterns in the SOM. With these modifications, the SOM presents a versatile and efficient tool for investigating properties of associative memory. 421 Acknowledgements: Discussions with John Hopfield and Pentti Kanerva are gratefully acknowledged. This work: was supported by DARPA contract # 86-A227500-000. REFERENCES [1] McCulloch, W. S. & Pitts, W. (1943), Bull. Math. Biophys. 5, 115-133. [2] Hebb, D. O. (1949) The Organization of Behavior. John Wiley, New York. [3] Anderson, J. A., Solverstein, J. W., Ritz, S. A. & Jones, R. S. (1977) Psych. Rev., 84, 412-451. [4] Hopfield, J. J. (1982) Proc. Natn'l. Acad. Sci. USA 79 2554-2558. [5] Kirkpatrick, S. & Sherringtoo, D. (1978) Phys Rev. 174384-4405. [6] Little, W. A. & Shaw, G. L.(l978)Math. Biosci. 39, 281-289. [7] Nakano, K. (1972), Association - A model of associative memory. IEEE Trans. Sys. Man Cyber.2, [8] Willshaw, D. 1., Buneman, O. P. & Longuet-Higgins, H. c., (1969) Nature, 222 960-962. [9] Lee, Y. c.; Doolen, G.; Chen. H. H.; Sun, G. Z.; Maxwell, T.; Lee, H. Y.; & Giles, L. (1985) Physica , 22D, 276-306. [10] Baldi, P., and Venkatesh. S. S .• (1987) Phys. Rev. Lett. 58, 913-916. [11] Kanerva, P. (1984) Self-propagating Search: A Unified Theory of Memory, Stanford University Ph.D. Thesis, and Bradford Books (MIT Press). In press (1987 est). [12] Chou, P. A., The capacity of Kanerva's Associative Memory these proceedings. [13] McEliece, R. J., Posner, E. C., Rodemich, E. R., & Venkatesh, S. S. (1986), IEEE Trans. on Information Theory. [14] Amit, D. J., Gutfreund, H. & Sompolinsky, H. (1985) Phys. Rev. Lett. 55, 1530-1533. [15] Shannon, C. E., (1948), Bell Syst. Tech. J., 27, 379,623 (Reprinted in Shannon and Weaver 1949) . [16] Kleinfeld, D. & Pendergraft, D. B., (1987) Biophys. J. 51, 47-53. [17] Keeler, J. D. (1986), Comparison of Sparsely Distributed Memory and Hopfield-type Neural Network Models. RIACS Technical Report 86-31, also submitted to J. Cog. Sci. [18] Keeler, J. D. (1987) Physics Letters 124A, 53-58. [19] Abu-Mostafa, Y. & St. Jacques, (1985), IEEE Trans. on Info. Theor., 31, 461. [18] Keeler, J. D .• Basins of Attraction of Neural Network Models AIP Conf. Proc. #151, Ed: John Denker, Neural Networks for Computing, Snowbird Utah, (1986). [20] Peretto, P. & J.J. Niez, (1986) BioI. Cybem., 54. 53-63. [21] Keeler, J. D., Ph. D. Dissertation. Collective phenomena of coupled lattice maps: Reactiondiffusion systems and neural networks. Department of Physics. University of California, San Diego, (1987). [22] Kleinfeld, D. (1986). Proc. Nat. Acad. Sci. 83 9469-9473. [23] Sompolinsky, H. & Kanter, I. (1986). Physical Review Letters. [24] Lashley, K. S. (1951). Cerebral Mechanisms in Behavior. Edited by Jeffress, L. A. Wiley, New York, 112-136. [25] Hopfield, J. 1. & Tank, D. W. (1987). ICNN San Diego preprint. [26] Grossberg, S. & Stone, G. (1986). Psychological Review, 93, 46-74 [27] Marr, D. (1969). A Journal of Phisiology, 202, 437-470. [28] Grossberg, S. (1976). Biological Cybernetics 23, 121-134. [29] Kohonen, T. (1984) Self-organization and associative memory. Springer-Verlag, Berlin. [30] Rumelhart, D. E. & Zipser, D. J. Cognitive Sci .. 9, (1985), 75. [31] Baum, E., Moody J., Wilczek F. (1987). Preprint for Biological Cybernetics
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338 The Connectivity Analysis of Simple Association - orHow Many Connections Do You Need! Dan Hammerstrom * Oregon Graduate Center, Beaverton, OR 97006 ABSTRACT The efficient realization, using current silicon technology, of Very Large Connection Networks (VLCN) with more than a billion connections requires that these networks exhibit a high degree of communication locality. Real neural networks exhibit significant locality, yet most connectionist/neural network models have little. In this paper, the connectivity requirements of a simple associative network are analyzed using communication theory. Several techniques based on communication theory are presented that improve the robustness of the network in the face of sparse, local interconnect structures. Also discussed are some potential problems when information is distributed too widely. INTRODUCTION Connectionist/neural network researchers are learning to program networks that exhibit a broad range of cognitive behavior. Unfortunately, existing computer systems are limited in their ability to emulate such networks efficiently. The cost of emulating a network, whether with special purpose, highly parallel, silicon-based architectures, or with traditional parallel architectures, is directly proportional to the number of connections in the network. This number tends to increase geometrically as the number of nodes increases. Even with large, massively parallel architectures, connections take time and silicon area. Many existing neural network models scale poorly in learning time and connections, precluding large implementations. The connectivity 'costs of a network are directly related to its locality. A network exhibits locality 01 communication 1 if most of its processing elements connect to other physically adjacent processing elements in any reasonable mapping of the elements onto a planar surface. There is much evidence that real neural networks exhibit locality2. In this paper, a technique is presented for analyzing the effects of locality on the process of association. These networks use a complex node similar to the higher-order learning units of Maxwell et al. 3 NETWORK MODEL The network model used in this paper is now defined (see Figure 1). Definition 1: A recursive neural network, called a c-graph is a graph structure, r( V,E, e), where: • There is a set of CNs (network nodes), V, whose outputs can take a range of positive real values, Vi, between 0 and 1. There are N. nodes in the set. • There is a set of codons, E, that can take a range of positive real values, eij (for codon j of node i), between 0 and 1. There are Ne codons dedicated to each CN (the output of each codon is only used by its local CN), so there are a total of Ne N. codons in the network. The fan-in or order of a codon is Ie. It is assumed that leis the same for each codon, and Ne is the same for each CN. *This work was supported in part by the Semiconductor Research Corporation contract no. 86-10-097, and jointly by the Office of Naval Research and Air Force Office of Scientific Research, ONR contract no. NOOO14 87 K 0259. © American Institute of Physics 1988 339 I codon j e Figure 1 - A ON • Cijk E C is a set of connections of ONs to codons, 1 <i ,k<N. and 1 <j <Ne , Cijk can take two values {O,l} indicating the existence of a connection from ON k to codon j of ON i . 0 Definition 2: The value of ON i is Vi = F[8+~eijl J-l (1) The function, F, is a continuous non-linear, monotonic function, such as the sigmoid function. 0 Definition 9: Define a mapping, D(i,j,x)_y, where x is an input vector to rand y is the Ie element input vector of codon j of ON i. That is, y has as its elements those elements of Zk of x where Cijk=1, \;/ k. 0 The D function indicates the subset of x seen by codon j of ON i. Different input vectors may map to the same codon vectors, e.g., D(i,j,x)-y and D(i,j,Zj-y, where x~7. Definition 4: The codon values eij are determined as follows. Let X( m) be input vector m of the M learned input vectors for ON i. For codon eij of ON i, let Tij be the set of I cdimensional vectors such that lij(m)E Tij , and D(i,j,X(m))-lij(m). That is, each vector, lij( m) in Tij consists of those subvectors of X( m) that are in codon ii's receptive field. The variable 1 indexes the L ( i ,i) vectors of Tij . The number of distinct vectors in Tij may be less than the total number of learned vectors (L(i,j)<M). Though the X(m) are distinct, the subsets, lij(m), need not be, since there is a possible many to one mapping of the x vectors onto each vector lij. Let Xl be the subset of vectors where vi=l (ON i is supposed to output a 1), and .xo be those vectors where vi=O, then define 0#(/) - .izeof {D(i,i ,Z'( m)) " .,-q} (2) for q=O,1, and \;/ m that map to this I. That is, ni~(I) is the number of x vectors that map 340 into "Iij{l) where tlj-O and ni}{I) is the number of 7 vectors that map into "Iii ( I), where tI;-1. The compreaaion of a codon for a vector "Iii(1) then is defined as n.1.(/) He.·( I) = __ I.:....;;J -'--IJ nj}(I)+nj~(I) (3) (Hqj(l)=O when both nt, nO-O.) The output of codon I), eii' is the maximum-likelyhood decoding (4) Where He indicates the likely hood of tlj-l when a vector 7 that maps to , is input, and' is that vector 1'(') where min[d.(1'('),y)] \I I, D(i,j,7)-V, and 7 is the current input vector. In other words, , is that vector (of the set of subset learned vectors that codon ij receives) that is closest (using distance measure d.) to V (the subset input vector). 0 The output of a codon is the "most-likely" output according to its inputs. For example, when there is no code compression at a codon, eji-1, if the "closest" (in terms of some measure of vector distance, e.g. Hamming distance) subvector in the receptive field of the codon belongs to a learned vector where the CN is to output a 1. The codons described here are very similar to those proposed by Marr 4 and implement ne!'Lrest-neighbor classification. It is assumed that codon function is determined statically prior to network operation, that is, the desired categories have already been learned. To measure performance, network capacity is used. Definition 5: The input noiae, Or, is the average d. between an input vector and the closest (minimum d.) learned vector, where d. is a measure of the "difference" between two vectors - for bit vectors this can be Hamming distance. The output noise, 0 0 , is the average distance between network output and the learned output vector associated with the closest learned input vector. The in/ormation gain, Gr, is just Gt = -10.[ ~~ I (5) o Definition 6: The capacity of a network is the maximum number of learned vectors such that the information gain, Gr, is strictly positive (>0). 0 COMMUNICATION ANALOGY Consider a single connection network node, or CN. (The remainder of this paper will be restricted to a single CN.) Assume that the CN output value space is restricted to two values, 0 and 1. Therefore, the CN must decide whether the input it sees belongs to the class of "0" codes, those codes for which it remains off, or the class of "I" codes, those codes for which it becomes active. The inputs it sees in its receptive field constitute a subset of the input vectors (the D( ... ) function) to the network. It is also assumed that the CN is an ideal I-NN (Nearest Neighbor) classifier or feature detector. That is, given a particular set of learned vectors, the CN will classify an arbitrary input according to the class of the nearest (using d. as a measure of distance) learned vector. This situation is equivalent to the case where a single CN has a single codon whose receptive field size is equivalent to that of the CN. Imagine a sender who wishes to send one bit of information over a noisy channel. The sender has a probabilistic encoder that choses a code word (learned vector) according to some probability distribution. The receiver knows this code set, though it has no knowledge of which bit is being sent. Noise is added to the code word during its transmission over the 341 channel, which is analogous to applying an input vector to a network's inputs, where the vector lies within some learned vector's region. The "noise" is represented by the distance ( d,,) between the input vector and the associated learned vector. The code word sent over the channel consists of those bits that are seen in the receptive field of the ON being modeled. In the associative mapping of input vectors to output vectors, each ON must respond with the appropriate output (0 or 1) for the associated learned output vector. Therefore, a ON is a decoder that estimates in which class the received code word belongs. This is a classic block encoding problem, where increasing the field size is equivalent to increasing code length. As the receptive field size increases, the performance of the decoder improves in the presence of noise. Using communication theory then, the trade-off between interconnection costs as they relate to field size and the functionality of a node as it relates to the correctness of its decision making process (output errors) can be characterized. As the receptive field size of a node increases, so does the redundancy of the input, though this is dependent on the particular codes being used for the learned vectors, since there are situations where increasing the field size provides no additional information. There is a point of diminishing returns, where each additional bit provides ever less reduction in output error. Another factor is that interconnection costs increase exponentially with field size. The result of these two trends is a cost performance measure that has a single global maximum value. In other words, given a set of learned vectors and their probabilities, and a set of interconnection costs, a "best" receptive field size can be determined, beyond which, increasing connectivity brings diminishing returns. SINGLE CODON, WITH NO CODE COMPRESSION A single neural element with a single codon and with no code compression can be modelled exactly as a communication channel (see Figure 2). Each network node is assumed to have a single codon whose receptive field size is equal to that of the receptive field size of the node. sender I I ~ I nOIsy I Ch.nne11~1 : ~ I recelver encoder transmitter receiver decoder ON Figure 2 - A Transmission Channel 342 The operation of the channel is as follows. A bit is input into the channel encoder, which selects a random code of length N and transmits that code over the channel. The receiver then, using nearest neighbor classification, decides if the original message was either a 0 or a 1. Let M be the number of code words used by the encoder. The rate* then indicates the density of the code space. o Definition 7: The rate, R, of a communication channel is R = 10gM N (6) The block length, N, corresponds directly to the receptive field size of the codon, i.e., N=/e. The derivations in later sections use a related measure: Definition 8: The code utilization, b, is the number of learned vectors assigned to a particular code or (7) b can be written in terms of R b = 2N(R-l) (8) As b approaches 1, code compression increases. b is essentially unbounded, since M may be significantly larger than 2N. 0 The decode error (information loss) due to code compression is a random variable that depends on the compression rate and the a priori probabilities, therefore, it will be different with different learned vector sets and codons within a set. As the average code utilization for all codons approaches 1, code compression occurs more often and codon decode error is unavoidable. Let Zi be the vector output of the encoder, and the input to the channel, where each element of Zi is either a 1 or o. Let Vi be the vector output of the channel, and the input to the decoder, where each element is either a 1 or a o. The Noisy Channel Coding Theorem is now presented for a general case, where the individual M input codes are to be distinguished. The result is then extended to a CN, where, even though M input codes are used, the ON need only distinguish those codes where it must output a 1 from those where it must output a o. The theorem is from Gallager (5.6.1)5. Random codes are assumed throughout. Theorem 1: Let a discrete memoryless channel have transition probabilities PNU/k) and, for any positive integer N and positive number R, consider the ensemble of (N,R) block codes in which each letter of each code word is independently selected according to the probability assignment Q(k). Then, for each message m, l<m< fe NR l and all p, O<p<l, the ensemble average probability of decoding error using maximum-likelyhood decoding satisfies (9) where ·In the definitions given here and the theorems below, the notation of Gallager 6 is used. Many of the definitions and theorems are also from Gallager. 343 [ ] l+P Eo(p,Q)=-ln~ ~1 Q(k)PU/kp!p i-il k-il (10) o These results are now adjusted ror our special case. Theorem 2: For a single CN, the average channel error rate ror random code vectors is Pc.,.~2q(l-q )Pe•m (11) where q=Q(k) \I k is the probability or an input vector bit being a 1. 0 These results cover a wide range or models. A more easily computable expression can be derived by recognizing some or the restrictions inherent in the CN model. First, assume that all channel code bits are equally likely, that is, \I k, Q( k )=q, that the error model is the Binary Symmetric Channel (BSC), and that the errors are identically distributed and independent that is, each bit has the same probability, f, or being in error, independent or the code word and the bit position in the code word. A simplified version or the above theorem can be derived. Maximizing P gives the tightest bounds: Pc.,. < 0.5 maxPe(p) O$p~l where (letting codon input be the block length, N = I c) P,(p) :'> eXP{-f,IE,(P)-PR1} The minimum value or this expression is obtained when p=1 (for q=0.5): Eo; -log 2 [ (o.sV,+O.SVl-,)' 1 SINGLE-CODON WITH CODE COMPRESSION (12) (13) (14) Unfortunately, the implementation complexity of a codon grows exponentially with the size or the codon, which limits its practical size. An alternative is to approximate single codon function of a single CN with many smaller, overlapped codons. The goal is to maintain performance and reduce implementation costs, thus improving the cost/performance of the decoding process. As codons get smaller, the receptive field size becomes smaller relative to the number of CNs in the network. When this happens there is codon compression, or vector alia6ing, that introduces its own errors into the decoding process due to information loss. Networks can overcome this error by using multiple redundant codons (with overlapping receptive fields) that tend to correct the compression error. Compression occurs when two code words requiring different decoder output share the same representation (within the receptive field or the codon). The following theorem gives the probability of incorrect codon output with and without compression error. Theorem 9: For a BSC model where q=0.5, the codon receptive field is Ic, the code utilization is b, and the channel bits are selected randomly and independently, the probability of a codon decoding error when b > 1 is approximately Pc.,. < (l-f)"Pc- [1-(I-f)" ]0.5 (15) where the expected compression error per codon is approximated by 344 Pc = 0.5 (16) and from equations 13-14, when 6<1 P,,,, < exp { - j, [-log [ [(O .• V.+O .• Vl-' J' I-RI} (17) Proof is given in Hammerstrom6 . 0 As 6 grows, Pc approaches 0.5 asymptotically. Thus, the performance of a single codon degrades rapidly in the presence of even small amounts of compression. MULTIPLE CODONS WITH CODE COMPRESSION The use or mUltiple small codons is more efficient than a few large codons, but there are some fundamental performance constraints. When a codon is split into two or more smaller codons (and the original receptive field is subdivided accordingly), there are several effects to be considered. First, the error rate of each new codon increases due to a decrease in receptive field size (the codon's block code length). The second effect is that the code utilization, II, will increase for each codon, since the same number of learned vectors is mapped into a smaller receptive field. This change also increases the error rate per codon due to code compression. In fact, as the individual codon receptive fields get smaller, significant code compression occurs. For higher-order input codes, there is an added error that occurs when the order of the individual codons is decreased (since random codes are being assumed, this effect is not considered here). The third effect is the mass action of large numbers of codons. Even though individual codons may be in error, if the majority are correct, then the ON will have correct output. This effect decreases the total error rate. Assume that each ON has more than one codon, c>1. The union of the receptive fields for these codons is the receptive field for the ON with no no restrictions on the degree of overlap of the various codon receptive fields within or between ONs. For a ON with a large number of codons, the codon overlap will generally be random and uniformly distributed. Also assume that the transmission errors seen by different receptive fields are independent. Now consider what happens to a codon's compression error rate (ignoring transmission error for the time being) when a codon is replaced by two or more smaller co dons covering the same receptive field. This replacement process can continue until there are only 1 .. codons, which, incidentally, is analogous to most current neural models. For a multiple codon ON, assume that each codon votes a 1 or o. The summation unit then totals this information and outputs a 1 if the majority of codons vote for a 1, etc. Theorem 4: The probability of a ON error due to compression error is 1 00 J.2 Pc = "'\7?'; J e 2 dy 21r c!2-cp.-l!2 (18) V cP.(i-p.) where Pc is given in equation 16 and q=0.5. Pc incorporates the two effects of moving to mUltiple smaller codons and adding more codons. Using equation 17 gives the total error probability (per bit), PeN: (19) Proof is in Hammerstrom6 . 0 345 For networks that perform association as defined in this paper, the connection weights rapidly approach a single uniform value as the size of the network grows. In information theoretic terms, the information content of those weights approaches zero as the compression increases. Why then do simple non-conjunctive networks (1-codon equivalent) work at alI? In the next section I define connectivity cost constraints and show that the answer to the first question is that the general associative structures defined here do not scale costeffectively and more importantly that there are limits to the degree of distribution of information. CONNECTIVITY COSTS It is much easier to assess costs if some implementation medium is assumed. I have chosen standard silicon, which is a two dimensional surface where ON's and codons take up surface area according to their receptive field sizes. In addition, there is area devoted to the metal lines that interconnect the ONs. A specific VLSI technology need not be assumed, since the comparisons are relative, thus keeping ONs, codons, and metal in the proper proportions, according to a standard metal width, m. (which also includes the inter-metal pitch). For the analyses performed here, it is assumed that m, levels of metal are possible. In the previous section I established the relationship of network performance, in terms of the transmission error rate, E, and the network capacity, M. In this section I present an implementation cost, which is total silicon area, A. This figure can then be used to derive a cost/performance figure that can be used to compare such factors as codon size and receptive field size. There are two components to the total area: A ON, the area of a ON, and AMI, the area of the metal interconnect between ONs. AON consists of the silicon area requirements of the codons for all ONs. The metal area for local, intra-ON interconnect is considered to be much smaller than that of the codons themselves and of that of the more global, inter-ON interconnect, and is not considered here. The area per ON is roughly m. 2 AON = cfeme(-) m, (20) where me is the maximum number of vectors that each codon must distinguish, for 6>1, me = 2". Theorem 5: Assume a rectangular, un6ounded* grid of ONs (all ONs are equi-distant from their four nearest neighbors), where each ON has a bounded receptive field of its nON nearest ONs, where "ON is the receptive field size for the ON, nON = C~e , where c is the number of codons, and R is the intra-ON redundancy, that is, the ratio of inputs to synapses (e.g., when R=l each ON input is used once at the ON, when R=2 each input is used on the average at two sites). The metal area required to support each ON's receptive field is (proof is giving by Hammerstrom6 ): [ "ON3 3"ON ~ 21 [ m.j2 AMI = ----w-+ 2 +9"ON m, (21) The total area per ON, A, then is ·Another implementation IItrategy ill to place &II eNII along a diagonal, which givell n 2 area. However, thill technique only works ror a bounded number or eNII and when dendritic computation can be lipread over a large area, which limits the range or p08llible eN implementationll. The theorem IItated here covers an infinite plane or eNII each with a bounded receptive Held. 346 (22) o Even with the assumption of maximum locality, the total metal interconnect area increases as the cube of the per CN receptive field size! SINGLE CN SIMULATION What do the bounds tell us about CN connectivity requirements? From simulations, increasing the CN's receptive field size improves the performance (increases capacity), but there is also an increasing cost, which increases faster than the performance! Another observation is that redundancy is quite effective as a means for increasing the effectiveness of a CN with constrained connectivity. (There are some limits to R, since it can reach a point where the intra-CN connectivity approaches that of inter-CN for some situations.) With a fixed nON, increasing cost-effectiveness (A 1m) is possible by increasing both order and redundancy. In order to verify the derived bounds, I also wrote a discrete event simulation of a CN, where a random set of learned vectors were chosen and the CN's codons were programmed according to the model presented earlier. Learned vectors were chosen randomly and subjected to random noise, L The CN then attempted to categorize these inputs into two major groups (CN output = 1 and CN output = 0). For the most part the analytic bounds agreed with the simulation, though they tended to be optimistic in slightly underestimating the error. These differences can be easily explained by the simplifying assumptions that were made to make the analytic bounds mathematically tractable. DISTRmUTED VS. LOCALIZED Throughout this paper, it has been tacitly assumed that representations are distributed across a number of CNs, and that any single CN participates in a number of representations. In a local representation each CN represents a single concept or feature. It is the distribution of representation that makes the CN's decode job difficult, since it is the cause of the code compression problem. There has been much debate in the connectionist/neuromodelling community as to the advantages and disadvantages of each approach; the interested reader is referred to Hinton7 , Baum et al. 8, and BallardQ • Some of the results derived here are relevant to this debate. A1s the distribution of representation increases, the compression per CN increases accordingly. It was shown above that the mean error in a codon's response quickly approaches 0.5, independent of the input noise. This result also holds at the CN level. For each individual CN, this error can be offset by adding more codons, but this is expensive and tends to obviate one of the arguments in favor of distributed representations, that is, the multi-use advantage, where fewer CNs are needed because of more complex, redundant encodings. A1s the degree of distribution increases, the required connectivity and the code compression increases, so the added information that each codon adds to its CN's decoding process goes to zero (equivalent to all weights approaching a uniform value). SUMMARY AND CONCLUSIONS In this paper a single CN (node) performance model was developed that was based on Communication Theory. Likewise, an implementation cost model was derived. The communication model introduced the codon as a higher-order decoding element and showed that for small codons (much less than total CN fan-in, or convergence) code compression, or vector aliasing, within the codon's receptive field is a severe problem for 347 large networks. As code compression increases, the information added by any individual codon to the CN's decoding task rapidly approaches zero. The cost model showed that for 2-dimensional silicon, the area required for inter-node metal connectivity grows as the cube of a CN's fan-in. The combination of these two trends indicates that past a certain point, which is highly dependent on the probability structure of the learned vector space, increasing the fan-in of a CN (as is done, for example, when the distribution of representation is increased) yields diminishing returns in terms of total cost-performance. Though the rate of diminishing returns can be decreased by the use of redundant, higher-order connections. The next step is to apply these techniques to ensembles of nodes (CNs) operating in a competitive learning or feature extraction environment. REFERENCES [I] J. Bailey, "A VLSI Interconnect Structure for Neural Networks," Ph.D. Dissertation, Department of Computer SciencejEngineering, OGC. In Preparation. [2] V. B. Mountcastle, "An Organizing Principle for Cerebral Function: The Unit Module and the Distributed System," in The Mindful Brain, MIT Press, Cambridge, MA,1977. [3] T. Maxwell, C. L. Giles, Y. C. Lee and H. H. Chen, "Transformation Invariance Using High Order Correlations in Neural Net Architectures," Proceeding8 International Con! on SY8tem8, Man, and Cybernetic8, 1986. [4] D. Marr, "A Theory for Cerebral Neocortex," Proc. Roy. Soc. London, vol. 176(1970), pp. 161-234. [5] R. G. Gallager, Information Theory and Reliable Communication, John Wiley and Sons, New York, 1968. [6] D. Hammerstrom, "A Connectivity Analysis of Recursive, Auto-Associative Connection Networks," Tech. Report CS/E-86-009, Dept. of Computer SciencejEngineering, Oregon Graduate Center, Beaverton, Oregon, August 1986. [7] G. E . Hinton, "Distributed Representations," Technical Report CMU-CS-84-157, Computer Science Dept., Carnegie-Mellon University, Pittsburgh, PA 15213, 1984. [8] E. B. Baum, J. Moody and F . Wilczek, "Internal Representations for Associative Memory," Technical Report NSF-ITP-86-138, Institute for Theoretical Physics, Santa Barbara, CA, 1986. [9] D. H. Ballard, "Cortical Connections and Parallel Processing: Structure and Function," Technical Report 133, Computer Science Department, Rochester, NY, January 1985.
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432 Performance Measures for Associative Memories that Learn and Forget Anthony /(uh Department of Electrical Engineering University of Hawaii at Manoa Honolulu HI, 96822 ABSTRACT Recently, many modifications to the McCulloch/Pitts model have been proposed where both learning and forgetting occur. Given that the network never saturates (ceases to function effectively due to an overload of information), the learning updates can continue indefinitely. For these networks, we need to introduce performance measmes in addition to the information capacity to evaluate the different networks. We mathematically define quantities such as the plasticity of a network, the efficacy of an information vector, and the probability of network saturation. From these quantities we analytically compare different networks. 1. Introduction Work has recently been undertaken to quantitatively measure the computational aspects of network models that exhibit some of the attributes of neural networks. The McCulloch/Pitts model discussed in [1] was one of the earliest neural network models to be analyzed. Some computational properties of what we call a Hopfield Associative Memory Network (HAMN) :similar to the McCulloch/Pitts model was discussed by Hopfield in [2]. The HAMN can be measured quantitatively by defining and evaluating the information capacity as [2-6] have shown, but this network fails to exhibit more complex computational capabilities that neural network have due to its simplified structure. The HAMN belongs to a class of networks which we call static. In static networks the learning and recall procedures are separate. The network first learns a set of data and after learning is complete, recall occurs. In dynamic networks, as opposed to static networks, updated learning and associative recall are intermingled and continual. In many applications such as in adaptive communications systems, image processing, and speech recognition dynamic networks are needed to adaptively learn the changing information data. This paper formally develops and analyzes some dynamic models for neural networks. Some existing models [7-10] are analyzed, new models are developed, and measures are formulated for evaluating the performance of different dynamic networks. In [2-6]' the asymptotic information capacity of the HAMN is defined and evaluated. In [4-5]' this capacity is found by first assuming that the information vectors (Ns) to be stored have components that are chosen randomly and independently of all other components in all IVs. The information capacity then gives the maximum number of Ns that can be stored in the HAMN such that IVs can be recovered with high probability during retrieval. At or below capacity, the network with high probability, successfully recovers the desired IVs. Above capacity, the network quickly degrades and eventually fails to recover any of the desired IVs. This phenomena is sometimes referred to as the "forgetting catastrophe" [10]. In this paper we will refer to this phenomena as network saturation. There are two ways to avoid this phenomena. The first method involves learning a limited number of IVs such that this number is below capacity. After this leaming takes place, no more learning is allowed. Once learning has stopped, the network does not change (defined as static) and therefore lacks many of the interesting computational @ American Institute of Physics 1988 433 capabilities that adaptive learning and neural network models have. The second method is to incorporate some type oC forgetting mechanism in the learning structure so that the inCormation stored in the network can never exceed capacity. This type of network would be able to adapt to the changing statistics of the IVs and the network would only be able to recall the most recently learned IVs. This paper focuses on analyzing dynamic networks that adaptively learn new inCormation and do not exhibit network saturation phenomena by selectively Corgetting old data. The emphasis is on developing simple models and much oC the analysis is performed on a dynamic network that uses a modified Hebbian learning rule. Section 2 introduces and qualitatively discusses a number of network models that are classified as dynamic networks. This section also defines some pertinent measures Cor evaluating dynamic network models. These measures include the plasticity of a network, the probability oC network saturation, and the efficacy of stored IVs. A network with no plasticity cannot learn and a network with high plasticity has interconnection weights that exhibit large changes. The efficacy oC a stored IV as a function oC time is another important parameter as it is used in determining the rate at which a network forgets information. In section 3, we mathematically analyze a simple dynamic network referred to as the Attenuated Linear Updated Learning (AL UL) network that uses linear updating and a modified Hebbian rule. Quantities introduccd in section 3 are analytically dctcrmincd for the ALUL network. By adjusting the attenuation parameter of the AL UL network, the Corgetting factor is adjusted. It is shown that the optimal capacity for a large AL UL network in steady state defined by (2.13,3.1) is a factor of e less than the capacity of a HAMN. This is the tradeoff that must be paid for having dynamic capabilities. We also conjecture that no other network can perform better than this network when a worst case criterion is used. Finally, section 4 discusses further directions for this work along with possible applications in adaptive signal processing. 2. Dynamic Associative Memory Networks The network models discussed in this paper are based on the concept of associative memory. Associative memories are composed of a collection of interconnected elements that have data storage capabilities. Like other memory structures, there are two operations that occur in associative memories. In the learning operation (referred to as a write operation for conventional memories), inCormation is stored in the network structure. In the recall operation (referred to as a read operation for conventional memories), information is retrieved from the memory structure. Associative memories recall information on the basis of data content rather than by a specific address. The models that we consider will have learning and recall operations that are updated in discrete time with the activation state XU) consisting of N cells that take on the values {-l,1}. 2.1. Dynamic Network MeasureS General associative memory networks are described by two sets of equations. If we let XU) represent the activation state at time i and W( k) represent the weight matrix 01· interconnection state at time k then the activation or recall equation is described by X(j+ 1) = f (XU), W(k)), i? 0, k? 0, X(O) = X (2.1 ) where X is the data probe vector used for reca.ll. The learning algorithm or int.erconnection equation is described by W(k+ 1) = g(V(i),O::; i< k, W(O)) where {V( i)} are the information vectors (IV)s to be stored and W(O) is the initial state of the interconnection matrix. Usually the learning algorithm time scale is much longer than 434 the recall equation time scale so that W in (2.1) can be considered time invariant. Often (2.1) is viewed as the equation governing short term memory and (2.2) is the equation governing long term memory. From the Hebbian hypothesis we note that the data probe vectors should have an effect on the interconnection matrix W. If a number of data p!'Obe vectors recall an IV V( a') , the strength of recall of the IV V( i) should be increased by appropriate modification of W. If another IV is never recalled, it should gradually be forgotten by again adjusting terms of W. Following the analysis in [4,5] we assume that all components of IVs introduced are independent and identically distributed Bernoulli random variables with the probability of a 1 or -1 being chosen equal to ~. Our analysis focuses on learning algorithms. Before describing some dynamic learning algorithms we present some definitions. A network is defined as dynamic if given sorne period of time the rate of change of W is never nonzero. In addition we will primarily discuss networks where learning is gradual and updated at discrete times as shown in (2.2). By gradual, we want networks where each update usually consists of one IV being learned and/or forgotten. IVs that have been introduced recently should have a high probability of recovery. The probability of recall for one IV should also be a monotonic decreasing function of time, given that the IV is not repeated. The networks that we consider should also have a relatively low probability of network saturation. Quantitatively, we let e(k,l,i} be the event that an IV introduced at time l can be recovered at time k with a data probe vector which is of Hamming distance i f!'Om the desired IV. The efficacy of network recovery is then given as p(k,l,i) = Pr(e(k,l,i)). In the analysis performed we say a a vector V can recover V(I), if V(I) = 6(V) where 6(.) is a synchronous activation update of all cells in the network. The capacity for dynamic networks is then given by O(k,i,l) = maxm3-Pr(r(e(k,l,i),05:I<k)= m) > l-l O<i< N 2 (2.3) where r(X} gives the cardinality of the number of events that occur in the set X. Closely related to the capacity of a network is network saturation. Saturation occurs when the network is overloaded with IVs such that few or none of the IVs can be successfully recovered. When a network at time 0 starts to leal'll IVs, at some time l < i we have that O(l,i,l» OU,i,l). For k>1 the network saturation probability is defined by S(k,m) where S describes the probability that the network cannot recover m IVs. Another important measure in analyzing the performance of dynamic networks is t.he plasticity of the interconnections of the weight matrix W. Following definitions that are similar to [10], define N 2: 2: V AR{ Wi,j(k) - Wi,j(k-l)} h(k) = i". ii-I N(N-l) (2.4) as the incremental synaptic intensity and N 2: 2:V AR{ Wi,j(k)} H(k) = i"..;;= 1 N(N-l) (2.5) as the cumulative synaptic intensity. From these definitions we can define the plasticity of the network as P(k) = h(k) H(k) (2.6) When network plasticity is zero, the network does not change and no learning takes place. When plasticity is high, the network interconnections exhibit large changes. 435 When analyzing dynamic networks we are often interested if the network reaches a steady state. We say a dynamic network reaches steady state if limH(k) = H Ie--.oo (2.7) where H is a finite nonzero constant. If the IVs have stationary statistics and given that the learning operations are time invariant, then if a network reaches steady state, we have that limP(k) = P Ie-+oo (2.8) where P is a finite constant. It is also easily verified from (2.6) that if the plasticity converges to a nonzero constant in a dynamic network, then given the above conditions on the IVs and the learning operations the network will eventually reach steady state. Let us also define the synaptic state at time k for activation state V as s(k, V) = W(k)V (2.9) From the synaptic state, we Can define the SNR of V, which we show III section 3 is closely related to the efficacy of an IV and the capacity of the network. (E(s.(k V)))2 SNR(k, V,i) = ., VAR(si(k, V)) (2.1O) Another quantity that is important in measuring dynamic networks is the complexity of implementation. Quantities dealing with network complexity are discussed in [12] and this paper focuses on networks that are memory less. A network is memoryless if (2.2) can be expressed in the following form: W(k+ 1) = 9 #( W(k), V(k)) (2.11) Networks that are not memoryless have the disadvantage that all Ns need t.o be saved during all learning updates. The complexity of implementation is greatly increased in terms of space complexity and very likely increased in terms of time complexity. 2.2. Examples of Dynamic Associative Memory Networks The previous subsection discussed some quantities to measure dynamic networks. This subsection discusses some examples of dynamic associative memo!,y networks and qualitatively discusses advantages and disadvantages of different networks. All the networks considered have the memoryless propel·ty. The first network that we discuss is described by the following difference equation W(k+ 1) = a(k)W(k) + b(k)L(V(k)) (2.12) with W(O) being the initial value of weights before any learning has taken place. Networks with these learning rules will be labeled as Linear Updated Learning (LUL) networks and in addition if O<a(k)<l for k2::0 the network is labeled as an Attenuated Linear Updated Learning (ALUL) network. We will primarily deal with ALUL where O<a(k)<l and b(k) do not depend on the position in W. This model is a specialized version of Grossberg's Passive Decay LTM equation discussed in [11]. If the learning algorithm is of the conelation type then L(V(J.·)) = V(k)V(kf-1 (2.13) This learning scheme has similarities to the marginalist learning schemes introduced in [10]. One of the key parameters in the ALUL network is the value of the attenuation coefficient a. From simulations and intuition we know that if the attenuation coefficient is to high, the network will saturate and if the attenuation parameter is to low, the network will 436 forget all but the most recently introduced IVs. Fig. 1 uses Monte Carlo methods to show a plot of the number of IVs recoverable in a 64 cell network when a = 1, (the HAMN) as a function of the learning time scale. From this figure we clearly see that network saturation is exhibited and for the time k ~ 25 no IV are recoverable with high probability. Section 3 further analyzes the AL UL network and derives the value of different measUl'es introduced in section 2.1. Another learning scheme called bounded learning (BL) can be described by {V(k)V(k)T -I F(W(k)~A L(V(k)) = 0 F( W(J.:))<A (2.14) By setting the attenuation parameter a = 1 and letting F(W(k)) = ~a;<Wi.i(k) (2 .15) I,J this is identical to the learning with bounds scheme discussed in [10]. Unfortunately there is a serious drawbacks to this model. If A is too large the network will saturate with high probability. If A is set such that the probability of network saturation is low then the network has the characteristic of not learning for almost all values of k > k(A) = min I :7 F( W(I))~ A. Th~efore we have that the efficacy of netwOl'k recovery, p (k,1 ,0) ~ 0 for all J.: ~ I ~ k{A). In order for the (BL) scheme to be classified as dynamic learning, the attenuation parameter a must have values between 0 and 1. This learning scheme is just a more complex version of the learning scheme derived from (2.10,2 .11). Let us qualitatively analyze the learning scheme when a and b are constant. There are two cases to consider. When A> H, then the network is not affected by the bounds and the network behaves as the AL UL network. When A <H, then the network accepts IVs until the bound is reached. When the bound is reached, the network waits until the values of the interconnection matrix have attenuated to the prescribed levels where learning can continue. If A is judiciously chosen, BL with a < 1 provides a means for a network to avoid saturation. By holding an IV until H(k )<A, it is not too difficult to show that this learning scheme is equivalent to an AL UL network with b (k) time varying. A third learning scheme called refresh learning (RL) can be described by (2.12) with b(k)=I, W(O)=O, and a(k) = 1 -.5(kmod(l)) (2.16) This learning scheme learns a set of IV and periodically refreshes the weighting matrix so that all interconnections are O. RL can be classified as dynamic learning, but learning is not gradual during the periodic l'efresh cycle. Another problem with this learning scheme is that the efficacy of the IVs depend on where during the period they were learned. IVs learned late in a period are quickly forgotten where as IVs learned eady in a period have a longer time in which they are recoverable. In all the learning schemes introduced, the network has both learning and forgetting capabilities, A network introduced in [7,8] separates the learning and forgetting tasks by using the standard HAMN algorithm to learn IV and a random selective forgetting algorithm to unlearn excess information. The algorithm which we call random selective forgetting (RSF) can be described formally as follows. W(k+ 1) = Y(J.:) + L(V(k)) (2.17) where n(FU!::(k))) Y(k) = W(k) -Jl(k) 2..; (V(k,a')V(k,i)T -n(F(W(k)))I) (2.18) i= 1 437 Each of the vectors V( k, i) are obtained by choosing a random vector V in the same manner IVs are chosen and letting V be the initial state of the HAMN with interconnection matrix W(k). The recall operation described by (2.1) is repeated until the activation has settled into a local minimum state. V(k,i) is then assigned this state. /L(k) is the rate at which the randomly selected local minimum energy states are forgotten, W(k) is given by (2.15), and n (X) is a nonnegative integer valued function that is a monotonically increasing function of X. The analysis of the RSF algorithm is difficult, because the energy manifold that describes the energy of each activation state and the updates allowable for (2.1) must be well understood. There is a simple transformation between the weighting matrix and the energy of an activation state given below, E(X(k)) = -~~~Wi,jX;·(j)Xj(k) k>O i j (2.19) but aggregately analyzing all local minimum energy activation states is complex. Through computer simulations and simplified assumptions [7,8] have come up with a qualitative explanation of the RSF algorithm based on an eigenvalue approach. 3. Analysis of the ALUL Network Section 2 focused on defining properties and analytical measures for dynamic AMN along with presenting some examples of some learning algorithms for dynamic AMN. This section will focus on the analysis of one of the simpler algorithms, the ALUL network. From (2.12) we have that the time invariant ALUL network can be described by the following interconnection state equation. W(k+ 1) = aW(k) + bL(V(k)) (3.1 ) where a and b are nonnegative real numbers. Many of the measures introduced in section 2 can easily be determined for the AL UL network. To calculate the incremental synaptic intensity h (k) and the cumulative synaptic intensity H(k) let the initial condition of the interconnection state W",i(O) be independent of all other interconnections states and independent of all IVs. If E W",i(O) = 0 and V AR W .. ,j(O) = "Y then (3.2) and (3.3) In steady state when a < 1 we have that p = 2(1~) (3.4) From this simple relationship between the attenuation parameter a and the plasticity measure P, we can directly relate plasticity to other measures such as the capacity of the network. We define the steady state capacity as C(i,i)= lim C(k,i,i) for networks where k--o.o steady state exists. To analytically determine the capacity first assume that S(k, V(j)) = S(k-i) is a jointly Gaussian random vector. Further assume that Si(l) for 1~ i< N, 1~ 1< m are all independent and identically distributed. Then for N sufficiently large, f(a) = a2(k...,.-,l}(1~2), and 438 we have that SNR(k, VU)) = SNR(k-n = (N-l)f(a) I-f{a) = c{a )logN » 1 p{k,j,O) = 1~ ~l _ N 2 V21rC (a )logN j<k j<k (3.5) (3.6) Given a we first find the largest m= k-j>O where lim p(k,j,O) ~ 1. Note that N-oo ~~p{k,j,O)= 1 when c(a»2. By letting c(a)= 2 the maximum m is given when Solving for m we get that f(a) I-f (a) = 2logN N I [ 210gN 1 og (N + 21ogN)(1-a2) m = 1 -.......::.-------~ + 1 2 loga It is also possible to find the value of a that maximizes m. If we let f = 1 - a2, then (3.7) (3.8) I [ 2logN 1 og (N+ 2logN)f m ~ (3.9) f . . I h 2elogN h NTh· d m IS at a maximum va ue w en f ~ or w en m ~ . IS correspon s to N 2elogN a ~ 2m -l. Note that this is a factor of e less than the maximum number of Ns allowable 2m in a static HAMN [4,5], such that one of the Ns is recoverable. By following the analysis in [5], the independence assumption and the Gaussian assumptions used earlier can be removed. The arguments involve using results from exchangeability theory and normal approximation theory. A similar and somewhat more cumbersome analysis can be performed to show that in steady state the maximum capacity achievable is when a ~ 2m -l and given by 2m lim C(k,O,f) = ~ N N-oo 4e og (3.10) This again is a factor of e less than the maximum number of Ns allowable in a static HAMN [4,5]' such that all Ns are recoverable. Fig. 2 shows a Monte Carlo simulation of the number of Ns recoverable in a 64 cell network versus the learning time scale for a varying between .5 and .99. We can see that the network reaches approximate steady state when k:2: 35. The maximum capacity achievable is when a ~ .9 and the capacity is around 5. This is slightly more than the theoretical value predicted by the analysis just shown when we compare to Fig. 1. For smaller simulations conducted with larger networks the simulated capacity was closer to the predicted value. From the simulations and the analysis we observe that when a is too small Ns are forgotten at too high a rate and when 439 a is too high network saturation occurs. Using the same arguments, it is possible to analyze the capacity of the network and efficacy of rvs when k is small. Assuming zero initial conditions and a ~ 2m-l we can 2m summarize the learning behavior of the AL UL network. The learning behavior can be divided into three phases. In the first phase for k< N all Ns are remembered and 4elogN the characteristics of the network are similar to the HAMN below saturation. In the second phase some rvs are forgotten as the rate of forgetting becomes nonzero. During this phase the maximum capacity is reached as shown in fig. 2. At this capacity the network cannot dynamically recall all IVs so the network starts to forget more information then it receives. This continues until steady state is reached where the learning and forgetting rates are equal. If initial conditions are nonzero the network starts in phase 1 or the beginning of phase 2 if H( k) is below the value corresponding to the maximum capacity and at the end of phase 2 for larger H( k). The calculation of the network saturation probabilities S( k, m) is trivial for large networks when the capacity curves have been found. When m~ C(k,O,E) then S(k,m) ~ 0 otherwise S(k ,m) ~ 1. Before leaving this section let us briefly examine AL UL networks where a (k) and b (k) are time varying. An example of a time varying network is the marginalist learning scheme introduced in [10]. The network is defined by fixing the value of the SNR(k,k-l,i) = D(N) for all k. This value is fixed by setting a= 1 and varying b. Since the VARSi(k,V(k-l)) is a monotonic increasing function of k, b(k) must also be a monotonic increasing function of k. It is not too difficult to show that when k is large, the marginalist learning scheme is equivalent to the steady state AL UL defined by (3.1). The argument is based on noting that the steady state SNR depends not on the update time, but on the difference between the update time and when the rv was stored as is the case with the marginalist learning scheme. The optimal value of D( N) giving the highest capacity is when D(N) = 4elogN and where m = 4elogN' N b(k+ 1) = 2m b(k) 2m-l If performance is defined by a worst case criterion with the criterion being J(I,N) = min(C(k,O,E),k~/) (3.11) (3.12) then we conjecture that for I large, no AL UL as defined in (2.12,2.13) can have larger J(I,N) than the optimal ALUL defined by (3.1). If we consider average capacity, we note that the RL network has an average capacity of N which is larger than the optimal 810gN AL UL network defined in (3.1). However, for most envisioned applications a worst case criterion is a more accurate measure of performance than a criterion based on average capacity. 4. Summary This paper has introduced a number of simple dynamic neural network models and defined several measures to evaluate the performance of these models. All parameters for the steady state AL UL network described by (3.1) were evaluated and the attenuation parameter a giving the largest capacity was found. This capacity was found to be a factor of e less than the static HANIN capacity. Furthermore we conjectured that if we consider a worst case performance criteria that no AL UL network could perform better than the 440 optimal ALUL network defined by (3.1). Finally, a number of other dynamic models including BL, RL, and marginalist learning were stated to be equivalent to AL UL networks under certain conditions. The network models that were considered in this paper all have binary vector valued activation states and may be to simplistic to be considered in many signal processing application. By generalizing the analysis to more complicated models with analog vector valued activation states and continuous time updating it may be possible to use these generalized models in speech and image processing. A specific example would be a controller for a moving robot. The generalized network models would learn the input data by adaptively changing the interconnections of the network. Old data would be forgotten and data that was repeatedly being recalled would be reinforced. These network models could also be used when the input data statistics are nonstationary. References [I] W. S. McCulloch and W . Pitts, "A Logical Calculus of the Ideas Iminent in Nervous Activity", Bulletin of Mathematical Biophysics, 5, 115-133, 1943. [2) J. J. Hopfield, "Neural Networks and Physical Systems with Emergent Collective Computational Abilities ", Proc. Natl. Acad. Sci. USA 79, 2554-2558, 1982. [3J Y. S. Abu-Mostafa and J. M. St. Jacques, "The Information Capacity of the Hopfield Model", IEEE Trans. Inform. Theory, vol. IT-31, 461-464, 1985. [4) R. J. McEliece, E. C. Posner, E. R. Rodemich and S. S. Venkatesh, "The Capacity of 'the Hopfield Associative Memory", IEEE Trans. Inform. Theory, vol. IT-33, 461-482, 1987. [5J A. Kuh and B. W. Dickinson, "Information Capacity of Associative Memories ", to be published IEEE Trans. Inform. Theory. [6] D. J. Amit, H. Gutfreund, and H. Sompolinsky, "Spin-Glass Models of Nev.ral Networks", Phys. Rev. A, vol. 32, 1007-1018, 1985. [7J J. J. Hopfield, D. I. Feinstein, and R. G. Palmer, " 'Unlearning' has a StabIlizing effect in Collective Memories", Nature, vol. 304, 158-159, 1983. [8] R. J. Sasiela, "Forgetting as a way to Improve Neural-Net Behavior" , AIP Conference Proceedings 151, 386-392, 1986. [9] J. D. Keeler, "Basins of Attraction of Neural Network Models", AlP Conference Proceedings 151, 259-265, 1986. [10] J. P. Nadal, G. Toulouse, J. P. Changeux, and S. Dehaene, "Networks of Formal Neurons and Memory Palimpsests", Europhysics Let., Vol. 1,535-542, 1986. [11) S. Grossberg, "Nonlinear Neural Networks: Principles, Mechanisms, and Architectures ", Neural Networks in press. [12J S. S. Venkatesh and D. Psaltis, "Information Storage and Retrieval in Two Associative Nets ", California Institute of Technology Pasadena, Dept. of Elect. Eng., preprint, 1986. 441 "HAMN Capacity" 10 N=64, 1024 trials 8 > -aAverage # of IV - 6 0 ::ea: co C) CIS 4 "-co > < 2 0 0 10 20 30 40 Update Time Fig. 1 10 "ALUL Capacity" N=64, 1024 trials -Go a=.5 .. a=.7 ~ 8 -IIa=.90 0 .... a=.95 ... en .... a=.99 en 6 ~ 0 ::ea: 4 co C) CIS "-co > 2 < 0 0 10 20 30 40 Update Time Fig. 2
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554 STABILITY RESULTS FOR NEURAL NETWORKS A. N. Michell, J. A. FarreUi , and W. Porod2 Department of Electrical and Computer Engineering University of Notre Dame Notre Dame, IN 46556 ABSTRACT In the present paper we survey and utilize results from the qualitative theory of large scale interconnected dynamical systems in order to develop a qualitative theory for the Hopfield model of neural networks. In our approach we view such networks as an interconnection of many single neurons. Our results are phrased in terms of the qualitative properties of the individual neurons and in terms of the properties of the interconnecting structure of the neural networks. Aspects of neural networks which we address include asymptotic stability, exponential stability, and instability of an equilibrium; estimates of trajectory bounds; estimates of the domain of attraction of an asymptotically stable equilibrium; and stability of neural networks under structural perturbations. INTRODUCTION In recent years, neural networks have attracted considerable attention as candidates for novel computational systemsl- 3 . These types of large-scale dynamical systems, in analogy to biological structures, take advantage of distributed information processing and their inherent potential for parallel computation4,5. Clearly, the design of such neural-network-based computational systems entails a detailed understanding of the dynamics of large-scale dynamical systems. In particular, the stability and instability properties of the various equilibrium points in such networks are of interest, as well as the extent of associated domains of attraction (basins of attraction) and trajectory bounds. In the present paper, we apply and survey results from the qualitative theory oflarge scale interconnected dynamical systems6 - 9 in order to develop a qualitative theory for neural networks. We will concentrate here on the popular Hopfield model3 , however, this type of analysis may also be applied to other models. In particular, we will address the following problems: (i) determine the stability properties of a given equilibrium point; (ii) given that a specific equilibrium point of a neural network is asymptotically stable, establish an estimate for its domain of attraction; (iii) given a set of initial conditions and external inputs, establish estimates for corresponding trajectory bounds; (iv) give conditions for the instability of a given equilibrium point; (v) investigate stability properties under structural perturbations. The present paper contains local results. A more detailed treatment of local stability results can be found in Ref. 10, whereas global results are contained in Ref. 1l. In arriving at the results of the present paper, we make use of the method of analysis advanced in Ref. 6. Specifically, we view high dimensional neural network as an IThe work of A. N. Michel and J. A. Farrell was supported by NSF under grant ECS84-19918. 2The work of W. Porod was supported by ONR under grant NOOOI4-86-K-0506. © American Institute of Physics 1988 555 interconnection of individual subsystems (neurons). This interconnected systems viewpoint makes our results distinct from others derived in the literature1,12. Our results are phrased in terms of the qualitative properties of the free subsystems (individual neurons, disconnected from the network) and in terms of the properties of the interconnecting structure of the neural network. As such, these results may constitute useful design tools. This approach makes possible the systematic analysis of high dimensional complex systems and it frequently enables one to circumvent difficulties encountered in the analysis of such systems by conventional methods. The structure of this paper is as follows. We start out by defining the Hopfield model and we then introduce the interconnected systems viewpoint. We then present representative stability results, including estimates of trajectory bounds and of domains of attraction, results for instability, and conditions for stability under structural perturbations. Finally, we present concluding remarks. THE HOPFIELD MODEL FOR NEURAL NETWORKS In the present paper we consider neural networks of the Hopfield type3 • Such systems can be represented by equations of the form N Ui = ..... biUi + I:Aij Gj(Uj) + Ui(t), for i = 1, ... ,N, j=1 (1) where Aij = *"Ui(t) = l~g) and bi = *.. As usual, Ci > O,Tij i:;,RijfR = (-00,00),':. = ~ +E.f=IITiil, Ri > O,Ii: R+ = [0,00) ~ R,Ii is continuous, Ui = ~,Gi : R ~ (-1,1), Gi is continuously differentiable and strictly monotonically increasing (Le., Gi( uD > Gi( u~') if and only if u~ > u~'), UiGi( Ui) > 0 for all Ui ::j; 0, and Gi(O) = O. In (1), Ci denotes capacitance, Rij denotes resistance (possibly including a sign inversion due to an inverter), Gi (·) denotes an amplifier nonlinearity, and Ii(') denotes an external input. In the literature it is frequently assumed that Tij = Tji for all i,j = 1, ... , N and that Tii = 0 for all i = 1, ... , N. We will make these assumptions only when explicitly stated. We are interested in the qualitative behavior of solutions of (1) near equilibrium points (rest positions where Ui == 0, for i = 1, ... , N). By setting the external inputs Ui(t), i = 1, ... , N, equal to zero, we define U* = [ui, ... , u"NV fRN to be an equilibrium for (1) provided that -biui' + E.f=l Aij Gj(uj) = 0, for i = 1, ... ,N. The locations of such equilibria in RN are determined by the interconnection pattern of the neural network (i.e., by the parameters Aij, i,j = 1,. ", N) as well as by the parameters bi and the nature of the nonlinearities Gi(')' i = 1, ... ,N. Throughout, we will assume that a given equilibrium u* being analyzed is an isolated equilibrium for (1), i.e., there exists an r > 0 such that in the neighborhood B( u*, r) = {( u - u*)fRN : lu - u*1 < r} no equilibrium for (1), other than u = u*, exists. When analyzing the stability properties of a given equilibrium point, we will be able to assume, without loss of generality, that this equilibrium is located at the origin u = 0 of RN. If this is not the case, a trivial transformation can be employed which shifts the equilibrium point to the origin and which leaves the structure of (1) the same. 556 INTERCONNECTED SYSTEMS VIEWPOINT We will find it convenient to view system (1) as an interconnection of N free subsystems (or isolated sUbsystems) described by equations of the form Pi = -biPi + Aii Gi(Pi) + Ui(t). (2) Under this viewpoint, the interconnecting structure of the system (1) is given by N Gi(Xb" . ,xn ) ~ L AijGj(Xj), i = 1, ... ,N. (3) j=1 ii:i Following the method of analysis advanced in6 , we will establish stability results which are phrased in terms of the qualitative properties of the free subsystems (2) and in terms of the properties of the interconnecting structure given in (3). This method of analysis makes it often possible to circumvent difficulties that arise in the analysis of complex high-dimensional systems. Furthermore, results obtained in this manner frequently yield insight into the dynamic behavior of systems in terms of system components and interconnections. . GENERAL STABILITY CONDITIONS We demonstrate below an example of a result for exponential stability of an equilibrium point. The principal Lyapunov stability results for such systems are presented, e.g., in Chapter 5 of Ref. 7. We will utilize the following hypotheses in our first result. (A-I) For system (1), the external inputs are all zero, i.e., Ui(t) == 0, i = 1, ... , N. (A-2) For system (1), the interconnections satisfy the estimate for all Ixil < ri, Ix;1 < rj, i,j = 1, ... , N, where the ail are real constants. (A-3) There exists an N-vector a> ° (i.e., aT = (al, ... ,aN) and ai > 0, for all ~ = 1, ... ,N) such that the test matrix S = [Sij] is negative definite, where the bi are defined in (1) and the aij are given in (A-2). 557 We are now in a position to state and prove the following result. Theorem 1 The equilibrium x = 0 of the neural network (1) is exponentially stable if hypotheses (A-l), (A-2) and (A-3) are satisfied. Proof. For (1) we choose the Lyanpunov function (4) where the ai are given in (A-3). This function is clearly positive definite. The time deri vati ve of v along the solutions of (1) is given by N 1 N DV(1)(X) = 2: 2ai(2xd[-biXi + 2: Aij Gj(Xj)] i=1 j=1 where (A-l) has been invoked. In view of (A-2) we have N N DV(1)( x) < 2: ai( -bix~ + Xi 2: aijX j) i=1 j=1 for all IxI2 < r where r = mini(ri), IxI2 = (Ef:1 X~) 1/2, and the matrix R = [rij] is given by r;j = { ai( -bi + aii), t = J ai aij, i ::J j. But it follows that xT Rx = xT ( R ~ RT) X = xT Sx ::; )w(S) Ixl1 (5) where S is the matrix given in (A-3) and AM(S) denotes the largest eigenvalue of the real symmetric matrix S. Since S is by assumption negative definite, we have AM(S) < O. It follows from (4) and (5) that in some neighborhood of the origin x = 0, we have c1lxl~ ~ v(x) ~ c2lxl~ and DV(1)(X) ~ -c3Ixl~, where C1 = ! mini ai > 0, C2 = ! maxi ai > 0, and C3 = -AM(S) > O. Hence, the equilibrium x = ° of the neural network (1) is exponentially stable (c.f. Theorem 9.10 in Ref. 7). Consistent with the philosophy of viewing the neural network (1) as an interconnection of N free subsystems (2), we think of the Lyapunov function (4) as consisting of a weighted sum of Lyapunov functions for each free subsystem (2) (with Ui(t) == 0). The weighting vector a > 0 provides flexibility to emphasize the relative importance of the qualitative properties of the various individual subsystems. Hypothesis (A - 2) provides a measure of interaction between the various subsystems (3). Furthermore, it is emphasized that Theorem 1 does not require that the parameters Aij in (1) form a symmetric matrix. 558 WEAK COUPLING CONDITIONS The test matrix S given in hypothesis (A - 3) has off-diagonal terms which may be positive or nonpositive. For the special case where the off-diagonal terms of the test matrix S = [Sij] are non-negative, equivalent stability results may be obtained which are much easier to apply than Theorem 1. Such results are called weak-coupling conditions in the literature6,9. The conditions 8ij ~ 0 for all i ::J j may reflect properties of the system (1) or they may be the consequence of a majorization process. In the proof of the subsequent result, we will make use of some of the properties of M- matrices (see, for example, Chapter 2 in Ref. 6). In addition we will use the following assumptions. (A-4) For system (1), the nonlinearity Gi(Xi) satisfies the sector condition (A-S) The successive principal minors of the N X N test matrix D = [dij] are all positive where, the bi and Aij are defined in (1) and Ui2 is defined in (A - 4). Theorem 2 The equilibrium x = 0 of the neural network (1) is asymptotically stable if hypotheses (A-1), (A-4) and (A-5) are true. Proof. The proof proceeds10 along lines similar to the one for Theorem 1, this time with the following Lyapunov function N v(x) = L: Qilxd· (6) i=l The above Lyapunov function again reflects the interconnected nature of the whole system. Note that this Lyapunov function may be viewed as a generalized Hamming distance of the state vector from the origin. ESTIMATES OF TRAJECTORY BOUNDS In general, one is not only interested in questions concerning the stability of an equilibrium of the system (1), but also in performance. One way of assessing the qualitative properties of the neural system (1) is by investigating solution bounds near an equilibrium of interest. We present here such a result by assuming that the hypotheses of Theorem 2 are satisfied. In the following, we will not require that the external inputs Ui(t), i = 1, ... , N be zero. However, we will need to make the additional assumptions enumerated below. (A-6) Assume that there exist .xi > 0, for i = 1, ... , N, and an ( > 0 such that N L: j=1 i:/;j (~~) IAjil > ( > 0, i = 1, ... ,N where bi and Aij are defined in (1) and (Ti2 is defined in (A-4). (A-7) Assume that for system (1), N L: .xiIUi(t)1 ~ k for all t ~ 0 i=l for some constant k > 0 where the .xi, i = 1, ... , N are defined in (A-6). 559 In the proof of our next theorem, we will make use of a comparison result. We consider a scalar comparison equation of the form iJ = G(y) where y(R,G : B(r) R for some r > 0, and G is continuous on B(r) = {XfR: Ixl < r}. We can then prove the following auxiliary theorem: Let p(t) denote the maximal solution of the comparison equation with p(to) = Yo(B(r), t ~ to > O. If r(t), t ~ to ~ 0 is a continuous function such that r(to) $ Yo, and if r(t) satisfies the differential inequality Dr(t) = limk-+O+ t sup[r(t + k) - r(t)] $ G(r(t)) almost everywhere, then r(t) $ p(t) for t ~ to ~ 0, for as long as both r(t) and p(t) exist. For the proof of this result, as well as other comparison theorems, see e.g., Refs. 6 and 7. For the next theorem, we adopt the following notation. We let 6 = mini (Til where (Til is defined in (A - 4), we let c = (6 , where ( is given in (A-6), and we let ¢(t,to,xo) = [¢I(t,to,xo)'''',</>N(t,to,xo)]T denote the solution of (1) with ¢(to, to, xo) = Xo = (XlO,"" xNol for some to ~ O. We are now in a position to prove the following result, which provides bounds for the solution of(1). Theorem 3 Assume that hypotheses (A-6) and (A-7) are satisfied. Then ~ ~ I k) c(t t) k 11¢(t, to, xo)11 = L..." .xil¢i(t, to, xo) ::; (a - - e0 + -, t ~ to ~ 0 i=l C C provided that a > k/c and IIxoll = E~l .xilxiOI ~ a, where the .xi, i = 1,. ", N are given in (A-6) and k is given in (A-7). Proof. For (1) we choose the Lyapunov function N v(x) = L .xilxil· (7) i=l 560 Along the solutions of (1), we obtain N DV(l)(X) ~ AT Dw + z: Ai!Ui(t)\ (8) i=l where wT = [G1J;d\Xl\,'''' G'Z~N)lxN\]' A = (A}, ... ,ANf, and D = [dij] is the test matrix given in (A-5). Note that when (A-6) is satisfied, as in the present theorem, then (A-5) is automatically satisfied. Note also that w ~ 0 (Le., Wi ~ 0, i = 1, ... , N) and w = 0 if and only if x = O. Using manipulations involving (A-6), (A-7) and (8), it is easy to show that DV(l)(X) ~ -cv(x) + k. This ineqUality yields now the comparison equation iJ = -cy + k, whose unique solution is given by pet, to, Po) = (Po - ~) e-c(t-to) +~, for all t ~ to. H we let r = v, then we obtain from the comparison result N pet) ~ ret) = v(4)(t,to,xo)) = 2: Ail4>i(t,to,xo)1 = 114>(t,to,xo)\I, i=l i.e., the desired estimate is true, provided that Ir(to)\ = Ef:l Ai/XiOI = IIxoll ~ a and a> kjc. ESTIMATES OF DOMAINS OF ATTRACTION Neural networks of the type considered herein have many equilibrium points. If a given equilibrium is asymptotically stable, or exponentially stable, then the extent of this stability is of interest. As usual, we assume that x = 0 is the equilibrium of interest. If 4>(t, to, xo) denotes a solution of the network (1) with 4>(to, to, xo) = xo, then we would like to know for which points Xo it is true that 4>( t, to, xo) tends to the origin as t ---+ 00. The set of all such points Xo makes up the domain of attraction (the basin of attraction) of the equilibrium x = O. In general, one cannot determine such a domain in its entirety. However, several techniques have been devised to estimate subsets of a domain of attraction. We apply one such method to neural networks, making use of Theorem 1. This technique is applicable to our other results as well, by making appropriate modifications. We assume that the hypotheses (A-I), (A-2) and (A-3) are satisfied and for the free subsystem (2) we choose the Lyapunov function 1 2 Vi(Pi) = 2 Pi' (9) Then DVi(2) (Pi) ~ (-bi + aii)p~, \Pi/ < ri for some ri > O. If (A-3) is satisfied, we must have (-bi + aii) < 0 and DVi(2)(Pi) is negative definite over B(ri). Let Gvo; = {PifR : Vi(Pi) = !p~ < trl ~ Voi}. Then GVo; is contained in the domain of attraction of the equilibrium Pi = 0 for the free subsystem (2). To obtain an estimate for the domain of attraction of x = 0 for the whole neural network (1), we use the Lyapunov function N 1 N v(x) - '"' -"'·x~ - '"' o·v·(x·) -LJ2 ..... •• -LJ •••. i=l i=l It is now an easy matter to show that the set N C>. = {uRN: v(x) = LOiVi(Xi) < oX} i=l 561 (10) will be a subset of the domain of attraction of x = 0 for the neural network (1), where oX = min (OiVOi) = min (~Oir~) . l$.i$.N 1$.i$.N 2 • In order to obtain the best estimate of the domain of attraction of x = 0 by the present method, we must choose the 0i in an optimal fashion. The reader is referred to the literature9,l3,l4 where several methods to accomplish this are discussed. INSTABILITY RESULTS Some of the equilibrium points in a neural network may be unstable. We present here a sample instability theorem which may be viewed as a counterpart to Theorem 2. Instability results, formulated as counterparts to other stability results of the type considered herein may be obtained by making appropriate modifications. (A-B) For system (1), the interconnections satisfy the estimates XiAiiGi(Xi) < OiAiiX;, IXiAjjGj(xj)1 $ IxdlAijlO"j2lxil, if; j where OJ = O"il when Aii < 0 and Oi = O"i2 when Aii > 0 for all IXil < ri, and for alllXjl < Tj,i,j = 1, ... ,N. (A-9) The successive principal minors of the N x N test matrix D = [dij] given by are positive, where O"i = ~ - Au when ifFIl (i.e., stable subsystems) and O"i -!:; + Aji when ifFu (i.e., unstable subsystems) with F = FII U Fu and F = {I, ., . , N} and Fu f; </>. We are now in a position to prove the following result. Theorem 4 The equilibrium x = 0 of the neural network (1) is unstable if hypotheses (A-l), (A-8) and (A-g) are satisfied. If in addition, FII = </> (</> denotes the empty set), then the equilibrium x = 0 is completely unstable. 562 Proof. We choose the Lyapunov function (11) ifF .. ifF. where ai > 0, i = 1, ... ,N. Along the solutions of (1) we have (following the proof of Theorem 2), DV(l)(X) $ -aTDw for all x€B(r), r = miniri where aT = (a}, ... ,aN), D is defined in (A-9), and wT = [G1l;d IXll, ... , GNx~N) IXNI]. We conclude that DV(l)(X) is negative definite over B(r). Since every neighborhood of the origin x = ° contains at least one point x' where v(x') < 0, it follows that the equilibrium x = 0 for (1) is unstable. Moreover, when F, = </>, then the function v(x) is negative definite and the equilibrium x = 0 of (1) is in fact completely unstable (c.f. Chapter 5 in Ref. 7). STABILITY UNDER STRUCTURAL PERTURBATIONS In specific applications involving adaptive schemes for learning algorithms in neural networks, the interconnection patterns (and external inputs) are changed to yield an evolution of different sets of desired asymptotica.l1y stable equilibrium points with appropriate domains of attraction. The present diagonal dominance conditions (see, e.g., hypothesis (A-6)) can be used as constraints to guarantee that the desired equilibria always have the desired stability properties. To be more specific, we assume that a given neural network has been designed with a set of interconnections whose strengths can be varied from zero to some specified values. We express this by writing in place of (1), N Xi = -biXi + L:8ij Aij Gj(Xj) + Ui(t), for i = 1, ... ,N, j=l (12) where 0 $ 8ij $ 1. We also assume that in the given neural network things have been arranged in such a manner that for some given desired value ~ > 0, it is true that ~ = mini (!:; - 8iiAii). From what has been said previously, it should now be clear that if Ui( t) == 0, i = 1, ... ,N and if the diagonal dominance conditions ~ - t (~~) 18ij Aiji > 0, for i = 1, ... ,N j = 1 i:f;j (13) are satisfied for some Ai > 0, i = 1, ... , N, then the equilibrium x = ° for (12) will be asymptotically stable. It is important to recognize that condition (13) constitutes a single stability condition for the neural network under structural perturbations. Thus, the strengths of interconnections of the neural network may be rearranged in any manner to achieve some desired set of equilibrium points. If (13) is satisfied, then these equilibria will be asymptotically stable. (Stability under structural perturbations is nicely surveyed in Ref. 15.) 563 CONCLUDING REMARKS In the present paper we surveyed and applied results from the qualitative theory of large scale interconnected dynamical systems in order to develop a qualitative theory for neural networks of the Hopfield type. Our results are local and use as much information as possible in the analysis of a given eqUilibrium. In doing so, we established cri-teria for the exponential stability, asymptotic stability, and instability of an equilibrium in such networks. We also devised methods for estimating the domain of attraction of an asymptotically stable equilibrium and for estimating trajectory bounds for such networks. Furthermore, we showed that our stability results are applicable to systems under structural perturbations (e.g., as experienced in neural networks in adaptive learning schemes). In arriving at the above results, we viewed neural networks as an interconnection of many single neurons, and we phrased our results in terms of the qualitative properties of the free single neurons and in terms of the network interconnecting structure. This viewpoint is particularly well suited for the study of hierarchical structures which naturally lend themselves to implementations16 in VLSI. Furthermore, this type of approach makes it possible to circumvent difficulties which usually arise in the analysis and synthesis of complex high dimensional systems. REFERENCES [1] For a review, see, Neural Networks for Computing, J. S. Denker, Editor, American Institute of Physics Conference Proceedings 151, Snowbird, Utah, 1986. [2] J. J. Hopfield and D. W. Tank, Science 233, 625 (1986). [3] J. J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 79,2554 (1982), and ibid. 81,3088 (1984). [4] G. E. Hinton and J. A. Anderson, Editors, Parallel Models of Associative Memory, Erlbaum, 1981. [5] T. Kohonen, Self-Organization and Associative Memory, Springer-Verlag, 1984. [6] A. N. Michel and R. K. Miller, Qualitative Analysis of Large Scale Dynamical Systems, Academic Press, 1977. [7] R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, 1982. [8] I. W. Sandberg, Bell System Tech. J. 48, 35 (1969). [9] A. N. Michel, IEEE Trans. on Automatic Control 28, 639 (1983). [10] A. N. Michel, J. A. Farrell, and W. Porod, submitted for publication. [11] J.-H. Li, A. N. Michel, and W. Porod, IEEE Trans. Cire. and Syst., in press. [12] G. A. Carpenter, M. A. Cohen, and S. Grossberg, Science 235, 1226 (1987). [13] M. A. Pai, Power System Stability, Amsterdam, North Holland, 1981. [14] A. N. Michel, N. R. Sarabudla, and R. K. Miller, Circuits, Systems and Signal Processing 1, 171 (1982). [15] Lj. T. Grujic, A. A. Martynyuk and M. Ribbens-Pavella, Stability of Large-Scale Systems Under Structural and Singular Perturbations, Nauka Dumka, Kiev, 1984. [16] D. K. Ferry and W. Porod, Superlattices and Microstructures 2, 41 (1986).
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62 Centric Models of the Orientation Map in Primary Visual Cortex William Baxter Department of Computer Science, S.U.N.Y. at Buffalo, NY 14620 Bruce Dow Department of Physiology, S.U.N.Y. at Buffalo, NY 14620 Abstract In the visual cortex of the monkey the horizontal organization of the preferred orientations of orientation-selective cells follows two opposing rules: 1) neighbors tend to have similar orientation preferences, and 2) many different orientations are observed in a local region. Several orientation models which satisfy these constraints are found to differ in the spacing and the topological index of their singularities. Using the rate of orientation change as a measure, the models are compared to published experimental results. Introduction It has been known for some years that there exist orientation-sensitive neurons in the visual cortex of cats and mOnkeysl,2. These cells react to highly specific patterns of light occurring in narrowly circumscribed regiOns of the visual field, i.e., the cell's receptive field. The best patterns for such cells are typically not diffuse levels of illumination, but elongated bars or edges oriented at specific angles. An individual cell responds maximally to a bar at a particular orientation, called the preferred orientation. Its response declines as the bar or edge is rotated away from this preferred orientation. Orientation-sensitive cells have a highly regular organization in primary cortex3• Vertically, as an electrode proceeds into the depth of the cortex, the column of tissue contains cells that tend to have the same preferred orientation, at least in the upper layers. Horizontally, as an electrode progresses across the cortical surface, the preferred orientations change in a smooth, regular manner, so that the recorded orientations appear to rotate with distance. It is this horizontal structure we are concerned with, hereafter referred to as the orientation map. An orientation map is defined as a twodimensional surface in which every point has associated with it a preferred orientation ranging from 00 ... 1800. In discrete versions, such as the array of cells in the cortex or the discrete simulations in this paper, the orientation map will be considered to be a sampled version of the underlying continuous surface. The investigations of this paper are confined to the upper layers of macaque striate cortex. Detailed knowledge of the two-dimensional layout of the orientation map has implications for the architecture, development, and function of the visual cortex. The organization of orientation-sensitive cells reflects, to some degree, the organization of intracortical connections in striate cortex. Plausible orientation maps can be generated by models with lateral connections that are uniformly exhibited by all cells in the layer4,5, or by models which presume no specific intracortical connections, only appropriate patterns of afferent input6• In this paper, we examine models in which intracortical connections produce the orientation map but the orientation-controlling circuitry is not displayed by all cells. Rather, it derives from localized "centers" which are distributed across the cortical surface with uniform spacing7,8,9. ® American Institute of Physics 1988 The orientation map also represents a deformation in the retinotopy of primary visual cortex. Since the early sixties it has been known that V1 refiects a topographic map of the retina and hence the visual field 10. There is some global distortion of this mapping11 ,t2,13, but generally spatial relations between points in the visual field are maintained on the cortical surface. This well-known phenomenon is only accurate for a medium-grain description of V1, however. At a finer cellular level there is considerable scattering of receptive fields at a given cortical location 14. The notion of the hypercolumn3 proposes that such scattering permits each region of the visual field to be analyzed by a population of cells consisting of all the necessary orientations and with inputs from both eyes. A quantitative description of the orientation map will allow prediction of the distances between iso-orientation zones of a particular orientation, and suggest how much cortical machinery is being brought to bear on the analysis of a given feature at a given location in the visual field. Models of the Orientation Map Hubel and Wiesel's Parallel Stripe Model The classic model of the orientation map is the parallel stripe model first published by Hubel and Wiesel in 197215• This model has been reproduced several times in their publications3,16,17 and appears in many textbooks. The model consists of a series of parallel slabs, one slab for each orientation, which are postulated to be orthogonal to the ocular dominance stripes. The model predicts that a microelectrode advancing tangentially (i.e., horizontally) through the tissue should encounter steadily changing orientations. The rate of change, which is also called the orientation drift rate18, is determined by the angle of the electrode with respect to the array of orientation stripes. The parallel stripe model does not account for several phenomena reported in long tangential penetrations through striate cortex in macaque monkeysI7.19. First, as pointed out by Swindale20, the model predicts that some penetrations will have fiat or very low orientation drift rates over lateral distances of hundreds of micrometers. This is because an electrode advancing horizontally and perpendicular to the ocular dominance stripes (and therefore parallel to the orientation stripes) would be expected to remain within a single orientation column over a considerable distance with its orientation drift rate equal to zero. Such results have never been observed. Second, reversals in the direction of the orientation drift, from clockwise to counterclockwise or vice versa, are commonly seen, yet this phenomenon is not addressed by the parallel stripe model. Wavy stripes in the ocular dominace system21 do not by themselves introduce reversals. Third, there should be a negative correlation between the orientation drift rate and the ocularity "drift rate". That is, when orientation is changing rapidly, the electrode should be confined to a single ocular dominance stripe (low ocularity drift rate), whereas when ocularity is changing rapidly the electrode should be confined to a single orientation stripe (low orientation drift rate). This is clearly not evident in the recent studies of Uvingstone and Hubel17 (see especially their figs. 3b, 21 & 23), where both orientation and ocularity often have high drift rates in the same electrode track, i.e., they show a positive correlation. Anatomical studies with 2deoxyglucose also fail to show that the orientation and ocular dominance column systems are orthogonal22• 63 64 Centric Models and the Topological Index Another model, proposed by Braitenberg and Braitenberg in 19797, has the orientations arrayed radially around centers like spokes in a wheel The centers are spaced at distances of about O.5mm. This model produces reversals and also the sinusoidal progressions frequently encountered in horizontal penetrations. However this approach suggests other possibilities, in fact an entire class of centric models. The organizing centers form discontinuities in the otherwise smooth field of orientations. Different topological types of discontinuity are possible, characterized by their topological index23• The topological index is a parameter computed by taking a path around a discontinuity and recording the rotation of the field elements (figure 1). The value of the index indicates the amount of rotation; the sign indicates the direction of rotation. An index of 1 signifies that the orientations rotate through 3600; an index of 112 signifies a 1800 rotation. A positive index indicates that the orientations rotate in the same sense as a path taken around the singularity; a negative index indicates the reverse rotation. Topological singularities are stable under orthogonal transformations, so that if the field elements are each rotated 900 the index of the singUlarity remains unchanged. Thus a +1 singularity may have orientations radiating out from it like spokes from a wheel, or it may be at the center of a series of concentric circles. Only four types of discontinuities are considered here, +1, -1, +1/2, -1/2, since these are the most stable, i.e .. their neighborhoods are characterized by smooth change. \ : I \ , " \ • I " , \ I , , ......... ',\!,',', .......... ',\1' .... '_ ...... - - - - : : ~/~\,E : : - - - ... "'1\', ...... _ ... , ,\ , ...... ' / : \ '" , I \ I I \ +1 I I J J I I ~ I I ~ " , , / _/ I I I , , \ \ " ' ... , " , -" '--------+------- ........ , :- .... ,', I ..... , ' \ I \ \ , I I I I I -1 , , -, I ',.I I I • I I I I I figure 1. Topological singularities. A positive index indicates that the orientations rotate in the same direction as a path taken around the singularity; a negative index indicates the reverse rotation. Orientations rotate through 3600 around ±1 centers, 1800 around ±l12 centers. Cytochrome Oxidase Puffs At topological singularities the change in orientation is discontinuous, which violates the structure of a smoothly changing orientation map; modellers try to minimize discontinuities in their models in order to satisfy the smoothness constraint. Interestingly, in the upper layers of striate cortex of monkeys, zones with little or no orientation selectivity have been discovered. These zones are notable for their high cytochrome oxidase reactivity24 and have been referred to as cytochrome oxidase puffs, dots, spots, patches or blobs17,25,26,27. We will refer to them as puffs. If the organizing centers of centric models are located in the cytochrome oxidase puffs then the discontinuities in the orientation map are effectively eliminated (but see below). Braitenberg has indicated28 that the +1 centers of his model should correspond to the puffs. Dow and Bauer proposed a model8 with +1 and -1 centers in alternating puffs. Gotz proposed a similar model9 with alternating +1f2 and -1f2 centers in the puffs. The last two models manage to eliminate all discontinuities from the interpuff zones, but they assume a perfect rectangular lattice of cytochrome oxidase puffs. A Set of Centric Models There are two parameters for the models considered here. (1) Whether the positive singularities are placed in every puff or in alternate puffs; and (2) whether the singularities are ±1's or ±'-h's. This gives four centric models (figure 2): El Al Elh Alh + 1 centers in puffs. -1 centers in the interpuff zones. ooth +1 and -1 centers in the puffs, interdigitated in a checkerboard fashion. +112 centers in the puffs, -112 centers in the interpuff zones. ooth +lj2 and -lj2 centers in the puffs, as in At. The El model corresponds to the Braitenberg model transposed to a rectangular array rather than an hexagonal one, in accordance with the observed organization of the cytochrome oxidase regions27. In fact, the rectangular version of the Braitenberg model is pictured in figure 49 of27. The Al model was originally proposed by Dow and Bauer8 and is also pictured in an article by Mitchison29. The A1f2 model was proposed by Gotz9. It should be noted that the El and Al models are the same model rotated and scaled a bit; the Ph and A 1h have the same relationship. E1 At A 1h figure 2. The four centric models. Dark ellipses represent cytochrome oxidase puffs. Dots in interpuff zones of El & E1/2 indicate singularities at those points. 65 66 Simulations Simulated horizontal electrode recordings were made in the four models to compare their orientation drift rates with those of published recordings. In the computer simulations (figure 2) the interpuff distances were chosen to correspond to histological measurements27• Puff centers are separated by 500JL along their long axes, 350JL along the short axes. The density of the arrays was chosen to approximate the sampling frequency observed in Hubel and Wiesel's horizontal electrode recording experiments19, about 20 cells per millimeter. Therefore the cell density of the simulation arrays was about six times that shown in the figure. All of the models produce simulated electrode data that qualitatively resemble the published recording resUlts, e.g., they contain reversals, and runs of constantly changing orientations. The orientation drift rate and number of reversals vary in the different models. The models of figure 2 are shown in perfectly rectangular arrays. Some important characteristics of the models, such as the absence of discontinuites in interpuff zones, are dependent on this regularity. However, the real arrangement of cytochrome oxidase puffs is somewhat irregular, as in Horton's figure 327• A small set of puffs from the parafoveal region of Horton's figure was enlarged and each of the centric models was embedded in this irregular array. The E1 model and a typical simulated electrode track are shown in figure 3. Several problems are encountered when models developed in a regular lattice are implemented in the irregular lattice of the real system; the models have appreciably different properties. The -1 singularities in E1's interpuff zones have been reduced to _1/2's; the A1 and A1f2 models now have some interpuff discontinuities where before they had none. Quantitative Comparisons Measurement of the Orientation Drift Rate There are two sets of centric models in the computer simulations: a set in the perfectly rectangular array (figure 2) and a set in the irregular puff array (as in figure 3). At this point we can generate as many tracks in the simulation arrays as we wish. How can this information be compared to the published records? The orientation drift rate, or slope, is one basis for distinguishing between models. In real electrode tracks however, the data are rather noisy, perhaps from the measuring process or from inherent unevenness of the orientation map. The typical approach is to fit a straight line and use the slope of this line. Reversals in the tracks require that lines be fit piecewise, the approach used by Hubel and Wiese119• Because of the unevenness of the data it is not always clear what constitutes a reversal. Livingstone and Hubel17 report that the track in their figure 5 has only two reversals in 5 millimeters. Yet there seem to be numerous microreversals between the 1st and 3rd mj11jmeter of their track. At what point is a change in slope considered a true reversal rather than just noise? The approach used here was to use a local slope measure and ignore the problem of reversals - this permitted the fast calculation of slope by computer. A single electrode track, usually several millimeters long, was assigned a single slope, the average of the derivative taken at each point of the track. Since these are discrete samples, the local derivative must be approximated by taking measurements over a small neighborhood. How large should this neighborhood be? If too small it will be susceptible to noise in the orientation measures, if too large it will "flatten out" true reversals. Slope EI 913 .. + .. + 1 2 MM figure 3. A centric model in a realistic puff array (from27). A simulated electrode track and resulting data are shown. Only the El model is shown here, but other models were similarly embedded in this array. 67 68 measures using neighborhoods of several sizes were applied to six published horizontal electrode tracks from the foveal and parafoveal upper layers of macaque striate cortex: figures 5,6,7 from 17, figure 16 from3, figure 1 from3o. A neighborhood of O.lmm, which attempts to fit a line between virtually every pair of points, gave abnormally high slopes. Larger neighborhoods tended to give lower slopes, especially to those tracks which contained reversals. The smallest window that gave consistent measures for all six tracks was O.2mm; therefore this window was chosen for comparisons between published data and the centric models. This measure gave an average slope of 285 degrees per millimeter in the six published samples of track data, compared to Hubel & Wiesel's measure of 281 deg/mm for the penetrations in their 1974 paper19. Slope measures of the centric models The slope measure was applied to several thousand tracks at random locations and angles in the simulation arrays, and a slope was computed for each simulated electrode track. Average slopes of the models are shown in Table 1. Generally, models with ±1 centers have higher slopes than those with ±lh centers; models with centers in every puff have higher slopes than the alternate puff models. Thus EI showed the highest orientation drift rate, Al/2 the lowest, with A1 and E1f2 having intermediate rates. The E1 model, in both the rectangular and irregular arrays, produced the most realistic slope values. TABLE I Average slopes of the centric models EI Al Ph Alh RectangUlar array 312 216 198 117 Irregular array 289 216 202 144 Numbers are in degrees/mm. Slope measure (window = O.2mm) applied to 6 published records yielded an average slope of 285 degrees/mm. Discussion Constraints on the Orientation Map Our original definition of the orientation map permits each cell to have an orientation preference whose angle is completely independent of its neighbors. But this is much too general. Looking at the results of tangential electrode penetrations, there are two striking constraints in the data. The first of these is reflected in the smoothness of the graphs. Orientation changes in a regular manner as the electrode moves horizontally through the upper layers: neighboring cells have similar orientation preferences. Discontinuities do occur but are rare. The other constraint is the fact that the orientation is always changing with distance, although the rate of change may vary. Sequences of constant orientation are very rare and when they do occur they never carryon for any appreciable distance. This is one of the major reasons why the parallel stripe model is untenable. The two major constraints on the orientation map may be put informally as follows: 1. The smoothness constraint: neighboring points have similar orientation preferences. 2. The heterogeneity constraint: all orientations should be represented within a small region of the cortical surface. This second constraint is a bit stronger than the data imply. The experimental results only show that the orientations change regularly with distance, not that all orientations must be present within a region. But this constraint is important with respect to visual processing and the notion of hypercolumns3• These are opposing constraints: the first tends to minimize the slope, or orientation drift rate, while the second tends to maximize this rate. Thus the organization of the orientation map is analogous to physical systems that exhibit "frustration", that is, the elements must satisfy conflicting constraints31 • One of the properties of such systems is that there are many near-optimal solutions, no one of which is significantly better than the others. As a result, there are many plausible orientation maps: any map that satisfies these two constraints will generate qualitatively plausible simulated electrode tracks. This points out the need for quantitative comparisons between models and experimental results. Centric models and the two constraints What are some possible mechanisms of the constraints that generate the orientation map? Smoothness is a local property and could be attributed to the workings of individual cells. It seems to be a fundamental property of cortex that adjacent cells respond to similar stimuli. The heterogeneity requirement operates at a slightly larger scale, that of a hypercolumn rather than a minicolumn. While the first constraint may be modeled as a property of individual cells, the second constraint is distributed over a region of cells. How can such a collection of cells insure that its members cycle through all the required orientations? The topological singularities discussed earlier, by definition, include all orientations within a restricted region. By distributing these centers across the surface of the cortex, the heterogeneity constraint may be satisfied. In fact, the amount of orientation drift rate is a function of the density of this distribution (i.e., more centers per unit area give higher drift rates). It has been noted that the El and the Al organizations are the same topological model, but on different scales; the low drift rates of the At model may be increased by increasing the density of the + 1 centers to that of the El model. The same relationship holds for the E1I2 and A1f2 models. It is also possible to obtain realistic orientation drift rates by increasing the density of +1f2 centers, or by mixing +1's and +Ws. However, these alternatives increase the number of interpuff singularities. And given the possible combinations of centers, it may be more than coincidental that a set of + t centers at just the spacing of the cytochrome oxidase regions results in realistic orientation drift rates. Cortical Architecture and Types o/Circuitry Thus far, we have not addressed the issue of how the preferred orientations are generated. The mechanism is presently unknown, but attempts to depict it have traditionally been of a geometric nature, alluding to the dendritic morphologyl.8.28.32. More recently, computer simulations have shown that orientation-sensitive units may be obtained from asymmetries in the receptive fields of afferents6, or developed using 69 70 simple Hebbian rules for altering synaptic weights5• That is, given appropriate network parameters, orientation tuning arises an as inherent property of some neural networks. Centric models propose a quite different approach in which an originally untuned cell is "programmed" by a center located at some distance to respond to a specific orientation. So, for an individual cell, does orientation develop locally, or is it "imposed from without"? Both of these mechanisms may be in effect, acting synergistically to produce the final orientation map. The map may spontaneously form on the embryonic cortex, but with cells that are nonspecific and broadly tuned. The organization imposed by the centers could have two effects on this incipient map. First, the additional inft.uence from centers could "tighten up" the tuning curves, making the cells more specific. Second, the spacing of the centers specifies a distinct and uniform scale for the heterogeneity of the map. An unsupervised developing orientation map could have broad expanses of iso-orientation zones mixed with regions of rapidly changing orientations. The spacing of the puffs, hence the architecture of the cortex, insures that there is an appropriate variety of feature sensitive cells at each location. This has implications for cortical functioning: given the distances of lateral connectivity, for a cell of a given orientation, we can estimate how many other isoorientation zones of that same orientation the cell may be communicating with. For a given orientation, the E1 model has twice as many iso-orientation zones per unit area as At. Ever since the discovery of orientation-specific cells in visual cortex there have been attempts to relate the distribution of cell selectivities to architectural features of the cortex. Hubel and Wiesel originally suggested that the orientation slabs followed the organization of the ocular dominance slabs15• The Braitenbergs suggested in their original mode17 that the centers might be identified with the giant cells of Meynert. Later centric models have identified the centers with the cytochrome oxidase regiOns, again relating the orientation map to the ocular dominance array, since the puffs themselves are closely related to this array. While biologists have habitually related form to function, workers in machine vision have traditionally relied on general-purpose architectures to implement a variety of algorithms related to the processing of visual information33• More recently, many computer scientists designing artificial vision systems have turned their attention towards connectionist systems and neural networks. There is great interest in how the sensitivities to different features and how the selectivities to different values of those features may be embedded in the system architecture34.3S.36. Linsker has proposed (this volume) that the development of feature spaces is a natural concomitance of layered networks. providing a generic organizing principle for networks. Our work deals with more specific cortical architectonics, but we are convinced that the study of the cortical layout of feature maps will provide important insights for the design of artificial systems. References 1. D. Hubel & T. Wiesel. J. Physiol. (Lond.) 160, 106 (1962). 2. D. Hubel & T. Wiesel, J. Physiol. (Lond.) 195,225 (1968). 3. D. Hubel & T. Wiesel, Pmc. Roy. Soc. Lond. B 198, 1 (1977). 4. N. Swindale, Proc. Roy. Soc. Lond. B 215,211 (1982). 5. R.Linsker, Proc. Natl. Acad. Sci. USA 83, 8779 (1986). 6. R. Soodak. Proc. Natl. Acad. Sci. USA 84, 3936 (1987). 71 7. V. Braitenberg & c. Braitenberg, Biol. Cyber. 33, 179 (1979). 8. B. Dow & R. Bauer, Biol. Cyber. 49, 189 (1984). 9. K. Gotz, Biol. Cyber. 56, 107 (1987). 10. P. Daniel & D. Whitteridge, J. Physiol. (Lond.) 159,302 (1961). 11. B. Dow, R. Vautin & R. Bauer, J. Neurosci. 5, 890 (1985). 12. R.B. Tootell, M.S. Silverman, E. Switkes & R. DeValois, Science 218,902 (1982). 13. D.C. Van Essen, W.T. Newsome & J.H. Maunsell, Vision Research 24,429 (1984). 14. D. Hubel & T. Wiesel, J. Compo Neurol. 158, 295 (1974). 15. D. Hubel & T. Wiesel, J. Compo Neurol. 146,421 (1972). 16. D. Hubel, Nature 299. 515 (1982). 17. M. Livingstone & D. Hubel, J. Neurosci. 4,309 (1984). 18. R. Bauer, B. Dow, A. Snyder & R. Vautin, Exp. Brain Res. SO, 133 (1983). 19. D. Hubel & T. Wiesel, J. Compo Neurol. 158,267 (1974). 20. N. Swindale, in Models of the Visual Cortex, D. Rose & V. Dobson, eds., (W iley, 1985), p. 452. 21. S. LeVay, D. Hubel, & T. Wiesel, J. Compo Neurol. 159,559 (1975). 22. D. Hubel, T. Wiesel & M. Stryker, J. Compo Neurol. 177,361 (1978). 23. T. Elsdale & F. Wasoff, Wilhelm Roux's Archives 180, 121 (1976). 24. M.T. Wong-Riley, Brain Res. 162,201 (1979). 25. A. Humphrey & A. Hendrickson. J. Neurosci. 3,345 (1983). 26. E. Carroll & M. Wong-Riley, J. Compo Neurol. 222,1(1984). 27. J. Horton, Proc. Roy. Soc. Lond. B 304, 199 (1984). 28. V. Braitenberg. in Models of the Visual Cortex, p.479. 29. G. Mitchison, in Models of the Visual Cortex, p. 443. 30. C. Michael, Vision Research 25 415 (1985). 31. S. Kirkpatrick, M. Gelatt & M. Vecchio Science 220, 671 (1983). 32. S. Tieman & H. Hirsch, in Models of the Visual Cortex, p. 432. 33. D. Ballard & C. Brown Computer Vision (Prentice-Hall, NJ., 1982). 34. D. Ballard, G. Hinton, & T. Sejnowski, Nature 306, 21 (1983). 35. D. Ballard, Behav. and Brain Sci. 9, 67 (1986). 36. D. Walters, Proc. First Int. Conf. on Neural Networks (June 1987).
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114 A Computer Simulation of Olfactory Cortex With Functional Implications for Storage and Retrieval of Olfactory Information Matthew A. Wilson and James M. Bower Computation and Neural Systems Program Division of Biology, California Institute of Technology, Pasadena, CA 91125 ABSTRACT Based on anatomical and physiological data, we have developed a computer simulation of piriform (olfactory) cortex which is capable of reproducing spatial and temporal patterns of actual cortical activity under a variety of conditions. Using a simple Hebb-type learning rule in conjunction with the cortical dynamics which emerge from the anatomical and physiological organization of the model, the simulations are capable of establishing cortical representations for different input patterns. The basis of these representations lies in the interaction of sparsely distributed, highly divergent/convergent interconnections between modeled neurons. We have shown that different representations can be stored with minimal interference. and that following learning these representations are resistant to input degradation, allowing reconstruction of a representation following only a partial presentation of an original training stimulus. Further, we have demonstrated that the degree of overlap of cortical representations for different stimuli can also be modulated. For instance similar input patterns can be induced to generate distinct cortical representations (discrimination). while dissimilar inputs can be induced to generate overlapping representations (accommodation). Both features are presumably important in classifying olfactory stimuli. INTRODUCTION Piriform cortex is a primary olfactory cerebral cortical structure which receives second order input from the olfactory receptors via the olfactory bulb (Fig. 1). It is believed to play a significant role in the classification and storage of olfactory information1•2•3. For several years we have been using computer simulations as a tool for studying information processing within this cortex4•5. While we are ultimately interested in higher order functional questions, our fITst modeling objective was to construct a computer simulation which contained sufficient neurobiological detail to reproduce experimentally obtained cortical activity patterns. We believe this first step is crucial both to establish correspondences between the model and the cortex, and to assure that the model is capable of generating output that can be compared to data from actual physiological experiments. In the current case, having demonstrated that the behavior of the simulation at least approximates that of the actual cortex4 (Fig. 3), we are now using the model to explore the types of processing which could be carried out by this cortical structure. In particular, in this paper we will describe the ability of the simulated cortex to store and recall cortical activity patterns generated by stimulus various conditions. We believe this approach can be used to provide experimentally testable hypotheses concerning the functional organization of this cortex which would have been difficult to deduce solely from neurophysiological or neuroanatomical data. @ American Institute of Physics 1988 115 Olfactory Higher Cortical Areas 1'Hippocampus Receptors l 1 Piriform Cortex Entorhinal Olfactory I+and Other Cortex Bulb Olfactory Structures T LOT Fig. 1. Simplified block diagram of the olfactory system and closely related sbUctures. MODEL DESCRIPTION This model is largely instructed by the neurobiology of piriform cortex3. Axonal conduction velocities, time delays, and the general properties of neuronal integration and the major intrinsic neuronal connections approximate those currently described in the actual cortex. However, the simulation reduces both the number and complexity of the simulated neurons (see below). As additional infonnation concerning the these or other important features of the cortex is obtained it will be incorporated in the model. Bracketed numbers in the text refer to the relevent mathematical expressions found in the appendix. Neurons. The model contains three distinct populations of intrinsic cortical neurons, and a fourth set of cells which simulate cortical input from the olfactory bulb (Fig. 2). The intrinsic neurons consist of an excitatory population of pyramidal neurons (which are the principle neuronal type in this cortex), and two populations of inhibitory interneurons. In these simulations each population is modeled as 100 neurons arranged in a 10x10 array (the actual piriform cortex of the rat contains on the order of 106 neurons). The output of each modeled cell type consists of an all-or-none action potential which is generated when the membrane potential of the cell crosses a threshold [2.3]. This output reaches other neurons after a delay which is a function of the velocity of the fiber which connects them and the cortical distance from the originating neuron to each target neuron [2.0, 2.4]. When an action potential arrives at a destination cell it triggers a conductance change in a particular ionic channel type in that cell which has a characteristic time course, amplitude, and waveform [2.0, 2.1]. The effect of this conductance change on the transmembrane potential is to drive it towards the equilibrium potential of that channel. Na+, CI-, and K+ channels are included in the model. These channels are differentially activated by activity in synapses associated with different cell types (see below). 116 LOT Afferent Fiber Ceudelly Directed A .. oelatlOn Fiber Locel .... oclellon Flbe, Roatrelly Directed ... _llIlon Fiber locel FMdbeck InhlblUon Caudally DlrectecI Aasoclllion Fiber r Fig. 2. Schematic diagram of piriform cortex showing an excitatory pyramidal cell and two inhibitory intemeurons with their local interactions. Circles indicate sites of synaptic modifiability. Connection Patterns. In the olfactory system, olfactory receptors project to the olfactory bulb which, in turn, projects directly to the pirifonn cortex and other olfactory structures (Fig. 1). The input to the pirifonn cortex from the olfactory bulb is delivered via a fiber bundle known as the lateral olfactory tract (LOT). This fiber tract appears to make sparse, non-topographic, excitatory connections with pyramidal and feedforward inhibitory neurons across the extent of the cortex3,6. In the model this input is simulated as 100 independent cells each of which make random connections (p=O.05) with pyramidal and feedforward inhibitory neurons (Fig. 1 and 2). In addition to the input connections from the olfactory bulb, there is also an extensive set of connections between the neurons intrinsic to the cortex (Fig. 2). For example, the association fiber system arises from pyramidal cells and makes sparse, distributed excitatory connections with other pyramidal cells all across the cortex7,8.9 • In the model these connections are randomly distributed with 0.05 probability. In the model and in the actual cortex, pyramidal cells also make excitatory connections with nearby feedforward and feedback inhibitory cells. These intemeurons, in turn, make reciprocal inhibitory connections with the group of nearby pyramidal cells. The primary effect of the feedback inhibitory neurons is to inhibit pyramidal cell firing through a CI- mediated current shunting mechanism lO•ll•12. Feedforward intemeurons inhibit pyramidal cells via a long latency, long duration, K+ mediated hyperpolarizing potential12,13. Pyramidal cell axons also constitute the primary output of both the model and the actual pirifonn cortex7•14• 117 Synaptic Properties and Modification Rules. In the model, each synaptic connection has an associated weight which determines the peak amplitude of the conductance change induced in the postsynaptic cell following presynaptic activity [2.0]. To study learning in the model, synaptic weights associated with some of the fiber systems are modifiable in an activity-dependent fashion (Fig. 2). The basic modification rule in each case is Hebb-like; i.e. change in synaptic strength is proportional to presynaptic activity multiplied by the offset of the postsynaptic membrane potential from a baseline potential. This baseline potential is set slightly more positive than the CI- equilibrium potential associated with the shunting feedback inhibition. This means that synapses activated while a destination cell is in a depolarized or excited state are strengthened, while those activated during a period of inhibition are weakened. In the model, synapses which follow this rule include the association fiber connections between excitatory pyramidal neurons as well as the connections between inhibitory neurons and pyramidal neurons. Whether these synapses are modifiable in this way in the actual cortex is a subject of active research in our lab. However, the model does mimic the actual synaptic properties associated with the input pathway (LOT) which we have shown to undergo a transient increase in synaptic strength following activation which is independent of postsynaptic potential 15. This increase is not pennanent and the synaptic strength subsequently returns to its baseline value. Generation of Physiological Responses. Neurons in the model are represented as fIrst-order "leaky" integrators with multiple, time-varying inputs [1.0]. During simulation runs, membrane potentials and currents as well as the time of occurence of action potentials are stored for comparison with actual data. An explicit compartmental model (5 compartments) of the pyramidal cells is used to generate the spatial current distributions used for calculation of field potentials (evoked potentials, EEGs) [3.0,4.0]. Stimulus Characteristics. To compare the responses of the model to those of the actual cortex, we mimicked actual experimental stimulation protocols in the simulated cortex and contrasted the resulting intracellular and extracellular records. For example, shock stimuli applied to the LOT are often used to elicit characteristic cortical evoked potentials in vivo16,17,18. In the model we simulated this stimulus paradigm by simultaneously activating all 100 input fibers. Another measure of cortical activity used most successfully by Freeman and colleagues involves recording EEG activity from pirifonn cortex in behaving animals 19,20. These odor-like responses were generated in the model through steady, random stimulation of the input fibers. To study learning in the model, once physiological measures were established, it was required that we use more refined stimulation procedures. In the absence of any specific infonnation about actual input activity patterns along the LOT, we constructed each stimulus out of a randomly selected set of 10 out of the 100 input 118 fibers. Each stimulus episode consisted of a burst of activity in this subset of fibers with a duration of 10 msec at 25 msec intervals to simulate the 40 Hz periodicity of the actual olfactory bulb input. This pattern of activity was repeated in trials of 200 msec duration which roughly corresponds to the theta rhythm periodicity of bulbar activity and respiration21•22. Each trial was then presented 5 times for a total exposure time of 1 second (cortical time). During this period the Hebbtype learning rule could be used to modify the connection weights in an activitydependent fashion. Output Measure for Learning. Given that the sole output of the cortex is in the fonn of action potentials generated by the pyramidal cells, the output measure of the model was taken to be the vector of spike frequency for all pyramidal neurons over a 200 msec trial, with each element of the vector corresponding to the firing frequency of a single pyramidal cell. Figures 5 through 8 show the 10 by 10 array of pyramidal cells. The size of the box placed at each cell position represents the magnitude of the spike frequency for that cell. To evaluate learning effects, overlap comparisons between response pairs were made by taking the nonnalized dot product of their response vectors and expressing that value as a percent overlap (Fig. 4). Simulated ~\~f".-.lj Fig. 3. Simulated physiological responses of the model compared with actual cortical responses. Upper: Simulated intracellular response of a single cell to paired stimulation of the input system (LOn (left) compared with actual response (right) (Haberly & Bower: 84). Middle: Simulated extracellular response recorded at the cortical surface to stimulation of the LOT (left), compared with actual response (right) (Haberly:73b). Lower: Stimulated EEG response recorted at the cortical surface to odor-like input (left), for actual EEG see Freeman 1978. 119 Computational Requirements. All simulations were carried out on a Sun Microsystems 3/260 model microcomputer equipped with 8 Mbytes of memory and a floating point accelerator. Average time for a 200 msec simulation was 3 cpu minutes. RESULTS Physiological Responses As described above, our initial modeling objective was to accurately simulate a wide range of activity patterns recorded, by ourselves and others, in piriform cortex using various physiological procedures. Comparisons between actual and simulated records for several types of response are shown in figure 3. In general, the model replicated known physiological responses quite well (Wilson et al in preparation describes, in detail, the analysis of the physiological reSUlts). For example in response to shock stimulation of the input pathway (LOT), the model reproduces the principle characteristics of both the intracellular and locationdependent extracellular waveforms recorded in the actual cortex9,17,18 (Fig. 3). Percent Overlap with Final Response Pattern 100 60 0 5 Number of Trials Fig. 4. Convergence of the cortical response during training with a single stimulus with synaptic modification. 56% overlap 80% overlap • ••• •• •• • • •• • • •• • • • • • • • • • • •••• • • • • • • • • • • • • • •• • •• • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • •• • • • Full Stimulus 50% Simulus Full Stimulus 50% Simulus Before Training After Training Fig. S. Reconstruction of cortical response patterns with partially degraded stimuli. Left: Response, before training, to the full stimulus (left) and to the same stimulus with 50% of the input fibers inactivated (right). There is a 44% degradation in the response. Right: Response after ttaining, to the full stimulus (left), and to the same stimulus with 50% of the input fibers inactivated (right). As a result of ttaining, the degradation is now only 20%. 120 Trained on A Trained on B Retains A Response • •• • • •• • • • • • • • • ••• • • ••• • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Fig. 6. Storage of multiple patterns. Left Response to stimulus A afler training. Middle: Response to stimulus B afler training on A followed by training on B. Right: Response to stimulus A after training on A followed by training on B. When compared with the original response (left) there is an 85% congruence. Further, in response to odor-like stimulation the model exhibits 40 Hz oscillations which are characteristic of the EEG activity in olfactory cortex in awake, behaving animals19. Although beyond the scope of the present paper, the simulation also duplicates epileptiform9 and damped oscillatory16 type activity seen in the cortex under special stimulus or pharmacological conditions4. Learning Having simulated characteristic physiological responses, we wished to explore the capabilities of the model to store and recall information. Learning in this case is defined as the development of a consistent representation in the activity of the cortex for a particular input pattern with repeated stimulation and synaptic modification. Figure 4 shows how the network converges, with training, on a representation for a stimulus. Having demonstrated that, we studied three properties of learned responses - the reconstruction of trained cortical response patterns with partially degraded stimuli, the simultaneous storage of separate stimulus response patterns, and the modulation of cortical response patterns independent of relative stimulus characteristics. Reconstruction of Learned Cortical Response Patterns " with Partially Degraded Stimuli. We were interested in knowing what effect training would have on the sensitivity of cortical responses to fluctuations in the input signal. First we presented the model with a random stimulus A for one trial (without synaptic modification). On the next trial the model was presented with a degraded version of A in which half of the original 10 input fibers were inactivated. Comparison of the responses to these two stimuli in the naive cortex showed a 44% variation. Next, the model was trained on the full stimulus A for 1 second (with synaptic modification). Again, half of the input was removed and the model was presented with the degraded stimulus for 1 trial (without synaptic modification). In this case the dif121 27% overlap 46% overlap • • • • • • • • • • • • • • • •• • •• • • • • • • • • • • • • • • • • •• • • • • • • Stimulus A Stimulus B Stimulus A Stimulus B Before Training After Training Fig. 7. Results of merging cortical response patterns for dissimilar stimuli. Left: Response to stimulus A and stimulus B before training. Stimuli A and B do not activate any input fibers in common but still have a 27% overlap in cortical response patterns. Right: Response to stimulus A and stimulus B after training in the presence of a common modulatory input E 1. The overlap in cortical response patterns is now 46%. ference between cortical responses was only 20% (Fig. 5) showing that training increased the robustness of the response to degradation of the stimulus. Storage of Two Patterns. The model was frrst trained on a random stimulus A for 1 second. The response vector for this case was saved. Then, continuing with the weights obtained during this training, the model was trained on a new nonoverlapping (Le. different input fibers activated) stimulus B. Both stimulus A and stimulus B alone activated roughly 25% of the cortical pyramidal neurons with 25% overlap between the two responses. Following the second training period we assessed the amount of interference in recalling A introduced by training with B by presenting stimulus A again for a single trial (without synaptic modification). The variation between the response to A following additional training with B and the initially saved reponse to A alone was less than 15% (Fig. 6) demonstrating that learning B did not substantially interfere with the ability to recall A. Modulation of Cortical Response Patterns. It has been previously demonstrated that the stimulus evoked response of olfactory cortex can be modulated by factors not directly tied to stimulus qualities, such as the behavioral state of the animal 1,20,23. Accordingly we were interested in knowing whether the representations stored in the model could be modulated by the influence of such a "state" input. One potential role of a "state" input might be to merge the cortical response patterns for dissimilar stimuli; an effect we refer to as accomodation. To test this in the model, we presented it with a random input stimulus A for 1 trial. It was then presented with a random input stimulus B (non-overlapping input fibers). The amount of overlap in the cortical responses for these untrained cases was 27%. Next, the model was trained for 1 second on stimulus A in the presence of an additional random "state" stimulus El (activity in a set of 10 input fibers distinct 122 77% overlap 45% overlap ~----------~ r--~------~ • • • • • • • • •• • • • • • • • • • • • ••• • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • •• • • Stimulus A Stimulus B Stimulus A Stimulus B Before Training After Training Fig. 8. Results of differentiating cortical response patterns for similar stimuli. Left: Response to stimulus A and stimulus B before training. Stimuli A and B activate 75% of their input fibers in common and have a 77% overlap in cortical response patterns. Right: Respon~ to stimulus A and stimulus B after training A in the presence of modulatory input El and training B with a different modulatory input E2. The overlap in cortical response patterns is now 45%. from both A and B). The model was then trained on stimulus B in the presence of the same "state" stimulus El. After training, the model was presented with stimulus A alone for 1 trial and stimulus B alone for 1 trial. Results showed that now. even without the coincident E 1 input, the amount of overlap between A and B responses was found to have increased to 46% (Fig 7). The role of El in this case was to provide a common stimulus component during learning which reinforced shared components of the responses to input stimuli A and B. To test the ability of a state stimulus to induce differentiation of cortical response patterns for similar stimuli, we presented the model with a random input stimulus A for 1 trial, followed by 1 trial of a random input stimulus B (75% of the input fibers overlapping), The amount of overlap in the cortical responses for these untrained cases was 77%. Next, the model was trained for a period of 1 second on stimulus A in the presence of an additional random "state" stimulus El (a set of 10 input fibers not overlapping either A or B). It was then trained on input stimulus B in the presence of a different random "state" stimulus E2 (10 input fibers not overlapping either A, B, or El) After this training the model was presented with stimulus A alone for 1 trial and stimulus B alone for 1 trial. The amount of overlap was found to have decreased to 45% (Fig 8). In this situation EI and E2 provided a differential signal during learning which reinforced distinct components of the responses to input stimuli A and B. DISCUSSION PhYSiological Responses. Detailed discussion of the mechanisms underlying the simulated patterns of physiological activity in the cortex is beyond the scope of the current paper. However, the model has been of value in suggesting roles for 123 specific features of the cortex in generating physiologically recorded activity. For example, while actual input to the cortex from the olfactory bulb is modulated into 40 Hz bursts24 , continuous stimulation of the model allowed us to demonstrate the model's capability for intrinsic periodic activity independent of the complementary pattern of stimulation from the olfactory bulb. While a similar ability has also been demonstrated by models of Freeman25, by studying this oscillating property in the model we were able to associate these oscillatory characteristics with specific interactions of local and distant network properties (e.g. inhibitory and excitatory time constants and trans-cortical axonal conduction velocities). This result suggests underlying mechanisms for these oscillatory patterns which may be somewhat different than those previously proposed. Learning. The main subject of this paper is the examination of the learning capabilities of the cortical model. In this model, the apparently sparse, highly distributed pattern of connectivity characteristic of piriform cortex is fundamental to the way in which the model learns. Essentially, the highly distributed pattern of connections allows the model to develop stimulus-specific cortical response patterns by extracting correlations from randomly distributed input and association fiber activity. These correlations are, in effect, stored in the synaptic weights of the association fiber and local inhibitory connections. The model has also demonstrated robustness of a learned cortical response against degradation of the input signal. A key to this property is the action of sparsely distributed association fibers which provide reinforcment for previously established patterns of cortical activity. This property arises from the modification of synaptic weights due to correlations in activity between intra-cortical association fibers. As a result of this modification the activity of a subset of pyramidal neurons driven by a degraded input drives the remaining neurons in the response. In general, in the model, similar stimuli will map onto similar cortical responses and dissimilar stimuli will map onto dissimilar cortical responses. However, a presumably important function of the cortex is not simply to store sensory information, but to represent incoming stimuli as a function of the absolute stimulus qualities and the context in which the stimulus occurs. The fact that many of the structures that piriform cortex projects to (and receives projections from) may be involved in multimodal "state" generation14 is circumstantial evidence that such modulation may occur. We have demonstrated in the model that such a modulatory input can modify the representations generated by pairs of stimuli so as to push the representations of like stimuli apart and pull the representations of dissimilar stimuli together. It should be pointed out that this modulatory input was not an "instructive" signal which explicitly directed the course of the representation, but rather a "state" signal which did not require a priori knowledge of the representational structure. In the model, this modulatory phenomenon is a simple consequence of the degree of overlap in the combined (odor stimulus + modulator) stimulus. Both cases approached approximately 50% overlap in cortical responses reflecting the approximately 50% overlap in the combined stimuli for both cases. 124 Of interest was the use of the model's reconstructive capabilities to maintain the modulated response to each input stimulus even in the absence of the modulatory input. CA YEATS AND CONCLUSIONS Our approach to studying this system involves using computer simulation to investigate mechanisms of information processing which could be implemented given what is known about biological constraints. The significance of results presented here lies primarily in the finding that the structure of the model and the parameter settings which were appropriate for the reproduction of physiological responses were also appropriate for the proper convergence of a simple, biologically plausible learning rule under various conditions. Of course, the model we have developed is only an approximation to the actual cortex limited by our knowledge of its organization and the computing power available. For example, the actual piriform cortex of the rat contains on the order of 106 cells (compared with 1()2 in the simulations) with a sparsity of connection on the order of p=O.OOI (compared with p=0.05 in the simulations). Our continuing research effort will include explorations of the scaling properties of the network. Other assumptions made in the context of the current model include the assumption that the representation of information in piriform cortex is in the form of spatial distributions of rate-coded outputs. Information contained in the spatiotemporal patterns of activity was not analyzed, although preliminary observation suggests that this may be of significance. In fact, the dynamics of the model itself suggest that temporally encoded information in the input at various time scales may be resolvable by the cortex. Additionally, the output of the cortex was assumed to have spatial uniformity, Le. no differential weighting of information was made on the basis of spatial location in the cortex. But again, observation of the dynamics of the model, as well as the details of known anatomical distribution patterns for axonal ·connections, indicate that this is a major oversimplification. Preliminary evidence from the model would indicate that some form of hierarchical structuring of information along rostraVcaudal lines may occur. For example it may be that cells found in progressively more rostral locations would have increasingly non-specific odor responses. Further investigations of learning within the model will explore each of these issues more fully, with attempts to correlate simulated findings with actual recordings from awake, behaving animals. At the same time, new data pertaining to the structure of the cortex will be incorporated into the model as it emerges. ACKNOWLEDGEMENTS We wish to thank Dr. Lewis Haberly and Dr. Joshua Chover for their roles in the development and continued support of the modeling effort. We also wish to thank Dave Bilitch for his technical assistance. This work was supported by NIH grant NS22205, NSF grant EET-8700064, the Lockheed Corporation, and a fellowship from the ARCS foundation. APPENDIX dV, 1 [",E,-V, (r) ] = 1: lik(r) + --dl c'" i=1 r, (1.0) SOl1UJric Inregrarion n l'u .. number of input types V.(t) = membrane potential of i th cell lit (t ) .. current into cell i due to input type Ie Et - equilibrium potential associated with input type Ie Spilce Propagation and SynaptiC l"Pur A iji = (l-p :",")e -L., P. + P:"''' E, = resting potential r, = membrane leakage resistance c ... = membrane capacitance (1.1) goJ:(t) .. conductance due to input type Ie in cell i (2.0) (2.2) V) (r»T) , S)O .. )=O for A.=t .. r-ru, otherwise (2.3) L'j = Ii - j I~ nc61ls .. number of cells in the simulation ~ .. distance between adjacent cells di = duration of conductance change due to input type Ie Vi '" velocity of signals for input type Ie Et = latency for input type Ie Pt = spatial anenuation factor for input type Ie P:"''' .. minimum spatial anenuation for input type Ie ru, = refractory period Field Poren/ials nc61ls = number of cells in the simulation nugs = number of segments in the companmental model V k (r) .. approximate extracellular field potential at cell j I ... (r) = membrane current for segment n in cell i Dendriric Model (2.4) T) = threshold for cell j L,) OK distance from cell i to cell j ~')t = distribution of synaptic density for input type Ie w'} = synaptic weight from cell j to cell i goJ: (t) = conductance due to input type Ie in cell i F t (t) = conductance waveform for input type k ~J (I) = spike output of cell j at time t U (t) = unit step function Zr/ec = depth of recording site 1" = depth of segment n (3.0) x .. x location of the jth cell R. = extracellular resistance per unit length (4.0) (4.1) 125 126 (4.2) nc"",,, = number of different channels per segment V" (r) = membrane potential of nth segment l:'(r) = membrane current for segment n /" = length of segment n c,: = membrane capacitance for segment n r; = axial resistance for segment n r:' - membrane resistance for segment n BPI< (r) = conductance of channel c in segment n Ec = equilibrium potential associated with channel c I:%(r) = axial current between segment nil and n d" = diameter of segment n R". = membrane resistivity R j = intracellular resistiviry per unit length R. = extracellular resistance per unit length e", = capacitance per unit surface area REFERENCES 1. 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TOWARDS AN ORGANIZING PRINCIPLE FOR A LAYERED PERCEPTUAL NETWORK Ralph Linsker IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598 Abstract 485 An information-theoretic optimization principle is proposed for the development of each processing stage of a multilayered perceptual network. This principle of "maximum information preservation" states that the signal transformation that is to be realized at each stage is one that maximizes the information that the output signal values (from that stage) convey about the input signals values (to that stage), subject to certain constraints and in the presence of processing noise. The quantity being maximized is a Shannon information rate. I provide motivation for this principle and -- for some simple model cases -- derive some of its consequences, discuss an algorithmic implementation, and show how the principle may lead to biologically relevant neural architectural features such as topographic maps, map distortions, orientation selectivity, and extraction of spatial and temporal signal correlations. A possible connection between this information-theoretic principle and a principle of minimum entropy production in nonequilibrium thermodynamics is suggested. Introduction This paper describes some properties of a proposed information-theoretic organizing principle for the development of a layered perceptual network. The purpose of this paper is to provide an intuitive and qualitative understanding of how the principle leads to specific feature-analyzing properties and signal transformations in some simple model cases. More detailed analysis is required in order to apply the principle to cases involving more realistic patterns of signaling activity as well as specific constraints on network connectivity. This section gives a brief summary of the results that motivated the formulation of the organizing principle, which I call the principle of "maximum information preservation." In later sections the principle is stated and its consequences studied. In previous work l I analyzed the development of a layered network of model cells with feedforward connections whose strengths change in accordance with a Hebb-type synaptic modification rule. I found that this development process can produce cells that are selectively responsive to certain input features, and that these feature-analyzing properties become progressively more sophisticated as one proceeds to deeper cell layers. These properties include the analysis of contrast and of edge orientation, and are qualitatively similar to properties observed in the first several layers of the mammalian visual pathway.2 Why does this happen? Does a Hebb-type algorithm (which adjusts synaptic strengths depending upon correlations among signaling activities3) cause a developing perceptual network to optimize some property that is deeply connected with the mature network's functioning as an information processing system? © American Institute ofPhvsics 1988 486 Further analysis4.s has shown that a suitable Hebb-type rule causes a linear-response cell in a layered feedforward network (without lateral connections) to develop so that the statistical variance of its output activity (in response to an ensemble of inputs from the previous layer) is maximized, subject to certain constraints. The mature cell thus performs an operation similar to principal component analysis (PCA), an approach used in statistics to expose regularities (e.g., clustering) present in high-dimensional input data. (Oja6 had earlier demonstrated a particular form of Hebb-type rule that produces a model cell that implements PCA exactly.) Furthermore, given a linear device that transforms inputs into an output, and given any particular output value, one can use optimal estimation theory to make a "best estimate" of the input values that gave rise to that output. Of all such devices, I have found that an appropriate Hebb-type rule generates that device for which this "best estimate" comes closest to matching the input values. 4•s Under certain conditions, such a cell has the property that its output preserves the maximum amount of information about its input values.s Maximum Information Preservation The above results have suggested a possible organizing principle for the development of each layer of a multilayered perceptual network.s The principle can be applied even if the cells of the network respond to their inputs in a nonlinear fashion, and even if lateral as well as feedforward connections are present. (Feedback from later to earlier layers, however, is absent from this formulation.) This principle of "maximum information preservation" states that for a layer of cells L that is connected to and provides input to another layer M, the connections should develop so that the transformation of signals from L to M (in the presence of processing noise) has the property that the set of output values M conveys the maximum amount of information about the input values L, subject to various constraints on, e.g., the range of lateral connections and the processing power of each cell. The statistical properties of the ensemble of inputs L are assumed stationary, and the particular L-to-M transformation that achieves this maximization depends on those statistical properties. The quantity being maximized is a Shannon information rate. 7 An equivalent statement of this principle is: The L-to-M transformation is chosen so as to minimize the amount of information that would be conveyed by the input values L to someone who already knows the output values M. We shall regard the set of input signal values L (at a given time) as an input "message"; the message is processed to give an output message M. Each message is in general a set of real-valued signal activities. Because noise is introduced during the processing, a given input message may generate any of a range of different output messages when processed by the same set of connections. The Shannon information rate (i.e., the average information transmitted from L to M per message) is7 R = LL LMP(L,M) log [P(L,M)/P(L)P(M)]. (1) For a discrete message space, peL) [resp. P(M)] is the probability of the input (resp. output) message being L (resp. M), and P(L,M) is the joint probability of the input being L and the output being M. [For a continuous message space, probabilities are 487 replaced by probability densities, and sums (over states) by integrals.] This rate can be written as (2) where h == - LL P(L) log P(L) (3) is the average information conveyed by message Land (4) is the average information conveyed by message L to someone who already knows M. Since II. is fixed by the properties of the input ensemble, maximizing R means minimizing I LIM, as stated above. The information rate R can also be written as (5) where 1M and IMI L are defined by interchanging Land M in Eqns. 3 and 4. This form is heuristically useful, since it suggests that one can attempt to make R large by (if possible) simultaneously making 1M large and IMI L small. The term 1M is largest when each message M occurs with equal probability. The term 1"'1/. is smallest when each L is transformed into a unique M, and more generally is made small by "sharpening" the P(M I L) distribution, so that for each L, P(M I L) is near zero except for a small set of messages M. How can one gain insight into biologically relevant properties of the L M transformation that may follow from the principle of maximum information preservation (which we also call the "infomax" principle)? In a network, this L M transformation may be a function of the values of one or a few variables (such as a connection strength) for each of the allowed connections between and within layers, and for each cell. The search space is quite large, particularly from the standpoint of gaining an intuitive or qualitative understanding of network behavior. We shall therefore consider a simple model in which the dimensionalities of the Land M signal spaces are greatly reduced, yet one for which the infomax analysis exhibits features that may also be important under more general conditions relevant to biological and synthetic network development. The next four sections are organized as follows. (i) A model is introduced in which the Land M messages, and the L-to-M transformation, have simple forms. The infomax principle is found to be satisfied when some simple geometric conditions (on the transformation) are met. (ii) I relate this model to the analysis of signal processing and noise in an interconnection network. The formation of topographic maps is discussed. (iii) The model is applied to simplified versions of biologically relevant problems, such as the emergence of orientation selectivity. (iv) I show that the main properties of the infomax principle for this model can be realized by certain local algorithms that have been proposed to generate topographic maps using lateral interactions. 488 A Simple Geometric Model In this model, each input message L is described by a point in a low-dimensional vector space, and the output message M is one of a number of discrete states. For definiteness, we will take the L space to be two-dimensional (the extension to higher dimensionality is straightforward). The L M transformation consists of two steps. (i) A noise process alters L to a message L' lying within a neighborhood of radius v centered on L. (ii) The altered message L' is mapped deterministically onto one of the output messages M. A given L' M mapping corresponds to a partitioning of the L space into regions labeled by the output states M. (We do not exclude a priori the possibility that multiple disjoint regions may be labeled by the same M.) Let A denote the total area of the L state space. For each M, let A (M) denote the area of L space that is labeled by M. Let sCM) denote the total border length that the region(s) labeled M share with regions of unlike M -label. A point L lying within distance v of a border can be mapped onto either M-value (because of the noise process L - L'). Call this a "borderline" L. A point L that is more than a distance v from every border can only be mapped onto the M-value of the region containing it. Suppose v is sufficiently small that (for the partitionings of interest) the area occupied by borderline L states is small compared to the total area of the L space. Consider first the case in which peL) is uniform over L. Then the information rate R (using Eqn. 5) is given approximately (through terms of order v) by R = ~M[A(M)/A] 10g[A(M)/A] - (yv/A) ~Ms(M). (6) To see this, note that P(M) = A(M)/ A and that P(M I L) log P(M I L) is zero except for borderline L (since 0 log 0 = 1 log 1 = 0). Here y is a positive number whose value depends upon the details of the noise process, which determines P(M I L) for borderline L as a function of distance from the border. For small v (low noise) the first term (1M) on the RHS of Eqn. 6 dominates. It is maximized when the A(M) [and hence the P(M)] values are equal for all M. The second term (with its minus sign), which equals ( -~'4IL)' is maximized when the sum of the border lengths of all M regions is minimized. This corresponds to "sharpening" the P(M I L) distribution in our earlier, more general, discussion. This suggests that the infomax solution is obtained by partitioning the L space into M-regions (one for each M value) that are of substantially equal area, with each M-region tending to have near-minimum border length. Although this simple analysis applies to the low-noise case, it is plausible that even when v is comparable to the spatial scale of the M regions, infomax will favor making the M regions have approximately the same extent in all directions (rather than be elongated), in order to "sharpen" p(MI L) and reduce the probability of the noise process mapping L onto many different M states. What if peL) is nonuniform? Then the same result (equal areas, minimum border) is obtained except that both the area and border-length elements must now be weighted by the local value of peL). Therefore the infomax principle tends to produce maps in which greater representation in the output space is given to regions of the input signal space that are activated more frequently. To see how lateral interactions within the M layer can affect these results, let us suppose that the L M mapping has three, not two, process steps: L L' 489 - M - M, where the first two steps are as above, and the third step changes the output M into any of a number of states M (which by definition comprise the "M-neighborhood" of M). We consider the case in which this M-neighborhood relation is symmetric. This type of "lateral interaction" between M states causes the infomax principle to favor solutions for which M regions sharing a border in L space are M-neighbors in the sense defined. For a simple example in which each state M has n M-neighbors (including itself), and each M-neighbor has an equal chance of being the final state (given M), infomax tends to favor each M-neighborhood having similar extent in all directions (in L space). Relation Between the Geometric Model and Network Properties The previous section dealt with certain classes of transformations from one message space to another, and made no specific reference to the implementation of these transformations by an interconnected network of processor cells. Here we show how some of the features discussed in the previous section are related to network properties. For simplicity suppose that we have a two-dimensional layer of uniformly distributed cells, and that the signal activity of each cell at any given time is either 1 (active) or 0 (quiet). We need to specify the ensemble of input patterns. Let us first consider a simple case in which each pattern consists of a disk of activity of fixed radius, but arbitrary center position, against a quiet background. In this case the pattern is fully defined by specifying the coordinates of the disk center. In a two-dimensional L state space (previous section), each pattern would be represented by a point having those coordinates. Now suppose that each input pattern consists not of a sharply defined disk of activity, but of a "fuzzy" disk whose boundary (and center position) are not sharply defined. [Such a pattern could be generated by choosing (from a specified distribution) a position Xc as the nominal disk center, then setting the activity of the cell at position X to 1 with a probability that decreases with distance I x Xc I . ] Any such pattern can be described by giving the coordinates of the "center of activity" along with many other values describing (for example) various moments of the activity pattern relative to the center. For the noise process L L' we suppose that the activity of an L cell can be "misread" (by the cells of the M layer) with some probability. This set of distorted activity values is the "message" L'. We then suppose that the set of output activities M is a deterministic function of L'. We have constructed a situation in which (for an appropriate choice of noise level) two of the dimensions of the L state space -- namely, those defined by the disk center coordinates -- have large variance compared to the variance induced by the noise process, while the other dimensions have variance comparable to that induced by noise. In other words, the center position of a pattern is changed only a small amount by the noise process (compared to the typical difference between the center positions of two patterns), whereas the values of the other attributes of an input pattern differ as much from their noise-altered values as two typical input patterns differ from each other. (Those attributes are "lost in the noise. ") Since the distance between L states in our geometric model (previous section) corresponds to the likelihood of one L state being changed into the other by the noise 490 process, we can heuristically regard the L state space (for the present example) as a "slab" that is elongated in two dimensions and very thin in all other dimensions. (In general this space could have a much more complicated topology, and the noise process which we here treat as defining a simple metric structure on the L state space need not do so. These complications are beyond the scope of the present discussion.) This example, while simple. illustrates a feature that is key to understanding the operation of the infomax principle: The character of the ensemble statistics and of the noise process jointly determine which attributes of the input pattern are statistically most significant; that is, have largest variance relative to the variance induced by noise. We shall see that the infomax principle selects a number of these most significant attributes to be encoded by the L M transformation. We turn now to a description of the output state space M. We shall assume that this space is also of low dimensionality. For example, each M pattern may also be a disk of activity having a center defined within some tolerance. A discrete set of discriminable center-coordinate values can then be used as the M-region "labels" in our geometric model. Restricting the form of the output activity in this particular way restricts us to considering positional encodings L M, rather than encodings that make use of the shape of the output pattern, its detailed activity values, etc. However, this restriction on the form of the output does not determine which features of the input patterns are to be encoded, nor whether or not a topographic (neighbor-preserving) mapping is to be used. These properties will be seen to emerge from the operation of the infomax principle. In the previous section we saw that the infomax principle will tend to lead to a partitioning of the L space into M regions having equal areas [if peL) is uniform in the coordinates of the L disk center] and minimum border length. For the present case this means that the M regions will tend to "tile" the two long dimensions of the L state space "slab," and that a single M value will represent all points ill L space that differ only in their low-variance coordinates. If peL) is nonuniform, then the area of the M region at L will tend to be inversely proportional to peL). Furthermore, if there are local lateral connections between M cells, then (depending upon the particular form of such interaction) M states corresponding to nearby localized regions of layer-M activity can be M-neighbors in the sense of the previous section. In this case the mapping from the two high-variance coordinates of L space to M space will tend to be topographic. Examples: Orientation Selectivity and Temporal Feature Maps The simple example in the previous section illustrates how infomax can lead to topographic maps, and to map distortions [which provide greater M-space representation for regions of L having large peL)]. Let us now consider a case in which information about input features is positionally encoded in the output layer as a result of the infomax principle. Consider a model case in which an ensemble of patterns is presented to the input layer L. Each pattern consists of a rectangular bar of activity (of fixed length and width) against a quiet background. The bar's center position and orientation are chosen for each pattern from uniform distributions over some spatial interval for the position, and over all orientation angles (i.e., from 0° to 180°). The bar need not be sharply defined, but can be "fuzzy" in the sense described above. We assume, however, that all 491 properties that distinguish different patterns of the ensemble -- except for center position and orientation -- are "lost in the noise" in the sense we discussed. To simplify the representation of the solution, we further assume that only one coordinate is needed to describe the center position of the bar for the given ensemble. For example, the ensemble could consist of bar patterns all of which have the same y coordinate of center position, but differ in their x coordinate and in orientation 0. We can then represent each input state by a point in a rectangle (the L state space defined in a previous section) whose abscissa is the center-position coordinate x and whose ordinate is the angle 0. The horizontal sides of this rectangle are identified with each other, since orientations of 0 0 and 180 0 are identical. (The interior of the rectangle can thus be thought of as the surface of a horizontal cylinder.) The number Nx of different x positions that are discriminable is given by the range of x values in the input ensemble divided by the tolerance with which x can be measured (given the noise process L L'); similarly for No. The relative lengths Llx and MJ of the sides of the L state space rectangle are given by Llx/ MJ = Nj No. We discuss below the case in which Nx > > No; if No were> > Nx the roles of x and 0 in the resulting mappings would be reversed. There is one complicating feature that should be noted, although in the interest of clarity we will not include it in the present analysis. Two horizontal bar patterns that are displaced by a horizontal distance that is small compared with the bar length, are more likely to be rendered indiscriminable by the noise process than are two vertical bar patterns that are displaced by the same horizontal distance (which may be large compared with the bar's width). The Hamming distance, or number of binary activity values that need to be altered to change one such pattern into the other, is greater in the latter case than in the former. Therefore, the distance in L state space between the two UNORIENTED RECEPTIVE FIELDS Figure 1. Orientation Selectivity in a Simple Model: As the input domain size (see text) is reduced [from (a) upper left, to (b) upper right, to (c) lower left figure], infomax favors the emergence of an orientation-selective L M mapping. (d) Lower right figure shows a solution obtained by applying Kohonen's relaxation algorithm with 50 M-points (shown as dots) to this mapping problem. 492 states should be greater in the latter case. This leads to a "warped" rather than simple rectangular state space. We ignore this effect here, but it must be taken into account in a fuller treatment of the emergence of orientation selectivity. Consider now an L M transformation that consists of the three-step process (discussed above) (i) noise-induced L - L' ; (ii) deterministic L' - M'; (iii) lateral-interaction-induced M' - M. Step (ii) maps the two-dimensional L state space of points (x, 0) onto a one-dimensional M state space. For the present discussion, we .consider L' - M' maps satisfying the following Ansatz: Points corresponding to the M states are spaced uniformly, and in topographic order, along a helical line in L state space (which we recall is represented by the surface of a horizontal cylinder). The pitch of the helix (or the slope dO/dx) remains to be determined by the infomax principle. Each M-neighborhood of M states (previous section) then corresponds to an interval on such a helix. A state L' is mapped onto a state in a particular M-neighborhood if L' is closer (in L space) to the corresponding interval of the helix than to any other portion of the helix. We call this set of L states (for an M-neighborhood centered on M ) the "input domain" of M. It has rectangular shape and lies on the cylindrical surface of the L space. We have seen (previous sections) that infomax tends to produce maps having (i) equal M-region areas, (ii) topographic organization, and (iii) an input domain (for each M-neighborhood) that has similar extent in all directions (in L space). Our choice of Ansatz enforces (i) and (ii) explicitly. Criterion (iii) is satisfied by choosing dO / dx such that the input domain is square (for a given M-neighborhood size). Figure 1a (having dO/dx = 0) shows a map in which the output M encodes only information about bar center position x, and is independent of bar orientation o. The size of the M -neighborhood is relatively large in this case. The input domain of the state M denoted by the 'x' is shown enclosed by dotted lines. (The particular 0 value at which we chose to draw the M line in Fig. 1a is irrelevant.) For this M-neighborhood size, the length of the border of the input domain is as small as it can be. As the M -neighborhood size is reduced, the dotted lines move closer together. A vertically oblong input domain (which would result if we kept dO/dx = 0 ) would not satisfy the infomax criterion. The helix for which the input domain is square (for this smaller choice of M-neighborhood size) is shown in Fig. lb. The M states for this solution encode information about bar orientation as well as center position. If each M state corresponds to a localized output activity pattern centered at some position in a one-dimensional array of M cells, then this solution corresponds to orientation-selective cells organized in "orientation columns" (really "orientation intervals" in this one-dimensional model). A "labeling" of the linear array of cells according to whether their orientation preferences lie between 0 and 60, 60 and 120, or 120 and 180 degrees is indicated by the bold, light, and dotted line segments beneath the rectangle in Fig. 1 b (and 1c). As the M-neighborhood size is decreased still further, the mapping shown in Fig. Ie becomes favored over that of either Fig. 1a or lb. The "orientation columns" shown in the lower portion of Fig. 1 c are narrower than in Fig. 1 b. A more detailed analysis of the information rate function for various mappings confirms the main features we have here obtained by a simple geometric argument. The same type of analysis can be applied to different types of input pattern ensembles. To give just one other example, consider a network that receives an ensemble of simple patterns of acoustic input. Each such pattern consists of a tone of 493 some frequency that is sensed by two "ears" with some interaural time delay. Suppose that the initial network layers organize the information from each ear (separately) into tonotopic maps, and that (by means of connections having a range of different time delays) the signals received by both ears over some time interval appear as patterns of cell activity at some intermediate layer L. We can then apply the infomax principle to the signal transformation from layer L to the next layer M. The L state space can (as before) be represented as a rectangle, whose axes are now frequency and interaural delay (rather than spatial position and bar orientation). Apart from certain differences (the density of L states may be nonuniform, and states at the top and bottom of the rectangle are no longer identical), the infomax analysis can be carried out as it was for the simplified case of orientation selectivity. Local Algorithms The information rate (Eqn. I), which the infomax principle states is to be maximized subject to constraints (and possibly as part of an optimization function containing other cost terms not discussed here), has a very complicated mathematical form. How might this optimization process, or an approximation to it, be implemented by a network of cells and connections each of which has limited computational power? The geometric form in which we have cast the infomax principle for some very simple model cases, suggests how this might be accomplished. An algorithm due to Kohonen 8 demonstrates how topographic maps can emerge as a result of lateral interactions within the output layer. I applied this algorithm to a one-dimensional M layer and a two-dimensional L layer, using a Euclidean metric and imposing periodic boundary conditions on the short dimension of the L layer. A resulting map is shown in Fig. Id. This map is very similar to those of Figs. 1 band Ic, except for one reversal of direction. The reversal is not surprising, since the algorithm involves only local moves (of the M-points) while the infomax principle calls for a globally optimal solution. More generally, Kohonen's algorithm tends empirically8 to produce maps having the property that if one constructs the Voronoi diagram corresponding to the positions of the M-points (that is, assigns each point L to an M region based on which M-point L is closest to), one obtains a set of M regions that tend to have areas inversely proportional to P(L) , and neighborhoods (corresponding to our input domains) that tend to have similar extent in all directions rather than being elongated. The Kohonen algorithm makes no reference to noise, to information content, or even to an optimization principle. Nevertheless, it appears to implement, at least in a qualitative way, the geometric conditions that infomax imposes in some simple cases. This suggests that local algorithms along similar lines may be capable of implementing the infomax principle in more general situations. Our geometric formulation of the infomax principle also suggests a connection with an algorithm proposed by von der Malsburg and Willshaw9 to generate topographic maps. In their "tea trade" model, neighborhood relationships are postulated within the source and the target spaces, and the algorithm's operation leads to the establishment of a neighborhood-preserving mapping from source to target space. Such neighborhood relationships arise naturally in our analysis when the infomax principle is applied to our three-step L L' - M' - M transformation. The noise process induces a 494 neighborhood relation on the L space, and lateral connections in the M cell layer can induce a neighborhood relation on the M space. More recently, Durbin and Willshaw lO have devised an approach to solving certain geometric optimization problems (such as the traveling salesman problem) by a gradient descent method bearing some similarity to Kohonen's algorithm. There is a complementary relationship between the infomax principle and a local algorithm that may be found to implement it. On the one hand, the principle may explain what the algorithm is "for" -- that is, how the algorithm may contribute to the generation of a useful perceptual system. This in turn can shed light on the system-level role of lateral connections and synaptic modification mechanisms in biological networks. On the other hand, the existence of such a local algorithm is important for demonstrating that a network of relatively simple processors -- biological or synthetic -- can in fact find global near-maxima of the Shannon information rate. A Possible Connection Between Infomax and a Thermodynamic Principle The principle of "maximum preservation of information" can be viewed equivalently as a principle of "minimum dissipation of information." When the principle is satisfied, the loss of information from layer to layer is minimized, and the flow of information is in this sense as "nearly reversible" as the constraints allow. There is a resemblance between this principle and the principle of "minimum entropy production" II in nonequilibrium thermodynamics. It has been suggested by Prigogine and others that the latter principle is important for understanding self-organization in complex systems. There is also a resemblance, at the algorithmic level, between a Hebb-type modification rule and the autocatalytic processes l2 considered in certain models of evolution and natural selection. This raises the possibility that the connection I have drawn between synaptic modification rules and an information-theoretic optimization principle may be an example of a more general relationship that is important for the emergence of complex and apparently "goal-oriented If structures and behaviors from relatively simple local interactions, in both neural and non-neural systems. References [1] R. Linsker, Proc. Natl. Acad. Sci. USA 83,7508,8390,8779 (1986). [2] D. H. Hubel and T. N. Wiesel, Proc. Roy. Soc. London 8198,1 (1977). [3] D. O. Hebb, The Organization of Behavior (Wiley, N. Y., 1949). [4] R. Linsker, in: R. Cotterill (ed.), Computer Simulation in Brain Science (Copenhagen. 20-22 August 1986; Cambridge Univ. Press, in press), p. 416. [5] R. Linsker, Computer (March 1988, in press). [6] E. Oja,J. Math. Bioi. 15 , 267 (1982). [7] C. E. Shannon, Bell Syst. Tech. J. 27 . 623 (1948). [8] T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, N. Y .. 19S4). [9] C. von der Malsburg and D. J. Willshaw, Proc. Nat I. A cad. Sci. USA 74 , 5176 (1977). [10] R. Durbin and D. J. Willshaw, Nature 326,689 (1987). [11] P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stabili(v. and Fluctuations (Wiley-Interscience, N. Y., 1971). [12] M. Eigen and P. Schuster, Die Naturwissenschaften 64 , 541 (1977).
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592 A Trellis-Structured Neural Network* Thomas Petschet and Bradley W. Dickinson Princeton University, Department of Electrical Engineering Princeton, N J 08544 Abstract We have developed a neural network which consists of cooperatively interconnected Grossberg on-center off-surround subnets and which can be used to optimize a function related to the log likelihood function for decoding convolutional codes or more general FIR signal deconvolution problems. Connections in the network are confined to neighboring subnets, and it is representative of the types of networks which lend themselves to VLSI implementation. Analytical and experimental results for convergence and stability of the network have been found. The structure of the network can be used for distributed representation of data items while allowing for fault tolerance and replacement of faulty units. 1 Introd uction In order to study the behavior of locally interconnected networks, we have focused on a class of "trellis-structured" networks which are similar in structure to multilayer networks [5] but use symmetric connections and allow every neuron to be an output. We are studying such· locally interconnected neural networks because they have the potential to be of great practical interest. Globally interconnected networks, e.g., Hopfield networks [3], are difficult to implement in VLSI because they require many long wires. Locally connected networks, however, can be designed to use fewer and shorter wires. In this paper, we will describe a subclass of trellis-structured networks which optimize a function that, near the global minimum, has the form of the log likelihood function for decoding convolutional codes or more general finite impulse response signals. Convolutional codes, defined in section 2, provide an alternative representation scheme which can avoid the need for global connections. Our network, described in section 3, can perform maximum likelihood sequence estimation of convolutional coded sequences in the presence of noise. The performance of the system is optimal for low error rates. The specific application for this network was inspired by a signal decomposition network described by Hopfield and Tank [6]. However, in our network, there is an emphasis on local interconnections and a more complex neural model, the Grossberg on-center off-surround network [2], is used. A modified form of the Gorssberg model is defined in section 4. Section 5 presents the main theoretical results of this paper. Although the deconvolution network is simply a set of cooperatively interconnected ·Supported by the Office of N ava.l Research through grant N00014-83-K-0577 and by the National Science Foundation through grant ECS84-05460. tpermanent address: Siemens Corporate Research and Support, Inc., 105 College Road East, Princeton, N J 08540. @ American Institute of Physics 1988 593 on-center off-surround subnetworks, and absolute stability for the individual subnetworks has been proven [1], the cooperative interconnections between these subnets make a similar proof difficult and unlikely. We have been able, however, to prove equiasymptotic stability in the Lyapunov sense for this network given that the gain of the nonlinearity in each neuron is large. Section 6 will describe simulations of the network that were done to confirm the stability results. 2 Convolutional Codes and MLSE In an error correcting code, an input sequence is transformed from a b-dimensional input space to an M -dimensional output space, where M ~ b for error correction and/ or detection. In general, for the b-bit input vector U = (U1, . •• ,Ub) and the Mbit output vector V = (VI, ... , VM), we can write V = F( U1, . . . ,Ub). A convolutional code, however, is designed so that relatively short subsequences of the input vector are used to determine subsequences of the output vector. For example, for a rate 1/3 convolutional code (where M ~ 3b), with input subsequences oflength 3, we can write the output, V = (VI, ... , Vb) for Vi = (Vi,I, Vi,2, Vi,3), of the encoder as a convolution of the input vector U = (UI, ... , Ub, 0, 0) and three generator sequences go = (11 1) gi = (1 1 0) g2 = (0 1 1). This convolution can be written, using modulo-2 addition, as Vi= (1) k=max{I,i-2) In this example, each 3-bit output subsequence, Vi, of V depends only on three bits of the input vector, i.e., Vi = I( Ui-2, Ui-I, Ui). In general, for a rate l/n code, the constraint length, K, is the number of bits of the input vector that uniquely determine each n-bit output subsequence. In the absence of noise, any subsequences in the input vector separated by more than K bits (i.e., that do not overlap) will produce subsequences in the output vector that are independent of each other. If we view a convolutional code as a special case of block coding, this rate 1/3, K = 3 code converts a b-bit input word into a codeword of length 3(b + 2) where the 2 is added by introducing two zeros at the end of every input to "zero-out" the code. Equivalently, the coder can be viewed as embedding 2b memories into a 23{b+2L dimensional space. The minimum distance between valid memories or codewords in this space is the free distance of the code, which in this example is 7. This implies that the code is able to correct a minimum of three errors in the received signal. For a convolutional code with constraint length K, the encoder can be viewed as a finite state machine whose state at time i is determined by the K - 1 input bits, Ui-k, ... , Ui-I. The encoder can also be represented as a trellis graph such as the one shown in figure 1 for a K = 3, rate 1/3 code. In this example, since the constraint length is three, the two bits Ui-2 and Ui-I determine which of four possible states the encoder is in at time i. In the trellis graph, there is a set of four nodes arranged in a vertical column, which we call a stage, for each time step i. Each node is labeled with the associated values of Ui-2 and Ui-1. In general, for a rate l/n code, each stage of the trellis graph contains 2K -1 nodes, representing an equal number of possible states. A trellis graph which contains S stages therefore fully describes the operation of the encoder for time steps 1 through S. The graph is read from left to right and the upper edge leaving the right side of a node in stage i is followed if Ui is a zero; the lower edge 594 stage i-2 &000 111 101 &-010 stage i-1 stage i stage i+1 stage i+2 state 1 state 2 state 3 state 4 Figure 1: Part of the trellis-code representation for a rate 1/3, K = 3 convolutional code. if Ui is a one. The label on the edge determined by Ui is Vi, the output of the encoder given by equation 1 for the subsequence Ui-2, Ui-I, Ui. Decoding a noisy sequence that is the output of a convolutional coder plus noise is typically done using a maximum likelihood sequence estimation (MLSE) decoder which is designed to accept as input a possibly noisy convolutional coded sequence, R, and produce as output the maximum likelihood estimate, V, of the original sequence, V. If the set of possible n(b+2)-bit encoder output vectors is {Xm : m = 1, ... , 2n(b+2)} and Xm,i is the ith n-bit subsequence of Xm and ri is the ith n-bit subsequence of R then b V = argmax II P(ri I Xm,i) Xm i=l (2) That is, the decoder chooses the Xm that maximizes the conditional probability, given . X m , of the received sequence. A binary symmetric channel (BSC) is an often used transmission channel model in which the decoder produces output sequences formed from an alphabet containing two symbols and it is assumed that the probability of either of the symbols being affected by noise so that the other symbol is received is the same for both symbols. In the case of a BSC, the log of the conditional probability, P( ri I Xm,i), is a linear function of the Hamming distance between ri and Xm,i so that maximizing the right side of equation 2 is equivalent to choosing the Xm that has the most bits in common with R. Therefore, equation 2 can be rewritten as (3) where Xm,i,l is the lth bit of the ith subsequence of Xm and fa (b) is the indicator function: fa(b) = 1 if and only if a equals b. For the general case, maximum likelihood sequence estimation is very expensive since the number of possible input sequences is exponential in b. The Viterbi algorithm [7], fortunately, is able to take advantage of the structure of convolutional codes and their trellis graph representations to reduce the complexity of the decoder so that 595 it is only exponential in I( (in general K ~ b). An optimum version of the Viterbi algorithm examines all b stages in the trellis graph, but a more practical and very nearly optimum version typically examines approximately 5K stages, beginning at stage i, before making a decision about Ui. 3 A Network for MLSE Decoding The structure of the network that we have defined strongly reflects the structure of a trellis graph. The network usually consists of 5]( subnetworks, each containing 2K - 1 neurons. Each subnetwork corresponds to a stage in the trellis graph and each neuron to a state. Each stage is implemented as an "on-center off-surround" competitive network [2], described in more detail in the next section, which produces as output a contrast enhanced version of the input. This contrast enhancement creates a "winner take all" situation in which, under normal circumstances, only one neuron in each stage -the neuron receiving the input with greatest magnitude will be on. The activation pattern of the network after it reaches equilibrium indicates the decoded sequence as a sequence of "on" neurons in the network. If the j-th neuron in subnet i, Ni,i is on, then the node representing state j in stage i lies on the network's estimate of the most likely path. For a rate lin code, there is a symmetric cooperative connection between neurons Ni,j and Ni+1 ,k if there is an edge between the corresponding nodes in the trellis graph. If (xi,i,k,l, . .. , Xi,j,k,n) are the encoder output bits for the transition between these two nodes and (ri,!, ... , ri,n) are the received bits, then the connection weight for the symmetric cooperative connection between Ni,i and Ni+1,k is 1 n m "" k--"I. (X "" k/) ',J, L.J ri I ',J" n 1=1 ' If there is no edge between the nodes, then mi,i,k = o. (4) Intuitively, it is easiest to understand the action of the entire network by examining one stage. Consider the nodes in stage i of the trellis graph and assume that the conditional probabilities of the nodes in stages i - 1 and i + 1 are known. (All probabilities are conditional on the received sequence.) Then the conditional probability of each node in stage i is simply the sum of the probabilities of each node in stages i - 1 and i + 1 weighted by the conditional transition probabilities. If we look at stage i in the network, and let the outputs of the neighboring stages i - 1 and i + 1 be fixed with the output of each neuron corresponding to the "likelihood" of the corresponding state at that stage, then the final outputs of the neurons M,i will correspond to the "likelihood" of each of the corresponding states. At equilibrium, the neuron corresponding to the most likely state will have the largest output. 4 The Neural Model The "on-center off-surround" network[2] is used to model each stage in our network. This model allows the output of each neuron to take on a range of values, in this case between zero and one, and is designed to support contrast enhancement and competition between neurons. The model also guarantees that the final output of each neuron is a function of the relative intensity of its input as a fraction of the total input provided to the network. 596 Using the "on-center off-surround" model for each stage and the interconnection weights, mi,j,k, defined in equation 4, the differential equation that governs the instantaneous activity of the neurons in our deconvolution network with S stages and N states in each stage can be written as N Ui,j = -Aui,j + (B - Ui,j) (f(Ui,j) + I)mi-I,k,jf(Ui-I,k) + mi,j,kf(Ui+1,k)]) N klJ (5) - (C + Ui,j) L (f( Ui,k) + L[mi-I,k,t!( Ui-I,k) + mi,l,kf( Ui+1,k)]) k"lj 1=1 where f(x) = (1 + e-'\X)-I, oX is the gain of the nonlinearity, and A, B, and Care constants For the analysis to be presented in section 5, we note that equation 5 can be rewritten more compactly in a notation that is similar to the equation for additive analog neurons given in [4]: S N U' . Au" '"' '"'Cu' ·S· . k 1!(Uk I) 1'. .. k 1!(Uk I)) &,} &,} L..J L..J &,} &,}, , ,&,3" , (6) k=I/=I where, for 1 ~ I ~ N, S· " 1- 1 &,},&, S· " 11 = ~m ' I &,},&-, l.J &-I"q q S .. · 1/- ~m' I &,},&+ , l.J &,q, 1'. . ... - B &,},&,} 1'. .. . I = -C V I ~ J' &,},&, r Ti j i-I I = Bmi_l I j - C E mi-l I q '" , , q¢j " (7) q Si,j,k,1 = 0 V k ¢ {i - 1, i, i + 1} 1'. ... II - Bm"1 C ~ m' I &,3,&+ , I,), l.J &,q, q"lj To eliminate the need for global interconnections within a stage, we can add two summing elements to calculate N N N Xi = L f(Xi,j) and Ji = L L [mi-l,k,j!( Ui-l,k) + mi,j,kf( Ui+1,k)] (8) j=1 j=I k=I Using these two sums allows us to rewrite equation 5 as U· . = -Au' . + (B + C)(f(u· .) + L .) - U· ·(X· + J.) &,} &,} &,) &,) &,} & & (9) This form provides a more compact design for the network that is particularly suited to implementation as a digital filter or for use in simulations since it greatly reduces the calculations required, 5 Stability of the Network The end of section 3 described the desired operation of a single stage, given that the outputs of the neighboring stages are fixed. It is possible to show that in this situation a single stage is stable. To do this, fix f( Uk,/) for k E {i - 1, i + 1} so that equation 6 can be written in the form originally proposed by Grossberg [2]: N N Ui,j = -Auj,j + (B - Uj,j) (Ii,j + f(Ui,j)) - (Ui,j + C)(L Ii,k + L !(Ui,k)) (10) k=1 k=1 597 where Ii,; = 2:1'=1 [mi-l,k,jf( Ui-l,k) + mi,j,kf( Ui+I,k)]. Equation 10 is a special case of the more general nonlinear system Xi = ai(xi) (bi(Xi) - t Ci,kdk(Xk)) k=1 (11) where: (1) ai(xi) is continuous and ai(xd > 0 for Xi 2: OJ (2) bi(Xi) is continuous for Xi ~ OJ (3) Ci,k = Ck,ij and (4) di(Xi) ~ 0 for all Xi E (-00,00). Cohen and Grossberg [1] showed that such a system has a global Lyapunov function: (12) and that, therefore, such a system is equiasymptotically stable for all constants and functions satisfying the four constraints above. In our case, this means that a single stage has the desired behavior when the neighboring stages are fixed. IT we take the output of each neuron to correspond to the likelihood of the corresponding state then, if the two neighboring stages are fixed, stage i will converge to an equilibrium point where the neuron receiving the largest input will be on and the others will be off, just as it should according to section 2. It does not seem possible to use the Cohen-Grossberg stability proof for the entire system in equation 5. In fact, Cohen and Grossberg note that networks which allow cooperative interactions define systems for which no stability proof exists [1]. Since an exact stability proof seems unlikely, we have instead shown that in the limit as the gain, A, of the nonlinearity gets large the system is asymptotically stable. Using the notation in [4], define Vi = f(Ui) and a normalized nonlinearity J(.) such that J-l(Vi) = AUi. Then we can define an energy function for the deconvolution network to be 1 1 ( ) ~VIc'1 1 E - -T:. V.· ·Vi -A S ·· Vi d 2 L 1,.1,k,l 1,.1 k,l L A L 1,.1,k,l k,l 1 f (() ( i,j,k,l i,j k,l 2 (13) The time derivative of E is . L dVii ( L L --' -Au·· U· . S· · Vi T· · Vi E dt I,) 1,.1 1,.1,k,l k,l + 1,.1,k,l k,l i,i k,l k,l 1 ~ (VIc ,1 1 ) - "X L.i Si,j,k,l } l. f- (()d( k,l 2 (14) It is difficult to prove that E is nonpositive because of the last term in the parentheses. However, for large gain, this term can be shown to have a negligible effect on the derivative. It can be shown that for f(u) = (1 + C>'U)-I, Ii' l-I(()d( is bounded above 2 by log(2). In this deconvolution network, there are no connections between neurons unless they are in the same or neighboring stages, i.e., Si,i,k,l = 0 for Ii - kl > 1 and 1 is restricted so that 0 ~ 1 ~ S, so there are no more than 3S non-zero terms in the problematical summation. Therefore, we can write that 1 ~VIc ' 1 _ lim -, L Si,j,k,l f-l(() d( = 0 >'-00 A k,l t 598 Then, in the limit as ). -- 00, the terms in parentheses in equation 14 converge to Ui . . h li E· ~ dVi j. U· th h· I ·t thO III equatIOn 6, so t at m = L..J --' Ui. smg e c am ru e, we can reWrl e IS .\-+00 .. dt t,) as E.n~J = - t= (d~J )'( d~/-l (V;J») It can also be shown that that, if 1(·) is a monotonically increasing function then ~ I-I (Vi) > 0 for all Vi. This implies that for all u = (Ui,b . .• ,UN,S), lim>.-+oo E S; 0, and, therefore, for large gains, E as defined in equation 13 is a Lyapunov function for the system described by equation 5 and the network is equiasymtotically stable. If we apply a similar asymptotic argument to the energy function, equation 13 reduces to 1 E ~ T: . k IV.· . Vik I - - '2 L..J ',3" 1,3 , i,j,k,1 (15) which is the Lyapunov function for a network of discontinuous on-off neurons with interconnection matrix T. For the binary neuron case, it is fairly straight forward to show that the energy function has minima at the desired decoder outputs if we assume that only one neuron in each stage may be on and that Band C are appropriately chosen to favor this. However, since there are 0(52 N) terms in the disturbance summation in equation 15, convergence in this case is not as fast as for the derivative of the energy function in equation 13, which has only 0(5) terms in the summation. 6 Simulation Results The simulations presented in this section are for the rate 1/3, K = 3 convolutional code illustrated in figure 1. Since this code has a constraint length of 3, there are 4 possible states in each stage and an MLSE decoder would normally examine a minimum of 5K subsequences before making a decision, we will use a total of 16 stages. In these simulations, the first and last stage are fixed since we assume that we have prior knowledge or a decision about the first stage and zero knowledge about the last stage. The transmitted codeword is assumed to be all zeros. The simulation program reads the received sequence from standard input and uses it to define the interconnection matrix W according to equation 4. A relaxation subroutine is then called to simulate the performance of the network according to an Euler discretization of equation 5. Unit time is then defined as one RC time constant of the unforced system. All variables were defined to be single precision (32 bit) floating point numbers. Figure 2a shows the evolution of the network over two unit time intervals with the sampling time T = 0.02 when the received codeword contains no noise. To interpret the figure, recall that there are 16 stages of 4 neurons each. The output of each stage is a vertical set of 4 curves. The upper-left set is the output of the first stage; the upper-most curve is the output of the first neuron in the stage. For the first stage, the first neuron has a fixed output of 1 and the other neurons have a fixed output of o. The outputs of the neurons in the last stages are fixed at an intermediate value to represent zero a priori knowledge about these states. Notice that the network reaches an equilibrium point in which only the top neurons in each state (representing the "00" node in figure 1) are on and all others are off. This case illustrates that the network can correctly decode an unerrored input and that it does so rapidly, i.e., in about one time constant. In this case, with no errors in the input, the network performs the 599 o 2 0 2 0 2 0 2 o 10 0 10 0 10 0 10 ( a) (b) Figure 2: Evolution of the trellis network for (a) unerrored input, (b) input with burst errors: R is 000 000 000 000 000 000 000 000 111 000 000 000 000 000 000. A = 10., A = 1.0, B = 1.0, C = 0.75, T = 0.02. The initial conditions are XI,1 = 1., xI,.i = 0.0, X16,j = 0.2, all other Xi,j = 0.0. same function as Hopfield and Tank's network and does so quite well. Although we have not been able to prove it analytically, all our simulations support the conjecture that if xi,AO) = ~ for all i and j then the network will always converge to the global minimum. One of the more difficult decoding problems for this network is the correction of a burst of errors in a transition subsequence. Figure 2b shows the evolution of the network when three errors occur in the transition between stages 9 and 10. Note that 10 unit time intervals are shown since complete convergence takes much longer than in the first example. However, the network has correctly decoded many of the stages far from the burst error in a much shorter time. If the received codeword contains scattered errors, the convolutional decoder should be able to correct more than 3 errors. Such a case is shown in figure 3a in which the received codeword contains 7 errors. The system takes longest to converge around two transitions, 5-6 and 11-12. The first is in the midst of consecutive subsequences which each have one bit errors and the second transition contains two errors. To illustrate that the energy function shown in equation 13 is a good candidate for a Lyapunov function for this network, it is plotted in figure 3b for the three cases described above. The nonlinearity used in these simulations has a gain of ten, and, as predicted by the large gain limit, the energy decreases monotonically. To more thoroughly explore the behavior of the network, the simulation program was modified to test many possible error patterns. For one and two errors, the program exhaustively tested each possible error pattern. For three or more errors, the errors were generated randomly. For four or more errors, only those errored sequences for which the MLS estimate was the sequence of all zeros were tested. The results of this simulation are summarized in the column labeled "two-nearest" in figure 4. The performance of the network is optimum if no more than 3 errors are present in the received sequence, however for four or more errors, the network fails to correctly decode some sequences that the MLSE decoder can correctly decode. 600 ~~~~ 80 60 ~~~~ 40 E ~~~~ 20 errors ~~~E 0 -20 0 . 0 0.5 1.0 1.5 2.0 0 2 0 2 0 2 0 2 time (a) (b) Figure 3: (a) Evolution of the trellis network for input with distributed errors. The input, R, is 000 010 010 010 100 001 000 000 000 000 110 000 000 000 000. The constants and initial conditions are the same as in figure 2. (b) The energy function defined in equation 13 evaulated for the three simulations discussed. errored number of number of errors bits test vectors tvo-nearest four-nearest 0 1 0 0 1 39 0 0 2 500 0 0 3 500 0 0 4 500 7 0 5 500 33 20 6 500 72 68 7 500 132 103 Total 2500 244 191 Figure 4: Simulation results for a deconvolution network for a K = 3, rate 1/3 code. The network parameters were: .x = 15, A = 6, B = 1, C = 0.45, and T = 0.025. For locally interconnected networks, the major concern is the flow of information through the network. In the simulations presented until now, the neurons in each stage are connected only to neurons in neighboring stages. A modified form of the network was also simulated in which the neurons in each stage are connected to the neurons in the four nearest neighboring stages. To implement this network, the subroutine to initialize the connection weights was modified to assign a non-zero value to Wi,j,i+2,k. This is straight-forward since, for a code with a constraint length of three, there is a single path connecting two nodes a distance two apart. The results of this simulation are shown in the column labeled "four-nearest" in figure 4. It is easy to see that the network with the extra connections performs better 601 than the previous network. Most of the errors made by the nearest neighbor network occur for inputs in which the received subsequences ri and ri+1 or ri+2 contain a total of four or more errors. It appears that the network with the additional connections is, in effect, able to communicate around subsequences containing errors that block communications for the two-nearest neighbor network. 7 Summary and Conclusions We have presented a locally interconnected network which minimizes a function that is analogous to the log likelihood function near the global minimum. The results of simulations demonstrate that the network can successfully decode input sequences containing no noise at least as well as the globally connected Hopfield-Tank [6] decomposition network. Simulations also strongly support the conjecture that in the noiseless case, the network can be guaranteed to converge to the global minimum. In addition, for low error rates, the network can also decode noisy received sequences. We have been able to apply the Cohen-Grossberg proof of the stability of "oncenter off-surround" networks to show that each stage will maximize the desired local "likelihood" for noisy received sequences. We have also shown that, in the large gain limit, the network as a whole is stable and that the equilibrium points correspond to the MLSE decoder output. Simulations have verified this proof of stability even for relatively small gains. Unfortunately, a proof of strict Lyapunov stability is very difficult, and may not be possible, because of the cooperative connections in the network. This network demonstrates that it is possible to perform interesting functions even if only localized connections are allowed, although there may be some loss of performance. If we view the network as an associative memory, a trellis structured network that contains N S neurons can correctly recall 28 memories. Simulations of trellis networks strongly suggest that it is possible to guarantee a non-zero minimum radius of attraction for all memories. We are currently investigating the use of trellis structured layers in multilayer networks to explicitly provide the networks with the ability to tolerate errors and replace faulty neurons. References [1] M. Cohen and S. Grossberg, "Absolute stability of global pattern formation and parallel memory storage by competitive neural networks," IEEE Trans. Sys., Man, and Cyber., vol. 13, pp. 815-826, Sep.-Oct. 1983. [2] S. Grossberg, "How does a brain build a cognitive code," in Studies of Mind and Brain, pp. 1-52, D. Reidel Pub. Co., 1982. [3] J. Hopfield, "Neural networks and physical systems with emergent collective computational abilities," Proceedings of the National Academy of Sciences USA, vol. 79, pp. 2554-2558, 1982. [4] J. Hopfield, "Neurons with graded response have collective computational properties like those of two-state neurons," Proceeedings of the National Academy of Science, USA, vol. 81, pp. 3088-3092, May 1984. [5] J. McClelland and D. Rumelhart, Parallel Distributed Processing, Vol. 1. The MIT Press, 1986. [6] D. Tank and J. Hopfield, "Simple 'neural' optimization networks: an AID converter, signal decision circuit and a linear progra.mming circuit," IEEE Trans. on Circuits and Systems, vol. 33, pp. 533-541, May 1986. [7] A. Viterbi and J. Omura, Principles of Digital Communications and Coding. McGra.w-Hill, 1979.
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52 Supervised Learning of Probability Distributions by Neural Networks Eric B. Baum Jet Propulsion Laboratory, Pasadena CA 91109 Frank Wilczek t Department of Physics,Harvard University,Cambridge MA 02138 Abstract: We propose that the back propagation algorithm for supervised learning can be generalized, put on a satisfactory conceptual footing, and very likely made more efficient by defining the values of the output and input neurons as probabilities and varying the synaptic weights in the gradient direction of the log likelihood, rather than the 'error'. In the past thirty years many researchers have studied the question of supervised learning in 'neural'-like networks. Recently a learning algorithm called 'back propagation H - 4 or the 'generalized delta-rule' has been applied to numerous problems including the mapping of text to phonemes5 , the diagnosis of illnesses6 and the classification of sonar targets 7 • In these applications, it would often be natural to consider imperfect, or probabilistic information. We believe that by considering supervised learning from this slightly larger perspective, one can not only place back propagat Permanent address: Institute for Theoretical Physics, University of California, Santa Barbara CA 93106 © American Institute of Physics 1988 53 tion on a more rigorous and general basis, relating it to other well studied pattern recognition algorithms, but very likely improve its performance as well. The problem of supervised learning is to model some mapping between input vectors and output vectors presented to us by some real world phenomena. To be specific, coqsider the question of medical diagnosis. The input vector corresponds to the symptoms of the patient; the i-th component is defined to be 1 if symptom i is present and 0 if symptom i is absent. The output vector corresponds to the illnesses, so that its j-th component is 1 if the j-th illness is present and 0 otherwise. Given a data base consisting of a number of diagnosed cases, the goal is to construct (learn) a mapping which accounts for these examples and can be applied to diagnose new patients in a reliable way. One could hope, for instance, that such a learning algorithm might yield an expert system to simulate the performance of doctors. Little expert advice would be required for its design, which is advantageous both because experts' time is valuable and because experts often have extraodinary difficulty in describing how they make decisions. A feedforward neural network implements such a mapping between input vectors and output vectors. Such a network has a set of input nodes, one or several layers of intermediate nodes, and a layer of output nodes. The nodes are connected in a forward directed manner, so that the output of a node may be connected to the inputs of nodes in subsequent layers, but closed loops do not occur. See figure 1. The output of each node is assumed to be a bounded semilinear function of its inputs. That is, if Vj denotes the output of the j-th node and Wij denotes the weight associated with the connection of the output of the j-th node to the input of 54 the i-th, then the i-th neuron takes value Vi = g(L,i Wi:jV:j), where g is a bounded, differentiable function called the activation function. g(x) = 1/(1 + e- X ), called the logistic function, is frequently used. Given a fixed set of weights {Wi:j}, we set the input node values to equal some input vector, compute the value of the nodes layer by layer until we compute the output nodes, and so generate an output vector. Figure 1: A 5 layer network. Note bottleneck at layer 3. 55 Such networks have been studied because of analogies to neurobiology, because it may be easy to fabricate them in hardware, and because learning algorithms such as the Perceptron learning algorithm8 , Widrow- Hoff9, and backpropagation have been able to choose weights Wi,. that solve interesting problems. Given a set of input vectors sr, together with associated target values tj, back propagation attempts to adjust the weights so as to minimize the error E in achieving these target values, defined as E = E EJL = E(tj - oj)2 (1) JL JL,i where oj is the output of the j-th node when sJL is presented as input. Back propagation starts with randomly chosen Wi,. and then varies in the gradient direction of E until a local minimum is obtained. Although only a locally optimal set of weights is obtained, in a number of experiments the neural net so generated has performed surprisingly well not only on the training set but on subsequent data.4 - 6 This performance is probably the main reason for widespread interest in backpropagation. It seems to us natural, in the context of the medical diagnosis pro blem, the other real world problems to which backpropagation has been applied, and indeed in any mapping problem where one desires to generalize from a limited and noisy set of examples, to interpret the output vector in probabilistic terms. Such an interpretation is standard in the literature on pattern classification.1o Indeed, the examples might even be probabilistic themselves. That is to say it might not be certain whether symptom i was present in case /L or not. Let sr represent the probability symptom i is present in case /L, and let tj represent the probability disease j ocurred in case 56 fL. Consider for the moment the case where the tJ are 1 or 0, A so that the cases are in fact fully diagnosed. Let Ii (s, 0) be our prediction of the probability of disease i given input vector 5, where {; is some set of parameters determined by our learning algorithm. In the neural network case, the {; are the connection weights and Ii ( sl' , { Wi.i }) = oJ. Now lacking a priori knowledge of good 0, the best one can do is to choose the parameters {; to maximize the likelihood that the given set of examples should have occurred. 10 The formula for this likelihood, p, is immediate: or The extension of equation (2), and thus equation (3) to the case where the f are probabilities, taking values in [0,1]' is straight57 forward * 1 and yields log(p) = ~ [tjlog(Jj (s", 0)) + (1 - tj)log(1 - Ij (W, 0))] (4) p. ,3 Expressions of this sort often arise in physics and information theory and are generally interpreted as an entropy. 11 We may now vary the {O} in the gradient direction of the entropy. The back propagation algorithm generalizes immediately from minimizing 'Error' or 'Energy' to maximizing entropy or log likelihood, or indeed any other function of the outputs and the inputs 12 . Of course it remains true that the gradient can be computed by back propagation with essentially the same number of computations as are required to compute the output of the network. A backpropagation algorithm based on log-likelihood is not only more intuitively appealing than one based on an ad-hoc definition of error, but will make quite different and more accurate predictions as well. Consider e.g. training the net on an example which it already understands fairly well. Say tj = 0, and /j(80) = L Now, from eqn(l) BE/B/j = 2£, so using 'Error' as a * 1 We may see this by constructing an equivalent larger set of examples with the f taking only values 0 or 1 with the appropriate frequency. Thus assume the tj are rational numbers with denominator dj and numerator nj and let p = IIp.,j dj. What we mean by the set of examples {tp. : J-t = 1, ... , M} can be represented by considering a set of N = Mp examples {ij} where for each J-t, ij = 0 for p(J-t- 1) < v < pJ-t and 1 < vmod(dj) < (dj - nj), and ij = 1 otherwise. N ow applying equation (3) gives equation (4), up to an overall normalization. 58 criterion the net learns very little from this example, whereas, using eqn(3), Blog(p)/B!;j = 1/(1 - f), so the net continues to learn and can in fact converge to predict probabilities near 1. Indeed because back propagation using the standard 'Error' measure can not converge to generate outputs of 1 or 0, it has been customary in the literature4 to round the target values so that a target of 1 would be presented in the learning algorithm as some ad hoc number such as .8, whereas a target of 0 would be presented as .2. In the context of our general discussion it is natural to ask whether using a feedforward network and varying the weights is in fact the most effective alternative. Anderson and Abrahams 13 have discussed this issue from a Bayesian viewpoint. From this point of view, fitting output to input using normal distributions and varying the means and covariance matrix may seem to be more logical. Feedforward networks do however have several advantages for complex problems. Experience with neural networks has shown the importance of including hidden units wherein the network can form an internal representation of the world. If one simply uses normal distributions, any hidden variables included will simply integrate out in calculating an output. It will thus be necessary to include at least third order correlations to implement useful hidden variables. Unfortunately, the number of possible third order correlations is very large, so that there may be practical obstacles to such an approach. Indeed it is well known folklore in curve fitting and pattern classification that the number of parameters must be small compared to the size of the data set if any generalization to future cases is expected. 10 In feedforward nets the question takes a different form. There can be bottlenecks to information flow. Specifically, if the net is 59 constructed with an intermediate layer which is not bypassed by any connections (i.e. there are no connections from layers preceding to layers subsequent), and if furthermore the activation functions are chosen so that the values of each of the intermediate nodes tend towards either 1 or 0*2, then this layer serves as a bottleneck to information flow. No matter how many input nodes, output nodes, or free parameters there are in the net, the output will be constrained to take on no more than 21 different patterns, where I is the number of nodes in the bottleneck layer. Thus if I is small, some sort of 'generalization' must occur even if the number of weights is large. One plausible reason for the success of back propagation in adequately solving tasks, in spite of the fact that it finds only local minima, is its ability to vary a large number of parameters. This freedom may allow back propagation to escape from many putative traps and to find an acceptable solution. A good expert system, say for medical diagnosis, should not only give a diagnosis based on the available information, but should be able to suggest, in questionable cases, which lab tests might be performed to clarify matters. Actually back propagation inherently has such a capability. Back propagation involves calculation of 81og(p)/8wij. This information allows one to compute immediately 81og(p)/8sj . Those input nodes for which this partial derivative is large correspond to important experiments. In conclusion, we propose that back propagation can be generalized, put on a satisfactory conceptual footing, and very likely made more efficient, by defining the values of the output and in*2 Alternatively when necessary this can be enforced by adding an energy term to the log-likelihood to constrain the parameter variation so that the neuronal values are near either 1 or O. 60 put neurons as probabilities, and replacing the 'Error' by the loglikelihood. Acknowledgement: E. B. Baum was supported in part by DARPA through arrangement with NASA and by NSF grant DMB-840649, 802. F. Wilczek was supported in part by NSF grant PHY82-17853 References (1)Werbos,P, "Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences" , Harvard University Dissertation (1974) (2)Parker D. B., "Learning Logic" ,MIT Tech Report TR-47, Center for Computationl Research in Economics and Management Science, MIT, 1985 (3)Le Cun, Y., Proceedings of Cognitiva 85,p599-604, Paris (1985) (4)Rumelhart, D. E., Hinton, G. E., Williams, G. E., "Learning Internal Representations by Error Propagation", in "Parallel Distributed Processing" , vol 1, eds. Rumelhart, D. E., McClelland, J. L., MIT Press, Cambridge MA,( 1986) (5)Sejnowski, T. J., Rosenberg, C. R., Complex Systems, v 1, pp 145-168 (1987) (6)LeCun, Y., Address at 1987 Snowbird Conference on Neural Networks (7)Gorman, P., Sejnowski, T. J., "Learned Classification of Sonar Targets Using a Massively Parallel Network", in "Workshop on Neural Network Devices and Applications", JPLD-4406, (1987) pp224-237 (8)Rosenblatt, F., "Principles of Neurodynamics: Perceptrons and 61 the theory of brain mechanisms", Spartan Books, Washington DC (1962) (9)Widrow, B., Hoff, M. E., 1960 IRE WESCON Cony. Record, Part 4, 96-104 (1960) (10)Duda, R. 0., Hart, P. E., "Pattern Classification and Scene Analysis", John Wiley and Sons, N.Y., (1973) (11)Guiasu, S., "Information Theory with Applications", McGraw Hill, NY, (1977) (12)Baum,E.B., "Generalizing Back Propagation to Computation" , in "Neural Networks for Computing", AlP Conf. Proc. 151, Snowbird UT (1986)pp47-53 (13)Anderson, C.H., Abrahams, E., "The Bayes Connection" , Proceedings of the IEEE International Conference on Neural N etwor ks, San Diego,(1987)
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Stochastic Learning Networks and their Electronic Implementation Joshua Alspector*. Robert B. Allen. Victor Hut. and Srinagesh Satyanarayanat Bell Communications Research. Morristown. NJ 01960 We describe a family of learning algorithms that operate on a recurrent, symmetrically connected. neuromorphic network that. like the Boltzmann machine, settles in the presence of noise. These networks learn by modifying synaptic connection strengths on the basis of correlations seen locally by each synapse. We describe a version of the supervised learning algorithm for a network with analog activation functions. We also demonstrate unsupervised competitive learning with this approach. where weight saturation and decay play an important role. and describe preliminary experiments in reinforcement learning. where noise is used in the search procedure. We identify the above described phenomena as elements that can unify learning techniques at a physical microscopic level. These algorithms were chosen for ease of implementation in vlsi. We have designed a CMOS test chip in 2 micron rules that can speed up the learning about a millionfold over an equivalent simulation on a VAX lln80. The speedup is due to parallel analog computation for snmming and multiplying weights and activations. and the use of physical processes for generating random noise. The components of the test chip are a noise amplifier. a neuron amplifier. and a 300 transistor adaptive synapse. each of which is separately testable. These components are also integrated into a 6 neuron and 15 synapse network. Finally. we point out techniques for reducing the area of the electronic correlational synapse both in technology and design and show how the algorithms we study can be implemented naturally in electronic systems. 1. INTRODUCTION Ibere has been significant progress. in recent years. in modeling brain function as the collective behavior of highly interconnected networks of simple model neurons. This paper focuses on the issue of learning in these networks especially with regard to their implementation in an electronic system. Learning phenomena that have been studied include associative memoryllJ. supervised leaming by error correction(2) and by stochastic search(3). competitive learning(4) lS) reinforcement leamingI6). and other forms of unsupervised leaming(7). From the point of view of neural plausibility as well as electronic implementation. we particularly like learning algorithms that change synaptic connection strengths asynchronously and are based only on information available locally at the synapse. This is illustrated in Fig. 1. where a model synapse uses only the correlations of the neurons it connects and perhaps some weak global evaluation signal not specific to individual neurons to decide how to adjust its conductance. • Address for correspondence: J. Alspector, BeU Communications ReselllCh, 2E-378, 435 South St., Morristown, Nl 07960 / (201) 8294342/ [email protected] t Pennanent address: University of California, Belkeley, EE Department, Cory HaU, Belkeley, CA 94720 * PennllDeDt address: Columbia University, EE Department, S.W. Mudd Bldg., New Yolk, NY 10027 @ American Institute of Physics 1988 9 10 S, I S, J C.= <s 's > '1 i j <r> global scalar evaluation signal Hebb-type learning rule: If C ij Increases, (perhaps in the presence of r ) Increment W ij Fig. 1. A local correlational synapse. We believe that a stochastic search procedure is most compatible with this viewpoint. Statistical procedures based on noise form the communication pathways by which global optimization can take place based only on the interaction of neurons. Search is a necessary part of any learning procedure as the network attempts to find a connection strength matrix that solves a particular problem. Some learning procedures attack the search directly by gradient following through error (orrection[8J (9J but electronic implementation requires specifying which neurons are input, tudden and output in advanC'e and nece!;sitates global control of the error correction[2J procedure m a way that requires specific connectivity and ~ynch!'Ony at the neural Jevel. There is also the question of how such procedures would work with unsupervised methods and whether they might get stuck in local minima. Stochastic processes can also do gradient foUowing but they are better at avoiding minima, are compatible with asynchronous updates and local weight adjustments, and, as we show in this paper, can generalize well to less supervifM!d learning. The phenomena we studied are 1) analog activation, 2) noise, 3) semi-local Hebbian synaptic modification, and 4) weight decay and saturation. These techniques were applied to problems in supervised, unsupervised, and reinforcement learning. The goal of the study was to see if these diverse learning styles can be unified at the microscopic level with a small set of physically plausible and electronically implementable phenomena. The hope is to point the way for powerful electronic learning systems in the future by elucidating the conditions and the types of circuits that may be necessary. It may also be true that the conditions for electronic learning may 11 have some bearing on the general principles of biologicalleaming. 2. WCAL LEAltNlNG AND STOCHASl'IC SEARCH 2.1 Supervised Learning in Recurrent Networks with Analog Activations We have previously shown! 10] how the supervised learning procedure of the Boltzmann machine(3) can be implemented in an electronic system. This system works on a recurrent, symmetrically connected network which can be characterized as settling to a minimum in its Liapunov function(l]!II). While this architecture may stretch our criterion of neural plausibility, it does provide for stability and analyzability. The feedback connectivity provides a way for a supervised learning procedure to propagate information back through the network as the stochastic search proceeds. More plausible would be a randomly connected network where symmetry is a statistical approximation and inhibition damps oscillations, but symmetry is more efficient and weD matched to our choice of learning rule and search procedure. We have extended our electronic model of the Boltzmann machine to include analog activations. Fig. 2 shows the model of the neuron we used and its tanh or sigmoid transfer function. The net input consists of the usual weighted sum of activations from other neurons but, in the case of Boltzmann machine learning, these are added to a noise signal chosen from a variety of distributions so that the neuron performs the physical computation: activation =1 (neti FI (EwijSj+noise ):::tanh(gain*neti) Instead of counting the number of on-on and off-off cooccurrences of neurons which a synapse connects, the correlation rule now defines the value of a cooccurrence as: Cij=/i*/i where Ii is the activation of neuron i which is a real value from -1 to 1. Note that this rule effectively counts both on-on and off-off cooccurrences in the high gain limit. In this limit, for Gaussian noise, the cumulative probability distribution for the neuron to have activation + 1 (on) is close to sigmoidal. The effect of noise "jitter" is illustrated at the bottom of the figure. The weight change rule is still: if Cij+ > Cij- then increment Wij .... else decrement where the plus phase clamps the output neurons in their desired states while the minus phase allows them to run free. As· mentioned, we have studied a variety of noise distributions other than those based on the Boltzmann distribution. The 2-2-1 XOR problem was selected as a test case since it has been shown! 10] to be easily caught in local minima. The gain was manipulated in conditions with no noise or with noise sampled from one of three distributions. The Gaussian distribution is closest to true electronic thermal noise such as used in our implementation, but we also considered a cut-off uniform distribution and a Cauchy distribution with long noise tails for comparison. The inset to Fig. 3 shows a histogram of samples from the noise distributions used. The noise was multiplied by the temperature to 'jitter' the transfer function. Hence. the jitter decreased as the annealing schedule proceeded. 12 1;. Vnolse 1;. vout or f. (r. W II II + noise) I J high IIIln tr8nl'.. function wUh noll. 'line" 1;. Vln+ 1;. Vnolsl or r. WIJIJ + noise = ne~ Fig. 2. Electronic analog neuron. Fig. 3 shows average performance across 100 runs for the last 100 patterns of 2000 training pattern presentations. It can be seen that reducing the gain from a sharp step can improve learning in a small region of gain, even without noise. There seems to be an optimal gain level. However, the addition of noise for any distribution can substantially improve learning at all levels of gain. tl CLI ~ ~ u c 0 .,.j ~ ~ 8. 0 &: 1 0 . 9 0.8 0.7 0.60.5 -3 10 ~ ----~ ... Gaussian Unifona Cauchy HO Hoise ......-, .'.' .... u __ ... , .. ., -2 10 -1 10 Inverse Gain Fig. 3. Proportion correct vs. inverse gain. 1 1 10 13 2.2 Stochastic Competitive Learning We have studied how competitive leaming(4J[~) can be accomplished with stochastic local units. Mter the presentation of the input pattern. the network is annealed and the weight is increased between the winning cluster unit and the input units which are on. As shown in Fig. 4 this approach was applied to the dipole problem of Rumelhart and Zipser. A 4x4 pixel array input layer connects to a 2 unit competitive layer with recurrent inhibitory connections that are not adjusted. The inhibitory connections provide the competition by means of a winner-lake-all process as the network settles. The input patterns are dipoles only two input units are turned OIl at each pattern presentatiOll and they must be physically adjacent. either vertically or horizontally. In this way, the network learns about the connectedness of the space and eventually divides it into two equal spatial regions with each of the cluster units responding only to dipoles from one of the halves. Rumelhart and Zipser renormalized the weights after each pattern and picked the winning unit as the one with the highest activation. Instead of explicit nonnalization of the weights. we include a decay term proportional to the weight. The weights between the input layer and cluster layer are incremented for on-on correlations, but here there are no alternating phases so that even this gross synchrony is not necessary. Indeed. if small time constants are introduced to the weight updates. no external timing should be needed. winner-lake-all cluster layer input/ayer Pig. 4. Competitive learning network for the dipole problem. Fig. S shows the results of several runs. A 1 at the po~ition of an input unit means that unit 1 of the cluster layer has the larger weight leading to it from that position. A + between two units means the dipole from these two units excites unit 1. A 0 and - means that unit 0 is the winner in the complementary case. Note that adjacent l's should always have a + between them since both weights to unit 1 are stronger. H, however, there is a 1 next to a 0, then there is a tension in the dipole and a competition for dominance in the cluster layer. We define a figure of merit called "surface tension" which is the number of such dipoles in dispute. The smaller the number, the 14 better. Note in Runs A and B, the number is reduced to 4, the minimum possible value, after 2000 pattern presentations. The space is divided vertically and horizontally, respectively. Run C bas adopted a less favorable diagonal division with a surface tension of 6. Number of dipole pattern presentations 0 200 800 1400 2000 0-0-0-0 1+0-0+1 1+1+1+1 1+1+1+1 1+1+1+1 + + + + + + + + + + + + + + + 0-0-0-0 1+1+1+1 1+1+1-0 1+1+1+1 1+1+1+1 RUn A + + - + + - + - + - - - + 0-0-0-0 1+1-0-0 1-0-0-0 0-0-0-0 0-0-0-0 + - - 0-0-0-0 0-0-0-0 0-0-0-0 0-0-0-0 0-0-0-0 0-0-0-0 0-0-0-0 0-0-0+1 0-0-0-1 0-0-1+1 - - - + - - + + - - - + - - + + 0-0-0-0 0-0-0+1 0-0-1+1 0-0-1+1 0-0-1+1 Run B - - - + - + + - - + + - + + 0-0-0-0 1-0-1+1 0-0-1+1 0-0-1+1 0-0-1+1 + + + - + + - - + + - - + + 0-0-0-0 1+0+1+1 0-0+1+1 0-0+1+1 0-0+1+1 0-0-0-0 0+1+1+1 0+1+1+1 1+1+1+1 1+1+1+1 - + + + - + + + + + + + - + + + 0-0-0-0 0-1+1+1 0+1+1+1 0+1+1+1 0-0+1+1 Run C - + + + - + + + - - + + - - + + 0-0-0-0 0-1+1+1 0-0-0-0 0-0-0-0 0-0-0-1 - - + 0- 0-0-0 0-0-0-0 0-0-0-0 0-0-0-0 0-0-0-1 Fig. 5. Results of competitive learning runs on the dipole problem. Table 1 sbows the result of several competitive algorithms compared when averaged over 100 such runs. The deterministic algorithm of Rumelhart and Zipser gives an average surface tension of 4.6 while the stochastic procedure is almost as good. Note that noise is essential in belping the competitive layer settle. Without noise the surface tension is 9.8, sbowing that the winner-takeall procedure is not working properly. Competitive learning algorithm Stochastic net with decay - anneal: T=3H T=1.0 - no anneal: 70 @ T =1.0 Stochastic net with renonnallzation Deterministic, winner-take-all (Rumelhart & Zipser) "surface tension" 4.8 9.8 5.6 4.6 Table 1. Performance of competitive learning algorithms across 1 ()() runs. We also tried a procedure where, instead of decay, weights were renormalized. The model is that each neuron can support a maximum amount of weight leading into it. Biologically, this might be the area that other neurons can form synapses on, so that one synapse cannot increase its strength except at the expense of some of the others. Electronically, this can be implemented as 15 current emanating from a fixed clUTent source per neuron. As shown in Table 1, this works nearly as well as decay. Moreover, preliminary results show that renormalization is especiaUy effective when more then two cluster units are employed. Both of the stochastic algorithms, which can be implemented in an electronic synapse in nearly the same way as the supervised learning algorithm, divide the space just as the deterministic normalization procedure14J does. This suggests that our chip can do both styles of learning, supervised if one includes both phases and unsupervised if only the procedure of the minus phase is used. 1.3 Reiolorcelfteot Learning We have tried several approaches to reinforcement learning using the synaptic model of Fig. 1 where the evaluation signal is a scalar value available globally that represents how well the system performed on each trial. We applied this model to an xor problem with only one output unit. The reinforcement was r = 1 for the correct output and r = -1 otherwise. To the network, this was similar to supervised learning since for a single unit, the output state is fully specified by a scalar value. A major difference, however, is that we do not clamp the output unit in the desired state in order to compare plus and minus phases. This feature of supervised learning has the effect of adjusting weights to follow a gradient to the desired state. In the reinforcement learning described here, there is no plus phase. This has a satisfying aspect in that no overall synchrony is necessary to compare phases, but is also much slower at converging to a solution because the network has to search the solution space without the guidance of a teacher clamping the output units. This situation becomes much worse when there is more than one output unit. In that case, the probability of reinforcement goes down exponentially with the number of outputs. To test multiple outputs, we chose the simple replication problem whereby the output simply has to replicate the input. We chose the number of bidden units equal to the input (or output). 10 the absence of a teacher to clamp the outputs, the network has to find the answer by chance, guided only by a "critic" which rates its effort as "better" or "worse". This means the units must somehow search the space. We use the same stochastic units as in the supervised or unsupervised techniques, but now it is important to have the noise or the annealing temperature set to a proper level. If it is too high, the reinforcement received is random rather than directed by the weights in the network. If it is too low, the available states searched become too smaU and the probability of finding the right solution decreases. We tuned our annealing schedule by looking at a volatility measure defined at each neuron which is simply the fraction of the time the neuron activation is above zero. We then adjust the final anneal temperature so that this number is neither 0 or 1 (noise too low) nor 0.5 (noise too high). We used both a fixed annealing schedule for all neurons and a unit-specific schedule where the noise was proportional to the sum of weight magnitudes into the unit. A characteristic of reinforcement learning is that the percent correct initially increases but then decreases and often oscillates widely. To avoid this, we added a factor of (I - <r » multiplying the final temperature. This helped to stabilize the learning. In keeping with our simple model of the synapse, we chose a weight adjustment technique that consisted of correlating the states of the connected neurons with the global reinforcement signal. Each synapse measured the quantity R = rs;sj for each pattern presented. If R >0, then ~';j is incremented and it is decremented if R <0. We later refined this procedure by insisting that the reinforcement be greater than a recent average so that R = (r-<,. > hi Sj. This type of procedure 16 appears in previous work in a number of fonns.(12] (13) For r =±l only, this "excess reinforcement" is the same as our previous algorithm but differs if we make a comparison between short term and long tenn averages or use a graded reinforcement such as the negative of the sum squared error. Following a suggestion by G. Hinton, we also investigated a more complex technique whereby each synapse must store a time average of three quantities: <r>, <SiSj>, and <rsiSj>. The definition now is R = <rsiSj>-<r><SjSj> and the rule is the same as before. Statistically, this is the same as "excess reinforcement" if the latter is averaged over trials. For the results reported below the values were collected across 10 pattern presentations. A variation. which employed a continuous moving average, gave similar results. Table 2 summarizes the perfonnance on the xor and the replication task of these reinforcement learning techniques. As the table shows a variety of increasingly sophisticated weight adjustment rules were explored; nevertheless we were unable to obtain good results with the techniques described for more than S output units. In the third column, a small threshold had to be exceeded prior to weight adjustment. In the fourth column, unit-specific temperatures dependent on the sum of weights, were employed. The last column in the table refers to frequency dependent learning where we trained on a single pattern until the network produced a correct answer and then moved on to another pattern. This final procedure is one of several possible techniques related to 'shaping' in operant learning theory in which difficult patterns are presented more often to the network. network t=1 time-averaged +£=0.1 +T-I:W +freq xor 24-1 (0.60) 0.64 (0.70) 0.88 (0.76) 0.88 (0.92)0.99 (0.98) 1.00 2-2-1 (0.58) 0.57 (0.69) 0.74 (0.96) 1.00 (0.85) 1.00 (0.78) 0.88 eplication 2-2-2 (0.94)0.94 (0.46) 0.46 (0.91) 0.97 (0.87) 0.99 (0.97) 1.00 3-3-3 (0.15) 0.21 (0.31) 0.33 (0.31) 0.62 (0.37)0.37 (0.97) 1.00 444 (0.75) 1.00 S-S-S (0.13) 0.87 6-6-6 (0.02) 0.03 Table 2. Proportion correct performance of reinforcement learning after (2K) and 10K patterns. Our experiments. while incomplete, hint that reinforcement learning can also be implemented by the same type of local-global synapse that characterize the other learning paradigms. Noise is also necessary here for the random search procedure. 2. .. Sanunary of Study of hDdameatai Learning Par ... eters In summary, we see that the use of noise and our model of a local correlational synapse with a DOn-specific global evaluation signal are two important features in all the learning paradigms. Graded activation is somewhat less important. Weight decay seems to be quite important although saturation can substitute for it in unsupervised learning. Most interesting from our point of view is that all these phenomena are electronically implementable and therefore physically 17 plausible. Hopefully this means they are also related to true neural phenomena and therefore provide a basis for unifying the various approaches of learning at a microscopic level. 3. ELECTRONIC IMPLEMENTATION 3.1 The Supervised LearDiog Chip We have completed the design of the chip previously proposed.(IO] Its physical style of computation speeds up learning a millionfold over a computer simulation. Fig. 6 shows a block diagram of the neuron. It is a double differential amplifier. One branch forms a sum of the inputs from the differential outputs of aU other neurons with connections to it. The other adds noise from the noise amplifier. This first stage has low gain to preserve dynamic range at the summing nodes. The second stage has high gain and converts to a single ended output. This is fed to a switching arrangement whereby either this output state or some externally applied desired state is fed into the final set of inverter stages which provide for more gain and guaranteed digital complementarity . Sdlslrld Fig. 6. Block diagram of neuron. The noise amplifier is shown schematically in Fig. 7. Thermal noise, with an nns level of tens of microvolts, from the channel of an FET is fed into a 3 stage amplifier. Each stage provides a potential gain of 100 over the noise bandwidth. Low pass feedback in each stage stabilizes the DC output as well as controls gain and bandwidth by means of an externally controlled variable resistance for tuning the annealing cycle. Fig. 8 shows a block diagram of the synapse. The weight is stored in 5 flip-flops as a sign and magnitude binary number. These flip-flops control the conductance from the outputs of neuron i to the inputs of neuron j and vice-versa as shown in the figure. The conductance of the FETs are in the ratio 1 :2:4:8 to correspond to the value of the binary number while the sign bit determines whether the true or complementary lines connect. The flip-flops are arranged in a counter which is controUed by the correlation logic. If the plus phase correlations are greater than the minus phase, then the counter is incremented by a single unit If less, it is decremented. 18 Vcontrol I I l I >--.._V._.nOISI Fig. 7. Block diagram of noise amplifier. .r----... "ncrement correlation logic phase up. down. & set logic sgn o i------lhnl 2 3 Sj or I Sj or I nior~ " T---~'-~ __ ~--~ W I) or JI Fig. 8. Block diagram of synapse. Fig. 9 sbows the layout of a test chip. A 6 neuron, 15 synapse network may be seen in the lower left comer. Eacb neuron bas attacbed to it a noise amplifier to assure that the noise is uncorrelated. The network occupies an area about 2.5 mm on a side in 2 micron design rules. Eacb 300 transistor synapse occupies 400 by 600 microns. In contrast, a biological synapse occupies only about one square micron. The real miracle of biological learning is in the synapse wbere plasticity operates on a molecular level, not in the neuron. We can't bope to compete using transistors, bowevc:r small, especially in the digital domain. Aside from this small network, the rest of the chip is occupied with test structures of the various components. 3.1 Analog Synapse Analog circuit tecbni~ues can reduce the size of the synapse and increase its functionality. Several recent papers( 4] II~I have shown how to make a voltage controlled resistor in MOS technology. The voltage controlling the conductance representing the synaptic weight can be obtained by an analog charge integrator from the correlated activation of the neurons which the synapse in question connects. A charge integrator with a "leaky capacitor" bas a time constant 19 which can be used to make comparisons as a continuous time average over the last several trials. thereby' adding temporal information. One can envision this time constant as being adaptive as well. The charge integrator directly implements the analog Hebb-typel 16] correlation rules of section 2. ~.~ ~,~~~' ~ .. ~i~ 'i~ ~ ~~ilf'~~ .' ~., •• ' /., ~ "' ) '<" "~:~";" .. I I . · ~ii.:' .. . • • ./ . ' " A , ..• . \ ':": :" . _. ' . _ • . .. . . . •••••••••• * •• ~.* ••.•••• ~ ••••••••••• ~.~ •• :i. c·.. ..· If. ., • iii • ., •••• It ill ••••••••••••••••• ~.I ':;;::dU:;.;;;.UEEi Fig. 9. Chip layout. ......... ~ ~" , . • 1!~'.' i nr-':,~"·"";··;. -Jot-: : i ii r .. ·· .. ' .. ,: ~ II. :-"fO.,a'l.~"~;" •. ' ;~-. ~ .... ...... , •• ~I ' ... :t~"1I • ! : '. JjI! ••• i .. , .... " . II~.~ ... : .................. ~ .. ,;.:" ..... ' ..•.•...... ...• ',...... .. 1IIi1. ' :" :::'.l.. . .. ... . ,.;,:,... . s· ,· '.. • Ii iI • .. ....... • • . ". :::,':. •• 3.3 Tecbnologicalbnprovemeots for Flectronic Neural Networks It is still necessary to store the voltage which controls the analog conductance and we propose the EPROMll7] or EEPROM device for this. Such a device can hold the value of the weight in the same way that flip-flops do in the digital implementation of the synapse(lOJ. The process which creates this device has two polysilicon layers which are useful for making high valued capacitances in analog circuitry. In addition. the second polysilicon layer could be used to make CCD devices for charge storage and transport. Coupled with the charge storage on a floating gate(l8], this forms a compact. low power representation for weight values that apyroach biological values. Another useful addition would be a high valued stable resistive layerll9 . One 20 could thereby avoid space-wasting long-channel MOSFETs which are currently the only rea~ble way to achieve high resistance in MOS technology. Lastly, the addition of a diffusion step or two creates a Bi-CMOS process which adds high quality bipolar transistors useful in analog design. Furthermore, one gets the logarithmic dependence of voltage on current in bipolar technology in a natural, robust way, that is not subject to the variations inherent in using MOSFETs in the subthreshold region. This is especially useful in compressing the dynamic range in sensory processing[20J• 4. CONCLUSION We have shown how a simple adaptive synapse which measures correlations can account for a variety of learning styles in stochastic networks. By embellishing the standard CMOS process and using analog design techniques. a technology suitable for implementing such a synapse electronically can be developed. Noise is an important element in our formulation of learning. It can help a network settle, interpolate between discrete values of conductance during learning. and search a large solution space. Weight decay ("forgetting") and saturation are also important for stability. These phenomena not only unify diverse learning styles but are electronically implementabfe. ACKNOWLEDGMENT: This work has been influenced by many researchers. We would especially like to thank Andy Barto and Geoffrey Hinton for valuable discussions on reinforcement learning, Yannis Tsividis for contributing many ideas in analog circuit design, and Joel Gannett for timely releases of his vlsi verification software. 21 References 1. JJ. Hopfield, "Neural netwolks and physical systems with emergent coUective computational abilities", Proc. Natl. Acad. Sci. USA 79,2554-2558 (1982). 2. D.E. Rumelhart, G.E. Hinton, and RJ. Williams, "Learning internal representations by error propagation", in Paralld Distribuled Processing: Explorations in th~ Microstructur~ of Cognition. Vol. 1: Foundations. edited by D.E. Rumelhart and J.L. McClelland, (MrT Press, Cambridge, MA, 1986), p. 318. 3. D.H. Ackley, G.E. Hinton, and T J. Sejnowski, "A learning algorithm for Boltzmann machines", Cognitive Science 9, 147-169 (1985). 4. D.E. Rumelhart and D. apser, ''Feature dillCovery by competitive learning", Cognitive Science 9, 75-112 (1985). 5. s. Grossberg, "Adaptive pattern classification and universal recoding: Part L Parallel development and coding of neural feature detectors.", Biological Cybernetics 23, 121-134 (1976). 6. A.G. Barto, R.S. Sutton, and C.W. Anderson, "Neuronlike adaptive elements that can solve difficult learning control problems",1EEE Trans. Sys. Man Cyber. 13,835 (1983). 7. B.A. Pearlmutter and G.E. Hinton, "G-Maximization: An unsupervised learning procedure for discovering regularities", in N~ural Networks for Computing. edited by J.S. Denker, AIP Conference Proceedings 151, American Inst. of Physics, New Yolk (1986), p.333. 8. F. Rosenblatt, Principirs of Neurodyrramics: Perc~ptrons and the Th~ory of Brain Mechanisms (Spartan Books, Washington, D.C., 1961). 9. G. Widrowand M.E. Hoff, "Adaptive switching cirt:uits", Inst. of Radio Engineers, Western Electric Show and Convention. COftycntion Record, Part 4, ~104 (1960). 10. J. Alspeaor and R.B. Allen, "A neuromorphic vlsi learning system". in M~'aN:rd Rrs~arch in VLSl: Procudings ofth~ 1987 StQ1lfordConf~rtnu. edited by P. Losleben (MIT Press, Cambridge, MA.1987), pp. 313-349. 11. M.A. Cohen and S. Grossberg, "Absolute stability of global pattern formation and parallel memory storage by competitive neural networks", Trans. IEEE 13,815, (1983). 12. B. Widrow. N.K. Gupta, and S. Maitra, "Punish,IReward: Learning with a critic in adaptive threshold systems", IEEE Trans. on Sys. Man & Cyber., SMC-3, 455 (1973). 13. R.S. Sutton, "Temporal credit assignment in reinforcement learning", unpublished doctoral dissertation, U. Mass. Amherst, technical report COINS 84-02 (1984). ]4. Z. Czamul, "Design of voltage-controlled linear ttansconductance elements with a muched pair of FET transistors", IEEE Trans. Cire. Sys. 33, 1012, (1986). 15. M. Banu and Y. Tsividis, "Flouing voltage-controUed resistors in CMOS technology", Electron. Lett. 18,678-679 (1982). 16. D.O. Hebb, Th~ OrganizotiOlf ofBtMV;oT (Wiley, NY, 19(9). 17. D. Frohman-Bentchkowsky. HFAMOS - • new semiconductor charge storage device", Solid-State Electronics 17, 517 (1974). 18. J.P. Sage, K.. Thompson, and R.S. Withers, "An artificial neural network integrued circuit based on MNOS/CCD principles", in Nrural Networks for Computing. edited by J.S. Denker. AIP Conference Proceedings 151, American lost. of Physics, New York (1986), p.38 1. 19. A.P. ThaJcoor, J.L. Lamb. A. Moopenn, and J. Lambe, "Binary synaptic connections ba!ICd on memory switching in a-Si:H". in Neural N~"""orks for Computing. edited by J.S. Denker, AIP Conference Proceedings 151. American Inst. of Physics, New York (1986), p.426. 20. M.A. Sivilotti, M.A. Mahowald, and C.A. Mead, ~ReaJ-Time visual computations using analog CMOS processing arrays", in Advanud R~S('arch in VLSl: Prou~dings of thr 1987 Stanford Corrf~r~nu. edited by P. Losleben (MIT Press, Cambridge, MA, 1987), pp. 295-312.
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CONNECTING TO THE PAST Bruce A. MacDonald, Assistant Professor Knowledge Sciences Laboratory, Computer Science Department The University of Calgary, 2500 University Drive NW Calgary, Alberta T2N IN4 ABSTRACT 505 Recently there has been renewed interest in neural-like processing systems, evidenced for example in the two volumes Parallel Distributed Processing edited by Rumelhart and McClelland, and discussed as parallel distributed systems, connectionist models, neural nets, value passing systems and multiple context systems. Dissatisfaction with symbolic manipulation paradigms for artificial intelligence seems partly responsible for this attention, encouraged by the promise of massively parallel systems implemented in hardware. This paper relates simple neural-like systems based on multiple context to some other well-known formalisms-namely production systems, k-Iength sequence prediction, finite-state machines and Turing machines-and presents earlier sequence prediction results in a new light. 1 INTRODUCTION The revival of neural net research has been very strong, exemplified recently by Rumelhart and McClelland!, new journals and a number of meetingsG • The nets are also described as parallel distributed systems!, connectionist models2 , value passing systems3 and multiple context learning systems4,5,6,7,8,9. The symbolic manipulation paradigm for artificial intelligence does not seem to have been as successful as some hoped!, and there seems at last to be real promise of massively parallel systems implemented in hardware. However, in the flurry of new work it is important to consolidate new ideas and place them solidly alongside established ones. This paper relates simple neural-like systems to some other well-known notions-namely production systems, k-Iength sequence prediction, finite-state machines and Turing machines-and presents earlier results on the abilities of such networks in a new light. The general form of a connectionist systemlO is simplified to a three layer net with binary fixed weights in the hidden layer, thereby avoiding many of the difficulties-and challengesof the recent work on neural nets, The hidden unit weights are regularly patterned using a template. Sophisticated, expensive learning algorithms are avoided, and a simple method is used for determining output unit weights. In this way we gain some of the advantages of multilayered nets, while retaining some of the simplicity of two layer net training methods. Certainly nothing is lost in computational power-as I will explain-and the limitations of two layer nets are not carried over to the simplified three layer one. Biological systems may similarly avoid the need for learning algorithms such as the "simulated annealing" method commonly used in connectionist modelsll . For one thing, biological systems do not have the same clearly distinguished training phase. Briefly, the simplified netb is a production system implemented as three layers of neuron-like units; an output layer, an input layer, and a hidden layer for the productions themselves. Each hidden production unit potentially connects a predetermined set of inputs to any output. A k-Iength sequence predictor is formed once Ie levels of delay unit are introduced into the input layer. k-Iength predictors are unable to distinguish simple sequences such as ba . .. a and aa ... a since after Ie or more characters the system has forgotten whether an a or b appeared first. If the k-Iength predictor is augmented with "auxiliary" actions, it is able to learn this and other regular languages, since the auxiliary actions can be equivalent to states, and can be inputs to aAmong them the 1st International Conference on Neural Nets, San Diego,CA, June 21-24, 1987, and this con.ference. bRoughly equivalent to a single context system in Andreae's multiple context system4.5,6,7,8,9. See also MacDonald12 . @) American Institute of Physics 1988 506 Figure 1: The general form of a connectionist system 10 . (a) Form of a unit (a) Operations within a unit in~uts ;::; L'" excitation-.I 1:.. aCtiVation--W'" output weIghts ¥--== sum Typical F Typical f the production units enabling predictions to depend on previous states7 . By combining several augmented sequence predictors a Thring machine tape can be simulated along with a finite-state controller9 , giving the net the computational power of a Universal Turing machine. Relatively simple neural-like systems do not lack computational ability. Previous implementations 7,9 of this ability are production system equivalents to the simplified nets. 1.1 Organization of the paper The next section briefly reviews the general form of connectionist systems. Section 2 simplifies this, then section 3 explains that the result is equivalent to a production system dealing only with inputs and outputs of the net. Section 4 extends the simplified version, enabling it to learn to predict sequences. Section 5 explains how the computational power of the sequence predictor can be increased to that of a Thring machine if some input units receive auxiliary actions; in fact the system can learn to be a TUring machine. Section 6 discusses the possibility of a number of nets combining their outputs, forming an overall net with "association areas". 1.2 General form of a connectionist system Figure 1 shows the general form of a connectionist system unit, neuron or ce1l 10 . In the figure unit i has inputs, which are the outputs OJ of possibly all units in the network, and an output of its own, 0i' The net input excitation, net" is the weighted sum of inputs, where !Vij is the weight connecting the output from unit j as an input to unit i. The activation, ai of the unit is some function Fi of the net input excitation. Typically Fi is semilinear, that is non-decreasing and differentiable13 , and is the same function for all, or at least large groups of units. The output is a function fi of the activation; typically some kind of threshold function. I will assume that the quantities vary over discrete time steps, so for example the activation at time t + 1 is ai (t + 1) and is given by Fi((neti(t)). In general there is no restriction on the connections that may be made between units. Units not connected directly to inputs or outputs are hidden units. In more complex nets than those described in this paper, there may be more than one type of connection. Figure 2 shows a common connection topology, where there are three layers of units-input, hidden and output-with no cycles of connection. The net is trained by presenting it with input combinations, each along with the desired output combination. Once trained the system should produce the desired outputs given just Figure 2: The basic structure of a three layer connectionist system. input units hidden output units units 507 inputs. During training the weights are adjusted in some fashion that reduces the discrepancy between desired and actual output. The general method is lO : (1) where t; is the desired, "training" activation. Equation 1 is a general form of Hebb's classic rule for adjusting the weight between two units with high activations lO • The weight adjustment is the product of two functions, one that depends on the desired and actual activations--often just the difference-and another that depends on the input to that weight and the weight itself. As a simple example suppose 9 is the difference and h as just the output OJ. Then the weight change is the product of the output error and the input excitation to that weight: where the constant T} determines the learning rate. This is the Widrow-Hoff or Delta rule which may be used in nets without hidden units. 1o The important contribution of recent work on connectionist systems is how to implement equation 1 in hidden units; for which there are no training signals ti directly available . The Boltzmann learning method iteratively varies both weights and hidden unit training activations using the controlled, gradually decreasing randomizing method "simulated annealing" 14. Backpropagation13 is also iterative, performing gradient descent by propagating training signal errors back through the net to hidden units. I will avoid the need to determine training signals for hidden units, by fixing the weights of hidden units in section 2 below. 2 SIMPLIFIED SYSTEM Assume these simplifications are made to the general connectionist system of section 1.2: 1. The system has three layers, with the topology shown in Figure 2 (ie no cycles) 2. All hidden layer unit weights are fixed, say at unity or zero 3. Each unit is a linear threshold unit lO , which means the activation function for all units is the identity function, giving just net;, a weighted sum of the inputs, and the output function is a simple binary threshold of the form: ! output - I • threshold / activation 508 so that the output is binary; on or oft'. Hidden units will have thresholds requiring all inputs to be active for the output to be active (like an AND gate) while output units will have thresholds requiring only 1 or two active highly weighted inputs for an output to be generated (like an OR gate). This is in keeping with the production system view of the net, explained in section 3. 4. Learning-which now occurs only at the output unit weights-gives weight adjustments according to: Wij 1 if ai = OJ = 1 Wij 0 otherwise so that weights are turned on if their input and the unit output are on, and off otherwise. That is, Wij = ai A OJ. A simple example is given in Figure 3 in section 3 below. This simple form of net can be made probabilistic by replacing 4 with 4' below: 4'. Adjust weights so that Wij estimates the conditional probability of the unit i output being on when output j is on. That is, Wij = estimate of P(odoj). Then, assuming independence of the inputs to a unit, an output unit is turned on when the conditional probability of occurrence of that output exceeds the threshold of the output function. Once these simplifications are made, there is no need for learning in the hidden units. Also no iterative learning is required; weights are either assigned binary values, or estimate conditional probabilities. This paper presents some of the characteristics of the simplified net. Section 6 discusses the motivation for simplifying neural nets in this way. 3 PRODUCTION SYSTEMS The simplified net is a kind of simple production system. A production system comprises a global database, a set of production rules and a control system15 . The database for the net is the system it interacts with, providing inputs as reactions to outputs from t.he net. The hidden units of the network are the production rules, which have the form IF precondition THEN action The precondition is satisfied when the input excitation exceeds the threshold of a hidden unit. The actions are represented by the output units which the hidden production units activate. The control system of a production system chooses the rule whose action to perform, from the set of rules whose preconditions have been met. In a neural net the control system is distributed throughout the net in the output units. For example, the output units might form a winner-takeall net. In production systems more complex control involves forward and backward chaining to choose actions that seek goals. This is discussed elsewhere4.12.16. Figure 3 illust.rates a simple production implemented as a neural net. As the figure shows, the inputs to hidden units are just the elements of the precondition. When the appropriate input combination is present the associated hidden (production) unit is fired. Once weights have been leamed connecting hidden units to output units, firing a production results in output. The simplified neural net is directly equivalent to a production system whose elements are inputs and outputse . Some production systems have symbolic elements, such as variables, which can be given values by production actions. The neural net cannot directly implement this, since it can have outputs only from a predetermined set. However, we will see later that extensions t.o the framework enable this and other abilities. CThis might be referred to as a "sensory-motor" production system, since when implemented ill a l'eal system such as a robot, it deals only with sensed inputs and executable motor actions, which may include the auxiliary actions of section 4.3. Figure 3: A production implemented in a simplified neural net. (a) A production rule rr==~--r=======~~~==~ IF I cloudy I AND I pressure falling I THEN I it will rain I (b) The rule implemented as a hidden unit. The threshold of the hidden unit is 2 so it is. an AND gate. The threshold of the output unit is 1 so it is an OR gate. The learned weight will be 0 or 1 if the net is not probabilistic, otherwise it will be an estimate of P(it will rainlclouds AND pressure falling) weight It will rain 509 Figure 4: A net that predicts the next character in a sequence, based on only the last character. (a) The net. Production units (hidden units) have been combined with input units. For example this net could predict the sequence abcabcabc . . .. Productions have the form: IF last character is . .. THEN next character will be . . .. The learning rule is Wij = 1 if (inputj AND outputi). Output is ai = ~R WijOj (b) Learning procedure. input a b c neural net output 1. Clamp inputs and outputs to desired values 2. System calculates weight values a b c 3. Repeat 4 and 4 for all required input/output combinations 4 SEQUENCE PREDICTION A production system or neural net can predict sequences. Given examples of a repeating sequence, productions are learned which predict future events on the basis of recent ones. Figure 4 shows a trivially simple sequence predictor. It predicts the next character of a sequence based on the previous one. The figure also gives the details of the learning procedure for the simplified net. The net need be trained only once on each input combination, then it will "predict" as an output every character seen after the current one. The probabilistic form of the net would estimate conditional probabilities for the next character, conditional on the current one. Many 510 Figure 5: Using delayed inputs, a neural net can implement a k-length sequence predictor. (a) A net with the last three characters as input. input a':""'" -......:;;;;;;;::::~;:{ a' a b':-;-'----..", {:';;g 'J.,o e" {e'_~ 0 e-___ -" 2nd last hidden output (b) An example production. ~----------------------------------, IF last three characters were ~ THEN 0 a b c z presentations of each possible character pair would be needed to properly estimate the probabilities. The net would be learning the probability distribution of character pairs. A predictor like the one in Figure 4 can be extended to a general k-Iength17 predictor so long as inputs delayed by 1,2, ... , k steps are available. Then, as illustrated in Figure 5 for 3-length prediction, hidden production units represent all possible combinations of k symbols. Again output weights are trained to respond to previously seen input combinations, here of three characters. These delays can be provided by dedicated neural netsd , such as that shown in Figure 6. Note that the net is assumed to be synchronously updated, so that the input from feedback around units is not changed until one step after the output changes. There are various ways of implementing delay in neurons, and Andreae4 investigates some of them for the same purpose-delaying inputs-in a more detailed simulation of a similar net. 4.1 Other work on sequence prediction in neural nets Feldman and Ballard2 find connectionist systems initially not suited to representing changes with time. One form of change is sequence, and they suggest two methods for representing sequence in nets. The first is by units connected to each other in sequence so that sequential tasks are represented by firing these units in succession. The second method is to buffer the inputs in time so that inputs from the recent past are available as well as current inputs; that is, delayed inputs are available as suggested above. An important difference is the necessary length of the buffer; Feldman and Ballard suggest the buffer be long enough to hold a phrase of natural language, but I expect to use buffers no longer than about 7, after Andreae4 . Symbolic inputs can represent more complex information effectively giving the length seven buffers more information than the most recent seven simple inputs, as discussed in section 5. The method of back-propagation13 enables recurrent networks to learn sequential tasks in a dFeldman and Ballard2 give some dedicated neural net connections for a variety of flUlctions 511 Figure 6: Inputs can be delayed by dedicated neural subnets. A two stage delay is shown. (a) Delay network. (b) Timing diagram for (a). A B C D E --.r- 1.0 IL..-_-_-_-_-_-.:-_-.:-_-.:-_-_-_-_ ... _ tml _· _e_ -r-0.5 0.75 ~ 0.375 ....,L-___ _ __________ ~r--------r--------~------~L original signal delay of one step delay of two steps manner similar to the first suggestion in the last paragraph, where sequences of connected units represent sequenced events. In one example a net learns to complete a sequence of characters; when given the first two characters of a six character sequence the next four are output. Errors must be propagated around cycles in a recurrent net a number of times. Seriality may also be achieved by a sequence of states of distributed activation 18. An example is a net playing both sides of a tic-tac-toe game18 . The sequential nature of the net's behavior is derived from the sequential nature of the responses to the net's actions; tic-tac-toe moves. A net can model sequence internally by modeling a sequential part of its environment. For example, a tic-tac-toe playing net can have a model of its opponent. k-Iength sequence predictors are unable to learn sequences which do not repeat more frequently that every k characters. Their k-Iength context includes only information about the last k events. However, there are two ways in which information from before the kth last input can be retained in the net. The first method latches some inputs, while the second involves auxiliary actions. 4.2 Latch units Inputs can be latched and held indefinitely using the combination shown in Figure 7. Not all inputs would normally be latched. Andreae4 discusses this technique of "threading" latched events among non-latched events, giving the net both information arbitrarily far back in its input-output history and information from the immediate past. Briefly, the sequence ba . .. a can be distinguished from aa ... a if the first character is latched. However, this is an ad hoc solution to this probleme . 4-3 Auxiliary actions When an output is fed back into the net as an input signal, this enables the system to choose the next output at least partly based on the previous one, as indicated in Figure 8. If a particular fed back output is also one without external manifestation, or whose external manifestation is independent of the task being performed, then that output is an auxiliary action. It Las "The interested reader should refer to Andreae4 where more extensive analysis is given. 512 Figure 7: Threading. A latch circuit remembers an event until another comes along. This is a two input latch, e.g. for two letters a and b, but any number of units may be similarly connected. It is formed from a mutual inhibition layer, or winner-take-all connection, along with positive feedback to keep the selected output activated when the input disappears. a b---;~..!!J Figure 8: Auxiliary actions-the S outputs-are fed back to the inputs of a net, enabling the net to remember a state. Here both part of a net and an example of a production are shown. There are two types of action, characters and S actions. Sinputs S outputs character inputs character outputs IF S input is [§l] and character input is 0 THEN output character lliJ and S [ill no direct effect on the task the system is performing since it evokes no relevant inputs, and so can be used by the net as a symbolic action. If an auxiliary action is latched at the input then the symbolic information can be remembered indefinitely, being lost only when another auxiliary action of that kind is input and takes over the latch. Thus auxiliary actions can act like remembered states; the system performs an action to "remind" itself to be in a particular state. The figure illustrates this for a system that predicts characters and state changes given the previous character and state. An obvious candidate for auxiliary actions is speech. So the blank oval in the figure would represent the net's environment, through which its own speech actions are heard. Although it is externally manifested, speech has no direct effect on our physical interactions with the world. Its symbolic ability not only provides the power of auxiliary actions, but also includes other speakers in the interaction. 5 SIMULATING ABSTRACT AUTOMATA The example in Figure 8 gives the essence of simulating a finite state automaton with a production system or its neural net equivalent. It illustrates the transition function of an automaton; the new state and output are a function of the previous state and input. Thus a neural net can simulate a finite state automaton, so long as it has additional, auxiliary actions. A Thring machine is a finite state automaton controller plus an unbounded memory. A neural net could simulate a 'lUring machine in two ways, and both ways have been demonstrated with production system implementations-equivalent to neural nets----(;alled "multiple context learning systems"', briefly explained in section 6. The first Thring machine simulation 7 has the system simulate only the finite state controller, but is able to use an unbounded external memory fSee John Andreae's and his colleagues' work4,5,6,7,8,9,12,16 513 Figure 9: Multiple context learning system implementation as multiple neural nets. Each:3 layer net has the simplified form presented above, with a number of elaborations such as extra connections for goal-seeking by forward and backward chaining. Output channels from the real world, much like the paper of Turing's original work 19 . The second simnlat.ion[" 1 '2 embeds the memory in the multiple context learning system, along with a counter for accessing this simulated memory. Both learn all the productions-equivalent to learning output unit weights-required for the simulations. The second is able to add internal memory as required, up to a limit dependent on the size of the network (which can easily be large enough to allow 70 years of computation!). The second could also employ external memory as the first did. Briefly, the second simulation comprised multiple sequence predictors which predicted auxiliary actions for remembering the state of the controller, and the current memory position. The memory element is updated by relearning the production representing that element; the precondition is the address and the production action the stored item. 6 MULTIPLE SYSTEMS FORM ASSOCIATION AREAS A multiple context learning system is production system version of a multiple neural net, although a simple version has been implemented as a simulated net4 •20 . It effectively comprises several nets--or "association" areas-which may have outputs and inputs in common, as indicated in Figure 9. Hidden unit weights are specified by templates; one for each net. A template gives the inputs to have a zero weight for the hidden units of a net and the inputs to have a weight of unity. Delayed and latched inputs are also available. The actual outputs are selected from the combined predictions of the nets in a winner-take-all fashion. I see the design for real neural nets, say as controllers for real robots, requiring a large degree of predetermined connectivity. A robot controller could not be one three layer net wit.h every input connected to every hidden unit in turn connected to every output. There will need to be some connectivity constraints so the net reflects the functional specialization in the control requirements9 . The multiple context learning system has all the hidden layer connections predetermined, but allows output connections to be learned. This avoids the "credit assignment" problem and therefore also the need for learning algorithms such as Boltzmann learning and back-propagation. However, as the multiple context learning system has auxiliary actions, and delayed and latched inputs, it does not lack computational power. Future work in this area should investigate, for example, the ability of different kinds of nets to learn auxiliary act.ions. This may be difficult as symbolic actions may not be provided in training inputs and output.s. 9 For example a controller for a robot body would have to deal with vision, manipulation, motion, etc. 514 7 CONCLUSION This paper has presented a sImplified three layer connectionist model, with fixed weights for hidden units, delays and latches for inputs, sequence prediction ability, auxiliary "state" actions, and the ability to use internal and external memory. The result is able to learn to simulate a Turing machine. Simple neural-like systems do not lack computational power. ACKNOWLEDGEMENTS This work is supported by the Natural Sciences and Engineering Council of Canada. REFERENCES 1. Rumelhart,D.E. and McClelland,J .L. Parallel distributed processing. Volumes 1 and 2. MIT Press. (1986) 2. Feldman,J .A. and Ballard,D.H. Connectionist models and their properties. Cognitive Science 6, pp.205-254. (1982) 3. Fahlman,S.E. Three Flavors of Parallelism. Proc.4th Nat.Conf. CSCSI/SCSEIO, Saskatoon. (1982) 4. Andreae,J .H. Thinking with the teachable machine. Academic Press. (1977) 5. Andreae,J.H. Man-Machine Studies Progress Reports UC-DSE/1-28. Dept Electrical and Electronic Engineering, Univ. Canterbury, Christchurch, New Zealand. editor. (1972-87) (Also available from NTIS, 5285 Port Royal Rd, Springfield, VA 22161) 6. Andreae,J .H. and Andreae,P.M. Machine learning with a multiple context. Proc.9th Int.Conf.on Cybernetics and Society. Denver. October. pp.734-9. (1979) 7. Andreae,J.H. and Cleary,J.G. A new mechanism for a brain. Int.J.Man-Machine Studies 8(1): pp.89-1l9. (1976) 8. Andreae,P.M. and Andreae,J.H. A teachable machine in the real world. Int.J.Man-Machine Studies 10: pp.301-12. (1978) 9. MacDonald,B.A. and Andreae,J .H. The competence of a multiple context learning system. Int.J.Gen.Systems 7: pp.123-37. (1981) 10. Rumelhart,D.E., Hinton,G.E. and McClelland,J .L. A general framework for parallel distributed processing. chapter 2 in Rumelhart and McClellandl , pp.45-76. (1986) 11. Hinton,G.E. and Sejnowski,T.L. Learning and relearning in Boltzmann machines. chapter 7 in Rumelhart and McClelland l , pp.282-317. (1986) 12. MacDonald,B.A. Designing teachable robots. PhD thesis, University of Canterbury, Christchurch, New Zealand. (1984) 13. Rumelhart,D.E., Hinton,G.E. and Williams,R.J. Learning Internal Representations by Error Propagation. chapter 8 in Rumelhart and McClelland l , pp.318-362. (1986) 14. Ackley,D.H., Hinton,G.E. and Sejnowski,T.J. A Learning Algorithm for Boltzmann Machines. Cognitive Science 9, pp.147-169. (1985) 15. Nilsson,N.J. Principles of Artificial Intelligence. Tioga. (1980) 16. Andreae,J .H. and MacDonald,B~_A. Expert control for a robot body. Research Report 87/286/34 Dept. of Computer Science, University of Calgary, Alberta, Canada, T2N-1N4. (1987) 17. Witten,I.H. Approximate, non-deterministic modelling of behaviour sequences. Int. 1. General Systems, vol. 5 pp.1-12. (1979) 18. Rumelhart,D.E.,Smolensky,P.,McClelland,J.L. and Hinton,G.E. Schemata and Sequential thought Processes in PDP Models. chapter 14, vol 2 in Rumelhart and McClelland 1. pp.757. (1986) 19. Thring,A.M. On computable numbers, with an application to the entscheidungsproblem. Proc. London Math. Soc. vol 42(3). pp. 230-65. (1936) 20. Dowd,R.B. A digital simulation of mew-brain. Report no. UC-DSE/105 . pp.25-46. (1977)
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317 PARTITIONING OF SENSORY DATA BY A CORTICAL NETWORK1 Richard Granger, Jose Ambros-Ingerson, Howard Henry, Gary Lynch Center for the Neurobiology of Learning and Memory University of California Irvine, CA. 91717 SUMMARY To process sensory data, sensory brain areas must preserve information about both the similarities and differences among learned cues: without the latter, acuity would be lost, whereas without the former, degraded versions of a cue would be erroneously thought to be distinct cues, and would not be recognized. We have constructed a model of piriform cortex incorporating a large number of biophysical, anatomical and physiological parameters, such as two-step excitatory firing thresholds, necessary and sufficient conditions for long-term potentiation (LTP) of synapses, three distinct types of inhibitory currents (short IPSPs, long hyperpolarizing currents (LHP) and long cellspecific afterhyperpolarization (AHP)), sparse connectivity between bulb and layer-II cortex, caudally-flowing excitatory collateral fibers, nonlinear dendritic summation, etc. We have tested the model for its ability to learn similarity- and difference-preserving encodings of incoming sensory cueSj the biological characteristics of the model enable it to produce multiple encodings of each input cue in such a way that different readouts of the cell firing activity of the model preserve both similarity and difference'information. In particular, probabilistic quantal transmitter-release properties of piriform synapses give rise to probabilistic postsynaptic voltage levels which, in combination with the activity of local patches of inhibitory interneurons in layer II, differentially select bursting vs. single-pulsing layer-II cells. Time-locked firing to the theta rhythm (Larson and Lynch, 1986) enables distinct spatial patterns to be read out against a relatively quiescent background firing rate. Training trials using the physiological rules for induction of LTP yield stable layer-II-cell spatial firing patterns for learned cues. Multiple simulated olfactory input patterns (Le., those that share many chemical features) will give rise to strongly-overlapping bulb firing patterns, activating many shared lateral olfactory tract (LOT) axons innervating layer Ia of piriform cortex, which in tum yields highly overlapping layer-II-cell excitatory potentials, enabling this spatial layer-II-cell encoding to preserve the overlap (similarity) among similar inputs. At the same time, those synapses that are enhanced by the learning process cause stronger cell firing, yielding strong, cell-specific afterhyperpolarizing (AHP) currents. Local inhibitory intemeurons effectively select alternate cells to fire once strongly-firing cells have undergone AHP. These alternate cells then activate their caudally-flowing recurrent collaterals, activating distinct populations of synapses in caudal layer lb. Potentiation of these synapses in combination with those of still-active LOT axons selectively enhance the response of caudal cells that tend to accentuate the differences among even very-similar cues. Empirical tests of the computer simulation have shown that, after training, the initial spatial layer II cell firing responses to similar cues enhance the similarity of the cues, such that the overlap in response is equal to or greater than the overlap in lThis research was supported in part by the Office of Naval Research under grants NOOOl4-84-K-0391 and NOOOl4-87-K-0838 and by the National Science Foundation under grant IST-8S-12419. © American Institute of Physics 1988 318 input cell firing (in the bulb): e.g., two cues that overlap by 65% give rise to response patterns that overlap by 80% or more. Reciprocally, later cell firing patterns (after AHP), increasingly enhance the differences among even very-similar patterns, so that cues with 90% input overlap give rise to output responses that overlap by less than 10%. This difference-enhancing response can be measured with respect to its acuity; since 90% input overlaps are reduced to near zero response overlaps, it enables the structure to distinguish between even very-similar cues. On the other hand, the similarity-enhancing response is properly viewed as a partitioning mechanism, mapping quite-distinct input cues onto nearly-identical response patterns (or category indicators). We therefore use a statistical metric for the information value of categorizations to measure the value of partitionings produced by the piriform simulation network. INTRODUCTION The three primary dimensions along which network processing models vary are their learning rules, their performance rules and their architectural structures. In practice, performance rules are much the same across different models, usually being some variant of a 'weighted-sum' rule (in which a unit's output is calculated as some function of the sum ofits inputs multiplied by their 'synaptic' weights). Performance rules are usually either 'static' rules (calculating unit outputs and halting) or 'settling' rules (iteratively calculating outputs until a convergent solution is reached). Most learning rules are either variants of a 'correlation' rule, loosely based on Hebb's (1949) postulate; or a 'delta' rule, e.g., the perceptron rule (Rosenblatt, 1962), the adaline rule (Widrow and Hoff, 1960) or the generalized delta or 'backpropagation'rule (Parker, 1985; Rumelhart et al., 1986). Finally, architectures vary by and large with learning rules: e.g., multilayered feedforward nets require a generalized delta rule for convergence; bidirectional connections usually imply a variant of a Hebbian or correlation rule, etc. Architectures and learning and performance rules are typically arrived at for reasons of their convenient computational properties and analytical tractability. These rules are sometimes based in part on some results borrowed from neurobiology: e.g., 'units' in some network models are intended to correspond loosely to neurons, and 'weights' loosely to synapses; the notions of parallelism and distributed processing are based on metaphors derived from neural processes. An open question is how much of the rest of the rich literature of neurobiological results should or could profitably be incorporated into a network modeL From the point of view of constructing mechanisms to perform certain pre-specified computatonal functions (e.g., correlation, optimization), there are varying answers to this question. However, the goal of understanding brain circuit function introduces a fundamental problem: there are no known, pre-specified functions of any given cortical structures. We have constructed and studied a physiologically- and anatomically-accurate model of a particular brain structure, olfactory cortex, that is strictly based on biological data, with the goal of elucidating the local function of this circuit from its performance in a 'bottom-up' fashion. We measure our progress by the accuracy with which the model corresponds to known data, and predicts novel physiological results (see, e.g., Lynch and Granger, 1988; Lynch et al., 1988). Our initial analysis of the circuit reveals a mechanism consisting of a learning rule that is notably simple and restricted compared to most network models, a relatively novel architecture with some unusual properties, and a performance rule that is ex319 traordinarily complex compared to typical network-model performance rules. Taken together, these rules, derived directly from the known biology of the olfactory cortex, generate a coherent mechanism that has interesting computational properties. This paper describes the learning and performance rules and the architecture of the model; the relevant physiology and anatomy underlying these rules and structures, respectively; and an analysis of the coherent mechanism that results. LEARNING RULES DERIVED FROM LONG-TERM POTENTIATION Long-term potentiation (LTP) of synapses is a phenomenon in which a brief series of biochemical events gives rise to an enhancement of synaptic efficacy that is extraordinarily long-lasting (Bliss and L{lJmo, 1973; Lynch and Baudry, 1984; Staubli and Lynch, 1987); it is therefore a candidate mechanism underlying certain forms of learning, in which few training trials are required for long-lasting memory. The physiological characteristics of LTP form the basis for a straightforward network learning rule. It is known that simultaneous pre- and post-synaptic activity (i.e., intense depolarization) result in LTP (e.g., Wigstr{IJm et al., 1986). Since excitatory cells are embedded in a meshwork of inhibitory interneurons, the requisite induction of adequate levels of pre- and postsynaptic activity is achieved by stimulation of large numbers of afferents for prolonged periods, by voltage clamping the postsyna.ptic cell, or by chemically blocking the activity of inhibitory interneurons. In the intact animal, however, the q~estion of how simultaneous pre- and postsynaptic activity might be induced has been an open question. Recent work (Larson and Lynch, 1986) has shown that when hippocampal afferents are subjected to patterned stimulation with particular temporal and frequency parameters, inhibition is naturally eliminated within a specific time window, and LTP can arise as a result. Figure 1 shows that LTP naturally occurs using short (3-4 pulse) bursts of high-frequency (100Hz) stimulation with a 200ms interburst interval; only the second of a pair of two such bursts causes potentiation. This occurs because the normal short inhibitory currents (IPSPs), which prevent the first burst from depolarizing the postsynaptic cell sufficiently to produce LTP, are maximally refractory at 200ms after being stimulated, and therefore, although the second burst arrives against a hyperpolarized background resulting from the long hyperpolarizing currents (LHP) initiated by the first burst, the second burst does not initiate its own IPSPs, since they are then refractory. The studies leading to these conclusions were performed in in vitro hippocampal slices; LTP induced by this patterned stimulation technique in intact animals shows no measurable decrement prior to the time at which recording arrangements deteriora.te: more than a month in some cases (see Staubli and Lynch, 1987). PERFORMANCE RULES DERIVED FROM OLFACTORY PHYSIOLOGY AND BEHAVIOR From the above data we may infer that LTP itself depends on simultaneous pre- and postsynaptic activity, as Hebb postulated, but that a sufficient degree of the latter occurs only under particular conditions. Those conditions (patterned stimulation) suggest the beginnings of a performance rule for the network. Drawing this out requires a review of the inhibitory currents active in hippocampus and in piriform cortex. Three classes of such currents are known to be present: short IPSPs, long LHPs and extremely long, cell-specific afterhyperpolarization, or AHP (see Figure 2). Short IPSPs arise from both feedforward and feedback activation of inhibitory interneurons which in turn synapse 320 on excitatory cells (e.g., layer IT cells, which are primary excitatory cells in piriform). IPSPs develop more slowly than excitatory postsynaptic potentials (EPSPs) but quickly shunt the EPSP, thus reversing the depolarization that arises from EPSPs, and bringing the cell voltage down below its original resting potential. IPSPs last approximately 50lOOms, and then enter a refractory period during which they cannot be reactivated from about 100-30Oms after they have been once activated. Longer hyperpolarization (LHP) is presumably dependent on a distinct type of inhibitory interneuron or inhibitory receptor, and arises in much the same way; however, these cells are apparently not refractory once activated. LHP lasts for 300-500ms. Taken together, IPSPs and LHP constitute a form of high-pass frequency filter: 200ms after an input burst, a subsequent input will arrive against a background of hyperpolarization due to LHP, yet this input will not initiate its own IPSP due to the refractory period. If the input is a single pulse, its EPSP will fail to trigger the postsynaptic cell, since it will not be able to overcome the LHP-induced hyperpolarized potential of the cell. Yet if the input is a high-frequency burst, the pulses comprising the burst will give rise to different behavior. Ordinarily, the first EPSP would have been driven back to resting potential by its accompanying IPSP, before the second pulse in the burst could arrive. But when the IPSP is absent, the first EPSP is not driven rapidly down to resting potential, and the second pulse sums with it, raising the voltage of the postsynaptic cell and allowing voltage-dependent channels to open, thereby further depolarizing the cell, and causing it to spike (Figure 3). Hence these high-frequency bursts fire the cell, while single pulses or lower-frequency bursts would not do so. When these cells fire, then active synapses can be potentiated. The third inhibitory mechanism, AHP, is a current that causes an excitatory cell to become refractory after it has fired strongly or rapidly. This mechanism is therefore specific to those cells that have fired, unlike the first two mechanisms. AHP can prevent a cell from firing again for as long as 1000ms (1 second). It has long been observed that EEG waves in the hippocampi of learning animals are dominated by the theta rhythm, Le., activity occuring at about 4-8Hz. This is now seen to correspond to the optimal rate for firing postsynaptic cells and for enhancing synapses via LTP; i.e., this rhythmic aspect of the performance rules of these networks is suggested by the physiology ofLTP. The resulting activation patterns may take the following form: relatively synchronized cell firing occurring approximately once every 200ms, i.e., spatial patterns of induced activity occurring at the rate of one new spatial cell-firing pattern every 200ms. The cells most strongly participating in anyone firing pattern will not participate in subsequent patterns (at least the next 4-5 patterns, i.e., SOO-lOOOms), due to AHP. This raises the interesting possibility that different spatial patterns (at different times) may be conveying different information about their inputs. In summary, postsynaptic cells fire in pulses or bursts depending on the synaptically-weighted sums of their active axonal inputs; this firing is synchronized across the cells in a structure, giving rise to a spatial pattern of activity across these cells; once cells fire they will not fire again in subsequent patterns; each pattern (occuring at the theta rhythm, i.e., approximately once every 200ms) will therefore consist of extremely different spatial patterns of cell activity. Hence the 'output' of such a network is a sequence of spatial patterns. In an animal engaged in an olfactory discrimination learning task, the theta rhythm 321 dominates the animals behavior: the animals literally sniff at theta. We have been able to sustitute direct stimulation (in theta-burst mode) of the lateral olfactory tract (LOT), which is the input to the olfactory cortex, for odors: these 'electrical odors' are learned and discriminated by the animals, either from other electrical odors (via different stimulating electrodes) or from real odors. Furthermore, behavioral learning in this paradigm is accompanied by LTP of piriform synapses (Roman et al., 1987). This experimental paradigm thus provides us with a known set of behaviorally-relevant inputs to the olfactory cortex that give rise to synaptic potentiation that apparently underlies the learning of the stimuli. ARCHITECTURE OF OLFACTORY CORTEX Nasal receptor cells respond differentially to different chemicals; these cells topographically innervate the olfactory bulb, which is arranged such that combinations of specific spatial 'patches' of bulb characteristically respond to specific odors. Bulb also receives a number of centrifugal afferents from brain, most of which terminate on the inhibitory granule cells. The excitatory mitral cells in bulb send out axons that form the lateral olfactory tract (LOT), which constitutes the only major input to olfactory (piriform) cortex. This cortex in turn has some feedback connections to bulb via the anterior olfactory nucleus. Figure 4 illustrates the anatomy of the superficial layers of olfactory cortex: the LOT axons flow across layer la, synapsing with the dendrites of piriform layer-IT cells. Those cells in turn give rise to collateral axon outputs which flow, in layer Ib, parallel and subjacent to the LOT, in a predominantly rostral-to-caudal direction, eventually terminating in entorhinal cortex. Layer 180 is very sparsely connected; the probability of synapses between LOT axons and layer-IT cell dendrites is less than 0.10 (Lynch, 1986), and decreases caudally. Layer Ib (where collaterals synapse with dendrites) is also sparse, but its density increases caudally, as the number of collaterals increases; the overall connectivity density on layer-IT-cell dendrites is approximately constant throughout most of piriform. Layer IT also contains, in addition to the principal excitatory cells (modified stellates), inhibitory interneurons which synapse on excitatory cells within a specified radius, forming a 'patchwork' of cells affected by a particular inhibitory cell; the spheres of influence of inhibitory cells almost certainly overlap somewhat. There are approximately 50,000 LOT axons, 500,000 piriform layer IT cells, and a much smaller number of inhibitory cells that divide layer IT roughly into functional patches. (See Price, 1973; Luskin and Price, 1983; Krettek and Price, 1977; Price and Slotnick, 1983; Haberly and Price, 1977, 197880, 1978b). The layer IT cell collateral axons flow through layer ill for a distance before rising up to layer Ib (Haberly, 1985); taken in combination with the predominantly caudal directionality of these collaterals, this means that rostral piriform will be dominated by LOT inputs. Extreme caudal piriform (and all of lateral entorhinal cortex) is dominated by collaterals from more rostral cells; moving from rostral to caudal piriform, cells increasingly can be thought of as 'hybrid cells': cells receiving inputs from both the bulb (via the LOT) and from rostral piriform (via collateral axons). The architectural characteristics of rostral piriform is therefore quite different from that of caudal piriform, and differential analysis must be performed of rostral cells vs. hybrid cells, as will be seen later in the paper. 322 SIMULATION AND FORMAL ANALYSIS: INTRODUCTION We have conducted several simulations of olfactory cortex incorporating many of the physiological features discussed earlier. Two hundred layer IT cells are used with 100 input (LOT) lines and 200 collateral axons; both the LOT and collateral axons flow caudally. LOT axons connect with rostral dendrites with a probability of 0.2, which decreases linearly to 0.05 by the caudal end of the model. The connectivity is arranged randomly, subject to the constraint that the number of contacts for axons and dendrites is fixed within certain narrow b01llldaries (in the most severe case, each axon forms 20 synapses and each dendrite receives 20 contacts). The resulting matrix is thus hypergeometric in both dimensions. There are 20 simulated inhibitory interneurons, such that the layer IT cells are arranged in 20 overlapping patches, each within the influence of one such inhibitory cell. Inhibition rules are approximately as discussed above; i.e., the short IPSP is longer than an EPSP but only one fifth the length of the LHP; cell-specific AHP in tum is twice as long as LHP. Synaptic activity in the model is probabilistic and quantal: for any presynaptic activation, there is a fixed probability that the synapse will allow a certain amount of conductance to be contributed to the postsynaptic cell. Long-term potentiation was represented by a 40% increase in contact strength, as well as an increase in the probability of conductance being transmitted. These effects would be expected to arise, in situ, from modifying existing synapses as well as adding new ones (Lynch, 1986), two results obtained in electron microscopic studies (Lee et al., 1980). Only excitatory cell synapses are subject to LTP. LTP occurred when a cell was activated twice at a simulated 200ms interval: the first input 'primes' the synapse so that a subsequent burst input can drive it past a threshold value; following from the physiological results, previously potentiated synapses were much less different from "naive" synapses when driven at high frequency (see Lynch et al., 1988). The simulation used theta burst activation (i.e., bursts of pulses with the bursts occurring at 5Hz) of inputs during learning, and operated according to these synchronized fixed time steps, as discussed above. The network was trained on sets of "odors" , each of which was represented as a group of active LOT lines, as in the "electric odor" experiments already described. Usually three or four "components" were used in an odor, with each component consisting of a group of contiguous LOT lines. We assumed that the bulb normalized the output signal to about 20% of all LOT fibers. In some cases, more specific bulb rules were used and in particular inhibition was assumed to be greatest in areas surrounding an active bulb "patch" . The network exhibited several interesting behaviors. Learning, as expected, increased the robustness of the response to specific vectors; thus adding or subtracting LOT lines from a previously learned input did not, within limits, greatly change the response. The model, like most network simulations, dealt reasonably well with degraded or noisy known signals. An unexpected result developed after the network had learned a succession of cues. In experiments of this type, the simulation would begin to generate two quite distinct output signals within a given sampling episode; that is, a single previously learned cue would generate two successive responses in successive 'sniffs' presented to an "experienced" network. The first of these response patterns proved to be common to several signals while the second was specific to each learned signal. The 323 common signal was found to occur when the network had learned 3-5 inputs which had substantial overlap in their components (e.g., four odors that shared ::::::70% of their components). It appeared then that the network had begun to produce "category" or "clustering" responses, on the first sniff of a simulated odor, and "individual" or "differentiation" responses on subsequent sniffs of that same odor. When presented with a novel cue which contained elements shared with other, previously learned signals, the network produced the cluster response but no subsequent individual or specific output signal. Four to five cluster response patterns and 20 - 25 individual responses were produced in the network without distortion. In retrospect, it was clear that the model accomplished two necessary and in some senses opposing operations: 1) it detected similarities in the members of a cue category or cluster, and, 2) it nonetheless distinguished between cues that were quite similar. Its first response was to the similarity-based category and its second to the specific signal. ANALYSIS OF CATEGORIZATION IN ROSTRAL PIRIFORM Assume that a set of input cues (or 'simulated odors') XCI, Xf3 .. . X' differ from each other in the firing of dx LOT input lines; similarly, inputs Y CI , yf3 ... y' differ in dy lines, but that inputs from the sets X and Y differ from each other in D X,y > > d lines, such that the XS and the Ys form distinct natural categories. Then the performance of the network should give rise to output (layer II cell) firing patterns that are very similar among members of either category, but different for members of different categories; i.e., there should be a single spatial pattern of response for members of X, with little variation in response across members, and there should be a distinct spatial pattern of response for members of Y. Considering a matrix constructed by uniform selection of neurons, each with a hypergeometric distribution for its synapses, as an approximation of the bidimensional hypergeometric matrix described above, the following results can be derived. The expected value of el, the Hamming distance between responses for two input cues differing by 2d LOT lines (input Hamming distance of d) is: where No is the number of postsynaptic cells, each Si is the probability that a cell will have precisely i active contacts from one of the two cues, and I(i, j) is the probability that the number of contacts on the cell will increase (or decrease) from i to j with the change in d LOT lines; Le., changing from the first cue to the second. Hence, the first term denotes the probability of a cell decreasing its number of active contacts from above to below some threshold, (), such that that cell fired in response to one cue but not the other (and therefore is one of the cells that will contribute to the difference between responses to the two cues). Reciprocally, the second term is the probability that the cell increases its number of active synapses such that it is now over the threshold; this cell also will contribute to the difference in response. We restrict our analysis for now to rostral piriform, in which there are assumed to be few if any collateral axons. We will return to this issue in the next subsection. 324 The value for each Sa., the probability of a active contacts on a cell, is a hypergeometric function, since there are a fixed number of contacts anatomically between LOT and (rostral) piriform cells: where N is the number of LOT lines, A is the number of active (firing) LOT lines, n is the number of synapses per dendrite formed by the LOT, and a is the number of active such synapses. The formula can be read by noting that the first binomial indicates the number of ways of choosing a active synapses on the dendrite from the A active incoming LOT lines; for each of these, the next expression calculates the number of ways in which the remaining n - a (inactive) synapses on the dendrite are chosen from the N - A inactive incoming LOT lines; the probability of active synapses on a dendrite depends on the sparseness ofthe matrix (Le., the probability of connection between any given LOT line and dendrite); the solution must be normalized by the number of ways in which n synapses on a dendrite can be chosen from N incoming LOT lines. The probability of a cell changing its number of contacts from a to a is: I(a, a) = 2: 11-'= a.-4 where N, n, A, and a are as above, I is the "loss" or reduction in the number of active synapses, and 9 is the gain or increase. Hence the left expression is the probability of losing I active synapses by changing d LOT lines, and the right-hand expression is the probability of gaining 9 active synapses. The product of the expressions are summed over all the ways of choosing I and 9 such that the net change 9 - I is the desired difference a-a. If training on each cue induces only fractional LTP, then over trials, synapses contacted by any overlapping parts of the input cues should become stronger than those contacted only by unique parts of the cue. Comparing two cues from within a category, vs. two cues from between categories, there may be the same number of active synapses lost across the two cues in either case, but the expected strength of the synapses lost in the former case (within category) should be significantly lower than in the latter case (across categories). Hence, for a given threshold, the difference J between output firing patterns will be smaller for two within-category cues than for cues from two different categories. It is important to note that clustering is an operation that is quite distinct from stimulus generalization. Observing that an object is a car does not occur because of a comparison with a specific, previously learned car. Instead the category "car" emerges from the learning of many different cars and may be based on a "prototype" that has no necessary correspondence with a specific, real object. The same could be said of the network. It did not produce a categorical response when one cue had been learned 325 and second similar stimulus was presented. Category or cluster responses, as noted, required the learning of several exemplars of a similarity-based cluster. It is the process of extracting corrunonalities from the environment that defines clustering, not the simple noting of similarities between two cues. An essential question in clustering concerns the location of the boundaries of a given group; i.e., what degree of similarity must a set of cues possess to be grouped together? This issue has been discussed from any number of theoretical positions (e.g., information theory) i all these analyses incorporate the point that the breadth of a category must reflect the overall homogeneity or heterogeneity of the environment. In a world where things are quite similar, useful categories will necessarily be composed of objects with much in common. Suppose, for instance, that subjects were presented with a set of four distinct coffee cups of different colors, and asked later to recall the objects. The subjects might respond by listing the cups as a blue, red, yellow and green coffee cup, reflecting a relatively specific level of description in the hierarchy of objects that are coffee cups. In contrast, if presented with four different objects, a blue coffee cup, a drinking glass, a silver fork and a plastic spoon, the cup would be much more likely to be recalled as simply a cup, or a coffee cup, and rarely as a blue coffee cup; the specificity of encoding chosen depends on the overall heterogeneity of the environment. The categories formed by the simulation were quite appropriate when judged by an information theoretic measure, but how well it does across a wide range of possible worlds has not been addressed. ANALYSIS OF PROBLEMS ARISING FROM CAUDAL AXON FLOW The anatomical feature of directed flow of collateral axons gives rise to an immediate problem in principle. In essence, the more rostral cells that fire in response to an input, the more active inputs there are from these cells to the caudal cells, via collateral axons, such that the probability of caudal cell firing increases precipitously with probability of rostral cell firing. Conversely, reducing the number of rostral cells from firing, either by reducing the number of active input LOT axons or by raising the layer II cell firing threshold, prevents sufficient input to the caudal cells to enable their probability of firing to be much above zero. This problem can be stated formally, by making assumptions about the detailed nature of the connectivity of LOT and collateral axons in layer I as these axons proceed from rostral to caudal piriform. The probability of contact between LOT axons and layer-II-cell dendrites decreases caudally, as the number of collateral axons is increasing, given their rostral to caudal flow tendency. This situation is depicted in Figure 4. Assuming that probability of LOT contact tends to go to zero, we may adopt a labelling scheme for axons and synaptic contacts, as in the diagram, in which some combination of LOT axons (a:k) and collateral axons (h",,) contact any particular layer II cell dendrite (hn), each of which is itself the source of an additional collateral axon flowing to cells more caudal than itself. Then the cell firing function for layer II cell hn is: where the a:k denote LOT axon activity of those axons still with nonzero probability of contact for layer II cell hn, the hm. denote activity of layer II cells rostral of hn, 0 is 326 the cell firing threshold, Wnm is the synaptic strength between axon m and dendrite n, and H is the Heaviside step function, equal to 1 or 0 according to whether its argument is positive or negative. H we assume instead that probability of cell firing is a graded function rather than a step function, we may eliminate the H step function and calculate the firing of the cell (hn) from its inputs (hn,net) via the logistic: hn,net = L hm Wnm + L Zk Wnk m<n k~n 1 hn = ( ) 1 + e- khn,net + 9n Then we may expand the expression for firing of cell hn as follows: hn = [1 + e -(l:m<n hmwnm + l:k~n ZkWnk + 9)]-1 By assuming a fixed firing threshold, and varying the number of active input LOT lines, the probability of cell firing can be examined. Numerical simulation of the above expressions across a range of LOT spatial activation patterns demonstrates that probability of cell firing remains near zero until a critical number of LOT lines are active, at which point the probability flips to close to 100% (Figure 5). This means that, for any given firing threshold, given fewer than a certain amount of LOT input, practically no piriform cells will fire, whereas a slight increase in the number of active LOT lines will mean that practically all piriform cells should fire. This excruciating dependence of cell firing on amount of LOT input indicates that normalization of the size of the LOT input alone will be insufficient to stabilize the size of the layer II response; even slight variation of LOT activity in either direction has extreme consequences. A number of solutions are possible; in particular, the known local anatomy and physiology of layer II inhibitory intemeurons provides a mechanism for controlling the amount of layer II response. As discussed, inhibitory interneurons give rise to both feedforward (activated by LOT input) and feedback (activated by collateral axons) activity; the influence of any particuar interneuron is limited anatomically to a relatively small radius around itself within layer II, and the influence of multiple interneurons probably overlap to some extent. Nonetheless, the 'sphere of influence' of a particular inhibitory interneuron can be viewed as a local patch in layer II, within which the number of active excitatory cells is in large measure controlled by the activity of the inhibitory cell in that patch. If a number of excitatory cells are firing with varying depolarization levels within a patch in layer II, activation of the inhibitory cells by the excitatory cells will tend to weaken those excitatory cells that are less depolarized than the most strongly-firing cell within the patch, leading to a competition in which only those cells firing most strongly within a patch will burst, and these cells will, via the interneuron, suppress multiple firing of other cells within the patch. Thus the patch takes on some of the characteristics of a 'winner-take-all' network (Feldman, 1982): only the most strongly firing cells will be able to overcome inhibition sufficiently to burst, some additional cells will pulse once and then be overwhelmed by inhibition, and the rest of the cells in the patch will be silent, even though that patch may be receiving a large amount of excitatory input via LOT and collateral axon activity in layer I. 327 EMERGENT CATEGORIZATION BEHAVIOR IN THE MODEL The probabilistic quantal transmitter-release properties of piriform synapses described above give rise to probabilistic levels of postsynaptic depolarization. This inherent randomness of cell firing, in combination with activity oflocal inhibitory patches in layer IT, selects different sets of bursting and pulsing cells on different trials if no synaptic enhancement has taken place. The time-locked firing to the theta rhythm enables distinct spatial patterns of firing to be read out against a relatively quiescent background firing rate. Synaptic LTP enhances the conductances and alters the probabilistic nature of corrununication between a given axon and dendrite, which tends to overcome the randomness of the cell firing patterns in untrained cells, yielding a stable spatial pattern that will reliably appear in response to the same input in the future, and in fact will appear even in response to degraded or noisy versions of the input pattern. Furthermore, subsequent input patterns that differ in only minor respects from a learned LOT input pattern will contact many of the already-potentiated synapses from the original pattern, thereby tending to give rise to a very similar (and stable) output firing pattern. Thus as multiple cues sharing many overlapping LOT lines are learned, the layer IT cell responses to each of these cues will strongly resemble the responses to the others. Hence, the response(s) behave as though simply labelling a category of very-similar cues; sufficiently different cues will give rise to quite-different category responses. EMERGENT DIFFERENTIATION BEHAVIOR IN THE MODEL Potentiated synapses cause stronger depolarization and firing of those cells participating in a 'category' response to a learned cue. This increased depolarization causes strong, cell-specific afterhyperpolarization (AHP), effectively putting those cells into a relatively long-lasting (~ lsec) refractory period that prevents them from firing in response to the next few sampling sniffs of the cue. Then the inhibitory 'winner-take-all' behavior within patches effectively selects alternate cells to fire, once these stronglyfiring (learned) cells have undergone AHP. These alternates will be selected with some randomness, given the probabilistic release characteristics discussed above, since these cells will tend not to have potentiated synapses. These alternate cells then activate their caudally-flowing recurrent collaterals, activating distinct populations of synapses in caudal layer Th. Potentiation of these synapses in combination with those of stillactive LOT axons tends to 'recruit' stable subpopulations of caudal cells that are distinct for each simulated odor. They are distinct for each odor because first rostral cells are selected from the population of unpotentiated or weakly-potentiated cells (after the strongly potentiated cells have been removed via AHP)i hence they will at first tend to be selected randomly. Then, of the caudal cells that receive some activation from the weakening caudal LOT lines, those that also receive collateral innervation from these semi-randomly selected rostrals will be those that will tend to fire most strongly, and hence to be potentiated. The probability of a cell participating in the rostral semi-randomly selected groups for more than one odor (e.g., for two similar odors) is lower than the probability of cells being recruited by these two odors initially, since the population are those that receive not enough input from the LOT to have been recruited as a category cell and potentiated, yet receive enough input to fire as an alternate cell. The probability of any caudal cell then being recruited for more than one odor by these rostral cell collaterals 328 in combination with weakening caudal LOT lines is similarly low. The product of these two probabilities is of course lower still. Hence, the probability that any particular caudal cell potentiated as part of this process will participate in response to more than one odor is very low. This means that, when sampling (sniffing), the first pattern of cell firing will indicate similarity among learned odors, causing AHP of those patterns; thus later sniffs will generate patterns of firing that tend to be quite different for different odors, even when those odors are very similar. Empirical tests of the simulation have shown that odors consisting of 90%-overlapping LOT firing patterns will give rise to overlaps of between 85% and 95% in their initial layer IT spatial firing patterns, whereas these same cues give rise to layer IT patterns that overlap by less than 20% on 2nd and 3rd sniffs. The spatia-temporal pattern of layer IT firing over multiple samples thus can be taken as a strong differentiating mechanism for even very-similar cues, while the initial sniffresponse for those cues will nonetheless give rise to a spatial firing pattern that indicates the similarity of sets of learned cues, and therefore their 'category membership' in the clustering sense. CLUSTERING Incremental clustering of cues into similarity-based categories is a more subtle process than might be thought and while it is clear that the piriform simulation performs this function, we do not know how optimal its performance is in an information-theoretic sense, relative to some measure of the value or cost of information in the encoding. Building a categorical scheme is a non-monotonic, combinatorial problem: that is, each new item to be learned can have disproportionate effects on the existing scheme, and the number of potential categories (clusters) climbs factorially with the number of items to be categorized. Algorithmic solutions to problems of this type are computationally very expensive. Calculation of an ideal categorization scheme (with respect to particular cost measures in a performance task), using a hill-climbing algorithm derived from an information-theoretic measure of category value, applied to a problem involving 22 simulated odors, required more than 4 hours on a 68020-based processor. The simulation network reached the same answer as the game-theoretic program, but did so in seconds. It is worth mentioning again that the simulation did so while simultaneously learning unique encodings for the cues, as described above, which is itself a nontrivial task. Humans, on at least some tasks, may carry out clustering by building initial clusters and then merging or splitting them as more cues are presented. Thus far, the networks do not pass through successive categorization schema. However, experiments on human categorization have almost exclusively involved situations in which all cues were presented in rapid succession and category membership is taught explicitly, rather than developed independently by the subject. Hence, it is not clear from the experimental literature whether or not stable clusters develop in this way from stimuli presented at widely spaced intervals with no category membership information given, which is the problem corresponding to that given the network (and that is likely common in nature). It will be of interest to test categorizing skills of rats learning successive olfactory discriminations over several days. Using appropriately selected stimuli, it should be possible to determine if stable clusters are constructed and whether merging and splitting occurs over trials. 329 Any useful clustering device must utilize information about the heterogenity of the stimulus world in setting the heterogeneity of individual categories. Heterogeneity of categories refers to the degree of similarity that is used to determine if cues are to be grouped together or not. Several network parameters will influence category size and we are exploring how these influence the individuation function; one particularly interesting possibility involves a shifting threshold function, an idea used with great success by Cooper in his work on visual cortex. The problems presented to the simulation thus far involve a totally naive system, one that has had no "developmental" history. We are currently exploring a model in which early experiences are not learned by the network but instead set parameters for later ("adult") learning episodes. The idea is that early experience determines the heterogenity of the stimulus world and imprints this on the network, not by specific changes in synaptic strengths, but in a more general fashion. CONCLUSIONS Neurons have a nearly bewildering array of biophysical, chemical, electrophysiological and anatomical properties that control their behavior; an open question in neural network research is which of these properties need be incorporated into networks in order to simulate brain circuit function. The simulation described here incorporates an extreme amount of biological data, and in fact has given rise to novel physiological questions, which we have tested experimentally with results that are counterintuitive and previously unsuspected in the existing physiological literature (see, e.g., Lynch and Granger, 1988; Lynch et al., 1988). Incorporation of this mass of physiological parameters into the simulation gives rise to a coherent architecture and learning and performance rules, when interpreted in terms of computational function of the network, which generates a robust capability to encode multiple levels of information about learned stimuli. The coherence of the data in the model is useful in two ways: to provide a framework for understanding the purposes and interactions of many apparently-disparate biological properties of neurons, and to aid in the design of novel artificial network architectures inspired by biology, which may have useful computational functions. It is instructive to note that neurons are capable of many possible biophysical functions, yet early results from chronic recording of cells from olfactory cortex in animals actively engaged in learning many novel odors in an olfactory discrimination task clearly shows a particular operating mode of this cortical structure when it is actively in use by the animal (Larson et al., unpublished data). The rats in this task are very familiar with the testing paradigm and exhibit very raid learning, with no difficulty in acquiring large numbers of discriminations. Sampling, detection and responding occur in fractions of a second, indicating that the utilization of recognition memories in the olfactory system can be a rapid operation; it is not surprising, then, that the odor-coded units so far encountered in our physiological experiments have rapid and stereotyped responses. Given the dense innervation of the olfactory bulb by the brain, it is possible that the type of spatial encoding that appears to be responsible for the preliminary results of these chronic experiments would not appear in animals that were not engaged in active sampling or were confronted with unfamiliar problems. That is, the operation of the olfactory cortex might be as dependent upon the behavioral 'state' and behavioral history of the rat as upon the actual odors presented to it. It will be of interest to compare the results from well-trained freely-moving animals with those obtained using more restrictive testing conditions. 330 The temporal properties of synaptic currents and afterpotentials, results from simulations and chronic recording studies, taken together, suggest two useful caveats for biological models: • Cell firing in cortical structures (e.g., piriform, hippocampus and possibly neocortex) is linked to particular rhythms (theta in the case of piriform and hippocampus) during real learning behavior, and thus it is likely that the 'coding language' of these structures involves spatial cell firing patterns within a brief time window. This stands in contrast to other methods such as frequency coding that appears in other structures (such as peripheral sensory structures, e.g., retina and cochlea; see, e.g., Sivilotti et al., 1987). • Temporal sequences of spatial patterns may encode different types of information, such as hierarchical encodings of perceptions, in contrast with views in which either asynchronous 'cycling' activity occurs or a system yields a single punctate output and then halts. In particular, simulation of piriform gives rise to temporal sequences of spatial patterns of synchronized cell firing in layer IT, and the patterns change over time: the physiology and anatomy of the structure cause successive 'sniffs' of the same olfactory stimulus to give rise to a sequence of spatial patterns, each of which encodes successively more specific information about the stimulus, beginning with its similarity to other previouslylearned stimuli, and ending with a unique encoding of its characteristics. It is possible that both the early similarity-based 'cluster' information and the late unique encodings are used, for different purposes, by brain structures that receive these signals as output from piriform. ACKNOWLEDGEMENTS Much of the theoretical underpinning of this work depends critically on data generated by John Larson; we are grateful for his insightful advice and help. This work has benefited from discussions with Michel Baudry, Mark Gluck, and Ursula Staubli. Jose Ambros-Ingerson is supported by a fellowship from Hewlett-Packard, Mexico, administered by UC MEXUS. REFERENCES Bliss, T.V.P. and L~mo, T. (1973). Long-lasting potentiation of synaptic transmission in the dentate area of the anesthetized rabbit following stimulation of the perforant path. 1.Physiol.Lond. 232:357-374. Feldman, J.A. (1982). Dynamic connections in neural networks. Biological Cybernetics 46:27-39. Haberly, L.B. (1985). Neuronal circuitry in olfactory cortex: Anatomy and functional implications. Chemical Senses 10:219-238. Haberly, L.B. and J.L. Price (1977). The axonal projection patterns of the mitral and tufted cells of the olfactory bulb in the rat. Brain Res 129:152-157. 331 Haberly, L.B. and J.L. Price (1978a). Association and commissural fiber systems of the olfactory cortex of the rat. I. Systems originating in the piriform cortex and adjacent areas. J. Compo Neurol. 178:711-740. Haberly, L.B. and J.L. Price (1978b). Association and commissural fiber systems of the olfactory cortex of the rat. II. Systems originating in the olfactory peduncle. J. Compo Neurol. 181:781-808. Hebb, D.O. (1949). The Organization of Behavior. New York: Wiley. Krettek, J.E. and J.L. Price (1977). Projections from the amygdaloid complex and adjacent olfactory structures to the entorhinal cortex and to the subiculum in the rat and cat. J Comp NeuroI172:723-752. Larson, J. and G. Lynch (1986). Synaptic potentiation in hippocampus by patterned stimulation involves two events. Science 232:985-988. Lee, K., Schottler, F., Oliver, M. and Lynch, G. (1980). Brief bursts of high-frequency stimulation produce two types of structural change in rat hippocampus. J.Neurophysiol. 44:247-258. Lynch, G. and Baudry, M. (1984). The biochemistry of memory: a new and specific hypothesis. Science 224:1057-1063. Lynch, G. (1986). Synapses, circuits, and the beginnings of memory. Cambridge, Mass: MIT Press. Lynch, G., Larson, J., Staubli, U., and Baudry, M. (1987). New perspectives on the physiology, chemistry and pharmacology of memory. Drug Devel.Res. 10:295-315. Lynch, G., Granger, R., Levy, W. and Larson, J. (1988). Some possible functions of simple cortical networks suggested by computer modeling. In: Neural Models of Plasticity: Theoretical and Empirical Approaches, Byrne, J. and Berry, W.O. (Eds.), (in press). Lynch, G. and Granger, R. (1988). Simulation and analysis of a cortical network. The Psychology of Learning and Motivation, Vo1.22 (in press). Luskin, M.B. and J.L. Price (1983). The laminar distribution of intracortical fibers orginating in the olfactory cortex of the rat. J Comp NeuroI216:292-302. Parker, D.B. (1985). Learning-logic. MIT TR-47, Massachusetts Institute of Technology, Center for Computational Research in Economics and Management Science, Cambridge, Mass. Price, J.L. (1973). An autoradiographic study of complementary laminar patterns of termination of afferent fibers to the olfactory cortex. J.Comp.Neur. 150:87-108. Price, J .L. and B.M. Slotnick (1983). Dual olfactory representation in the rat thalamus: An anatomical and electrophysiological study. J Comp NeuroI215:63-77. Roman, F., Staubli, U. and Lynch, G. (1987). Evidence for synaptic potentiation in a cortical network during learning. Brain Res. 418:221-226. Rosenblatt, F. (1962). Principles ofneurodynamics. New York: Spartan. Rumelhart, D., Hinton, G. and Williams, R. (1986). Learning Internal Representations by Error Propagation. In D.Rumelhart and J.McClelland (Eds.), Parallel 332 Distributed Processing, Cambridge: MIT Press. Sivilotti, M.A., Mahowald, M.A. and Mead, C.A. (1987). Real-time visual computations using analog CMOS processing arrays. In: Advanced Research in VLSI (Ed. Paul Losleben), MIT Press, Cambridge. Staubli, U. and Lynch, G. (1987). Stable hippocampal long-term potentiation elicited by "theta" pattern stimulation. Brain Res. (in press). Widrow, G. and Hoff, M.E. (1960). Adaptive Switching Circuits. Institute of Radio Engineers, Western Electronic Convention Record, Part ., pp.96-104. Wigstrf/Sm, H., B. Gustaffson, Y.Y. Huang and W.C. Abraham (1986). Hippocampal long-term potentiation is induced by pairing single afferent volleys with intracellularly injected depolarizing current pulses. Acta Physiol Scand 126:317-319. 333 A SI HI! ID ....... I I 200 11\8 S2 ID In 2 8ec B I] 51 • • • • • ,... :::> e · . . . .......... . . . . .. 'V Ilen I] .s~ Il"" • • • I · . . . . . . . . .... . .. ,... M 1403 'V .... :-:.~ en .... 80 "" .A. • • .,.,....... •• · ...... . • • •• 10 20 30 40 50 60 70 TIME (MINUTES) C BEFORE AFTER SUPERIMPOSED Figure 1. LTP induction by short high-frequency bursts involves sequential "priming" and "consolidation" events. A) Sl and S2 represent separate groups of Shaffer/commissural fibers converging on a single CAl pyramidal neuron. The stimulation pattern employed consisted of pairs of bursts (each 4 pulses at 100Hz) given to Sl and S2 respectively, with a 200ms delay between them. The pairs were repeated 10 times at 2 sec intervals. B) Only the synapses activated by the delayed burst (52) showed LTP. The top panel shows measurements of amplitudes of intracellular EPSPs evoked by single pulses to Sl before and after patterned stimulation (given at 20 min into the experiment). The middle panel shows the amplitude of EPSPs evoked by 52. Bottom panel shows EP5P amplitudes for both pathways expressed as a percentage of their respective sizes before burst stimulation. C) Shown are records of EPSPs evoked by 51 and 52 five min. before and 40 min. after patterned burst stimulation. Calibration bar: Sm V, 5msec. (From Larson and Lynch, 1986). 334 12 -------1[> ----~t> <]----EPSP -20 msec No IPSP ~IOO msec CI LHP -.5sec K AHP -lsec K Figure 2. Onset and duration of events comprising stimulation of a layer IT cell in piriform cortex. Axonal stimulation via the lateral olfactory tract (LOT) activates feedforward EPSPs with rapid onset and short duration (:::::20msec) and two types of feedforward inhibition: short feedforward IPSPs with slower onset and somewhat longer duration (~10Omsec) than the EPSPs, and longer hyperpolarizing potentials (LHP) lasting :::::500msec. These two types of inhibition are not specific to firing cells; an additional, very long-lasting (:::::1sec) inhibitory afterhyperpolarizing current (AHP) is induced in a cell-specific fashion in those cells with intense firing activity. Finally, feedback EPSPs and IPSPs are induced by activation via recurrent collateral axons from layer IT cells. 335 FIRST SECOND TENTH Sl-AHP L S2 CONTROL ~S2 Figure 3. When short, high-frequency bursts are input to cells 200ms after an initial 'priming' event, the broadened EPSPs (see Figure 1) will allow the contributions ofthe second and subsequent pulses comprising the burst to sum with the depolarization of the first pulse, yielding higher postsynaptic depolarization sufficient to cause the cell to spike. (From Lynch, Larson, Staubli and Baudry, 1987). 336 NlTE1lIOll KIDDLI rOsntllOI to layer IV .nt.- po.t. " " . .. ,rob.bUity of LOT cont.ct p.r uoa. cI.cr ..... n ia conn.at __ ...... ~ ,robabUity of ••• oc. coa.t.cc p.r UOD ia coa.eaa.t- "n" acr ••••• lel.elv. contribution to .,lUna cell .aterior Aa.oc LOT ~.rior Figure 4. Organization of extrinsic and feedback inputs to layer-II cells of piriform cortex. The axons comprising the lateral olfactory tract (LOT), originating from the bulb, innervate distal dendrites, whereas the feedback collateral or associational fibers contact proximal dendrites. Layer II cells in anterior (rostral) piriform are depicted as being dominated by extrinsic (LOT) input, whereas feedback inputs are more prominent on cells in posterior (caudal) piriform. ~ .c m .c a ~ 0.. 337 Tapered Feedforward Firing Probabilities 10°r-----------------------~~~~~~~1 80 60 -Go StirrufusA ... Stirrufus B 4Stirrufus C 40 000Stirrufus 0 .... CumHypergmt 20 O~~~~~~~~~~~~~ o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Number Firing (of 40 Connections) Figure 5. Probability onayer-ll-cell firing as a function of number of LOT axons active, in the absence of local inhibitory patches. The hypergeometric function ('CumHypergmt ') specifies the probability o{layer II cell firing in the absence of caudally-directed feedback collaterals, i.e., assuming that all collaterals are equally probable to travel either rostrally or caudally. In this case, there is a smooth S-shaped function for probability of cell firing with increasing LOT activity, so that adjustment of global firing threshold (e.g., via nonspecific cholinergic inputs affecting all piriform inhibitory interneurons) can effectively normalize piriform layer II cell firing. However, when feedback axons are caudally directed, then probability steepens markedly, becoming a near step function, in which the probability of cell firing is exquisitely sensitive to the number of active inputs, across a range of empirically-tested LOT stimulation patterns (A - D in the figure). In this case, global adjustment of inhibition will fail to adequately normalize layer II cell firing: the probability of cell firing will always be either near zero or near 1.0; i.e., either nearly all cells will fire or almost none will fire. Local inhibitory control of 'patches' oflayer II solve this problem (refer to text).
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750 A DYNAMICAL APPROACH TO TEMPORAL PATTERN PROCESSING W. Scott Stornetta Stanford University, Physics Department, Stanford, Ca., 94305 Tad Hogg and B. A. Huberman Xerox Palo Alto Research Center, Palo Alto, Ca. 94304 ABSTRACT Recognizing patterns with temporal context is important for such tasks as speech recognition, motion detection and signature verification. We propose an architecture in which time serves as its own representation, and temporal context is encoded in the state of the nodes. We contrast this with the approach of replicating portions of the architecture to represent time. As one example of these ideas, we demonstrate an architecture with capacitive inputs serving as temporal feature detectors in an otherwise standard back propagation model. Experiments involving motion detection and word discrimination serve to illustrate novel features of the system. Finally, we discuss possible extensions of the architecture. INTRODUCTION Recent interest in connectionist, or "neural" networks has emphasized their ability to store, retrieve and process patterns1,2. For most applications, the patterns to be processed are static in the sense that they lack temporal context. Another important class consists of those problems that require the processing of temporal patterns. In these the information to be learned or processed is not a particular pattern but a sequence of patterns. Such problems include speech processing, signature verification, motion detection, and predictive signal processin,r-8. More precisely, temporal pattern processing means that the desired output depends not only on the current input but also on those preceding or following it as well. This implies that two identical inputs at different time steps might yield different desired outputs depending on what patterns precede or follow them. There is another feature characteristic of much temporal pattern processing. Here an entire sequence of patterns is recognized as a single distinct category, © American Institute of Physics 1988 generating a single output. A typical example of this would be the need to recognize words from a rapidly sampled acoustic signal. One should respond only once to the appearance of each word, even though the word consists of many samples. Thus, each input may not produce an output. With these features in mind, there are at least three additional issues which networks that process temporal patterns must address, above and beyond those that work with static patterns. The first is how to represent temporal context in the state of the network. The second is how to train at intermediate time steps before a temporal pattern is complete. The third issue is how to interpret the outputs during recognition, that is, how to tell when the sequence has been completed. Solutions to each of these issues require the construction of appropriate input and output representations. This paper is an attempt to address these issues, particularly the issue of representing temporal context in the state of the machine. We note in passing that the recognition of temporal sequences is distinct from the related problem of generating a sequence, given its first few members9.l O•11. TEMPORAL CLASSIFICATION With some exceptions10.12, in most previous work on temporal problems the systems record the temporal pattern by replicating part of the architecture for each time step. In some instances input nodes and their associated links are replicated3,4. In other cases only the weights or links are replicated, once for each of several time delays 7,8. In either case, this amounts to mapping the temporal pattern into a spatial one of much higher dimension before processing. These systems have generated significant and encouraging results. However, these approaches also have inherent drawbacks. First, by replicating portions of the architecture for each time step the amount of redundant computation is significantly increased. This problem becomes extreme when the signal is sampled very frequently4. :-.l' ext, by re lying on replications of the architecture for each time step, the system is quite inflexible to variations in the rate at which the data is presented or size of the temporal window. Any variability in the rate of the input signal can generate an input pattern which bears little or no resemblance to the trained pattern. Such variability is an important issue, for example, in speech recognition. Moreover, having a temporal window of any fixed length makes it manifestly impossible to detect contextual effects on time scales longer than the window size. An additional difficulty is that a misaligned signal, in its spatial representation, may have very little resemblance to the correctly aligned training signal. That is, these systems typically suffer from not being translationally invariant in time. ~etworks based on relaxation to equilibrium 11,13,14 also have difficulties for use with temporal problems. Such an approach removes any dependence on initial 751 752 conditions and hence is difficult to reconcile directly with temporal problems, which by their nature depend on inputs from earlier times. Also, if a temporal problem is to be handled in terms of relaxation to equilibrium, the equilibrium points themselves must be changing in time. A NON·REPLICATED, DYNAMIC ARCHITECTURE We believe that many of the difficulties mentioned above are tied to the attempt to map an inherently dynamical problem into a static problem of higher dimension. As an alternative, we propose to represent the history of the inputs in the state of the nodes of a system, rather than by adding additional units. Such an approach to capturing temporal context shows some very immediate advantages over the systems mentioned above. F'irst, it requires no replication of units for each distinct time step. Second, it does not fix in the architecture itself the window for temporal context or the presentation rate. These advantages are a direct result of the decision to let time serve as its own representation for temporal sequences, rather than creating additional spatial dimensions to represent time. In addition to providing a solution to the above problems, this system lends itself naturally to interpretation as an evolving dynamical system. Our approach allows one to think of the process of mapping an evolving input into a discrete sequence of outputs (such as mapping continuous speech input into a sequence of words) as a dynamical system moving from one attractor to another15. As a preliminary example of the application of these ideas, we introduce a system that captures the temporal context of input patterns without replicating units for each time step. We modify the conventional back propagation algorithm by making the input units capacitive. In contrast to the conventional architecture in which the input nodes are used simply to distribute the signal to the next layer, our system performs an additional computation. Specifically, let Xi be the value computed by an input node at time ti ' and Ii be the input signal to this node at the same time. Then the node computes successive values according to (1) where a is an input amplitude and d is a decay rate. Thus, the result computed by an input unit is the sum of the current input value multiplied by a, plus a fractional part, d, of the previously computed value of the input unit. In the absence of further input, this produces an exponential decay in the activation of the input nodes. The value for d is chosen so that this decay reaches lie of its original value in a time t characteristic of the time scale for the particular problem, i.e., d=e'tr, where r is the presentation rate. The value for a is chosen to produce a specified maximum value for X, given by alma/(1-d) . We note that Eq. (1) is equivalent to having a non-modifiable recurrent link with weight d on the input nodes, as illustrated in Fig. l. o 0 Fig. 1: Schematic architecture with capacitive inputs. The input nodes compute values according to Eq. (1). Hidden and output units are identical to standard back propagation nets. The processing which takes place at the input node can also be thought of in terms of an infinite impulse response (IIR) digital filter. The infinite impulse response of the filter allows input from the arbitrarily distant past to influence the current output of the filter, in contrast to methods which employ fixed windows, which can be viewed in terms of finite impulse response (FIR) filters. The capacitive node of Fig. 1 is equivalent to pre-processing the signal with a filter with transfer function a/(1-dz· 1). This system has the unique feature that a simple transformation of the parameters a and d allows it to respond in a near-optimal way to a signal which differs from the training signal in its rate. Consider a system initially trained at rate r with decay rate d and amplitude a. To make use of these weights for a different presentation rate, r~ one simply adjusts the values a 'and d'according to d' = dr/r' 1 - d' a' = a ""[:"d (2) (3) 753 754 These equations can be derived by the following argument. The general idea is that the values computed by the input nodes at the new rate should be as close as possible to those computed at the original rate. Specifically, suppose one wishes to change the sampling rate from r to nr, where n is an integer. Suppose that at a time to the computed value of the input node is Xo' If this node receives no additional input, then after m time steps, the computed value of the input node will be Xodm . For the more rapid sampling rate, Xodm should be the value obtained after nm time steps. Thus we require (4) which leads to Eq. (2) because n= r7r. Now suppose that an input I is presented m times in succession to an input node that is initially zero. After the mth presentation, the computed value of the input node is (5) Requiring this value to be equal to the corresponding value for the faster presentation rate after nm time steps leads to Eq. (3). These equations, then, make the computed values of the input nodes identical, independent of the presentation rate. Of course, this statement only holds exactly in the limit that the computed values of the input nodes change only infinitesimally from one time step to the next. Thus, in practice, one must insure that the signal is sampled frequently enough that the computed value of the input nodes is slowly changing. The point in weight space obtained after initial training at the rate r has two desirable properties. First, it can be trained on a signal at one sampling rate and then the values of the weights arrived at can be used as a near-optimal starting point to further train the system on the same signal but at a different sampling rate. Alternatively, the system can respond to temporal patterns which differ in rate from the training signal, without any retraining of the weights. These factors are a result of the choice of input representation, which essentially present the same pattern to the hidden unit and other layers, independent of sampling rate. These features highlight the fact that in this system the weights to some degree represent the temporal pattern independent of the rate of presentation. In contrast, in systems which use temporal windows, the weights obtained after training on a signal at one sampling rate would have little or no relation to the desired values of the weights for a differen.t sampling rate or window size. EXPERIMENTS As an illustration of this architecture and related algorithm, a three-layer, 15-30-2 system was trained to detect the leftward or rightward motion of a gaussian pulse moving across the field of input units with sudden changes in direction. The values of d and a were 0.7788 and 0.4424, respectively. These values were chosen to give a characteristic decay time of 4 time steps with a maximum value computed by the input nodes of 2.0. The pulse was of unit height with a half-width, 0, of 1.3. Figure 2 shows the input pulse as well as the values computed by the input nodes for leftward or rightward motion. Once trained at a velocity of 0.1 unit per sampling time, the velocity was varied over a wide range, from a factor of2 slower to a factor of2 faster as shown in Fig. 3. For small variations in velocity the system continued to correctly identify the type of motion. More impressive was its performance when the scaling relations given in Eqs. (2) and (3) were used to modify the amplitude and decay rate. In this case, acceptable performance was achieved over the entire range of velocities tested. This was without any additional retraining at the new rates. The difference in performance between the two curves also demonstrates that the excellent performance of the system is not an anomaly of the particular problem chosen, but characteristic of rescaling a and d according to Eqs. (2) and (3). We thus see that a simple use of capacitive links to store temporal context allows for motion detection at variable velocities. A second experiment involving speech data was performed to compare the system's performance to the time-delay-neural-network of Watrous and Shastri8. In their work, they trained a system to discriminate between suitably processed acoustic signals of the words "no" and "go." Once trained on a single utterance, the system was able to correctly identify other samples of these words from the same speaker. One drawback of their approach was that the weights did not converge to a fixed point. We were therefore particularly interested in whether our system could converge smoothly and rapidly to a stable solution, using the same data, and yet generalize as well as theirs did. This experiment also provided an opportunity to test a solution to the intermediate step training problem. The architecture was a 16-30-2 network. Each of the input nodes received an input signal corresponding to the energy (sampled every 2.5 milliseconds) as a function of time in one of 16 frequency channels. The input values were normalized to lie in the range 0.0 to 1.0. The values of d and a were 0.9944 and 0.022, respectively. These values were chosen to give a characteristic decay time comparable to the length of each word (they were nearly the same length), and a maximum value computed by the input nodes of 4.0. For an input signal that was part of the word "no", the training signal was (t.O, 0.0), while for the word "go" it was (0.0, 1.0). Thus the outputs that were compared to the training signal can be interpreted as evidence for one word or the other at each time step. The error shown in Fig. 4 is the sum of the squares of the 755 756 difference between the desired outputs and the computed outputs for each time step, for both words, after training up to the number ofiterations indicated along the x-axis. 2 2 a) input wavepacket 3 4 b) rightward motion 3 4 c) leftward motion 5 6 5 6 7 B 9 10 7 B 9 10 2 3 4 5 6 7 B 9 10 Fig. 2: a) Packet presented to input nodes. The x-axis represents the input nodes. b) Computed values from input nodes during rightward motion. c) Computed values during leftward motion. 100~ ______________ ~~~~ __ ~~~::::~ :"':w ' - - - -.................. ~I!"'.::..:/,j/.:). . 'i % 80 __ . ;~ c o 60 _ I) • • • • I~ r 40 +r e c t e r r 0 r 20 -Io I I .5 I I 1.0 v'lv I I 1.5 '0 I I 2.0 Fig. 3: Performance of motion detection experiment for various velocities. Dashed curve is performance without scaling and solid curve is with the scaling given in Eqs. (2) and (3). 125.0 100.0 75.0 50.0 25.0 0.0 0 500 1000 1500 2000 2500 iterations Fig. 4: Error in no/go discrimination as a function of the number of training iterations. Evidence for each word was obtained by summing the values of the respective nodes over time. This suggests a mechanism for signaling the completion of a sequence: when this sum crosses a certain threshold value, the sequence (in this case, the word) is considered recognized. Moreover, it may be possible to extend this mechanism to apply to the case of connected speech: after a word is recognized, the sums could be reset to zero, and the input nodes reinitialized. Once we had trained the system on a single utterance, we tested the perfor~ance of the resulting weights on additional utterances of the same speaker. 757 758 Preliminary results indicate an ability to correctly discriminate between "no" and "go." This suggests that the system has at least a limited ability to generalize in this task domain. DISCUSSION At a more general level, this paper raises and addresses some issues of representation. By choosing input and output representations in a particular way, we are able to make a static optimizer work on a temporal problem while still allowing time to serve as its own representation. In this broader context, one realizes that the choice of capacitive inputs for the input nodes was only one among many possible temporal feature detectors. Other possibilities include refractory units, derivative units and delayed spike units. Refractory units would compute a value which was some fraction of the current input. The fraction would decrease the more frequently and recently the node had been "on" in the recent past. A derivative unit would have a larger output the more rapidly a signal changed from one time step to the next. A delayed spike unit might have a transfer function of the form Itne-at, where t is the time since the presentation of the signal. This is similar to the function used by Tank and Hopfield7, but here it could serve a different purpose. The maximum value that a given input generated would be delayed by a certain amount of time. By similarly delaying the training signal, the system could be trained to recognize a given input in the context of signals not only preceding but also following it. An important point to note is that the transfer functions of each of these proposed temporal feature detectors could be rescaled in a manner similar to the capacitive nodes. This would preserve the property of the system that the weights contain information about the temporal sequence to some degree independent of the sampling rate. An even more ambitious possibility would be to have the system train the parameters, such as d in the capacitive node case. It may be feasible to do this in the same way that weights are trained, namely by taking the partial of the computed error with respect to the parameter in question. Such a system may be able to determine the relevant time scales of a temporal signal and adapt accordingly. ACKNOWLEDGEMENTS We are grateful for fruitful discllssions with Jeff Kephart and the help of Raymond Watrous in providing data from his own experiments. This work was partially supported by DARPA ISTO Contract # N00140-86-C-8996 and ONR Contract # N00014-82-0699_ 1. D. Rumelhart, ed., Parallel Distributed Processing, (:\'lIT Press, Cambridge, 1986). 2. J. Denker, ed., Neural Networks for Computing, AlP Conf. Proc.,151 (1986). 3. T. J. Sejnowski and C. R. Rosenberg, NETtalk: A Parallel Network that Learns to Read Aloud, Johns Hopkins Univ. Report No. JHU/EECS-86101 (1986). 4. J.L. McClelland and J.L. Elman, in Parallel Distributed Processing, vol. II, p. 58. 5. W. Keirstead and B.A. Huberman, Phys. Rev. Lett. 56,1094 (1986). 6. A. Lapedes and R. Farber, Nonlinear Signal Processing Using Neural Networks, Los Alamos preprint LA-uR-87-2662 (1987). 7. D. Tank and J. Hopfield, Proc. Nat. Acad. Sci., 84, 1896 (1987). 8. R. Watrous and L. Shastri, Proc. 9th Ann. Conf Cog. Sci. Soc., (Lawrence Erlbaum, Hillsdale, 1987), p. 518. 9. P. Kanerva, Self-Propagating Search: A Unified Theory of Memory, Stanford Univ. Report No. CSLI-84-7 (1984). 10. M.1. Jordan, Proc. 8th Ann. Conf. Cog. Sci. Soc., (Lawrence Erlbaum, Hillsdale, 1986), p. 531. 11. J. Hopfield,Proc. Nat. Acad. SCi., 79, 2554 (1982). 12. S. Grossberg, The Adaptive Brain, vol. II, ch. 6, (North-Holland, Amsterdam, 1987). 13. G. Hinton and T. J. Sejnowski, in Parallel Distributed Processing, vol. I, p. 282. 14. B. Gold, in Neural Networks for Computing, p. 158. 15. T. Hogg and B.A. Huberman, Phys. Rev. A32, 2338 (1985). 759
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348 Minkowski-r Back-Propaaation: Learnine in Connectionist Models with Non-Euclidian Error Silllais Stephen Jose Hanson and David J. Burr Bell Communications Research Morristown, New Jersey 07960 Abstract Many connectionist learning models are implemented using a gradient descent in a least squares error function of the output and teacher signal. The present model Fneralizes. in particular. back-propagation [1] by using Minkowski-r power metrics. For small r's a "city-block" error metric is approximated and for large r's the "maximum" or "supremum" metric is approached. while for r=2 the standard backpropagation model results. An implementation of Minkowski-r back-propagation is described. and several experiments are done which show that different values of r may be desirable for various purposes. Different r values may be appropriate for the reduction of the effects of outliers (noise). modeling the input space with more compact clusters. or modeling the statistics of a particular domain more naturally or in a way that may be more perceptually or psychologically meaningful (e.g. speech or vision). 1. Introduction The recent resurgence of connectionist models can be traced to their ability to do complex modeling of an input domain. It can be shown that neural-like networks containing a single hidden layer of non-linear activation units can learn to do a piece-wise linear partitioning of a feature space [2]. One result of such a partitioning is a complex gradient surface on which decisions about new input stimuli will be made. The generalization, categorization and clustering propenies of the network are therefore detennined by this mapping of input stimuli to this gradient swface in the output space. This gradient swface is a function of the conditional probability distributions of the output vectors given the input feature vectors as well as a function of the error relating the teacher signal and output. f'F'I an" c.,-i,. .... T.. ............ .. ., ~.r 01. 349 Presently many of the models have been implemented using least squares error. In this paper we describe a new model of gradient descent back-propagation [I] using Minkowski-r power error metrics. For small r's a "city-block" error measure (r=I) is approximated and for larger r's a "maximum" or supremum error measure is approached, while the standard case of Euclidian back-propagation is a special case with 1'*2. Fll"St we derive the general case and then discuss some of the implications of varying the power in the general metric. 2. Derivation of Minkowski-r Back-propagation The standard back-propagation is derived by minimizing least squares error as a function of connection weights within a completely connected layered network. The error for the Euclidian case is (for a single input-output pair), 1 .. 2 E = - L O'j-Yj) , 2 . J (1) where Y is the activation of a unit and y represents an independent teacher signal. The activation of a unit 0') is typically computed by nonnalizing the input from other units (x) over the interval (0,1) while compressing the high and low end of this range. A common function used for this normalization is the logistic, 1 Yj=--1 + e-Xt (2) The input to a unit (x) is found by summing products of the weights and corresponding activations from other units, (3) where Yle represents units in the fan in of unit i and Whi represents the strength of the connection between unit i and unit h. A gradient for the Euclidian or standard back-propagation case could be found by finding the partial of the error with respect to each weight, and can be expressed in this three tenn differential, 350 dE dE dyi dX; -.---dw/ti dyi ax; aw., (4) which from the equations before turns out to be, (5) Generalizing the error for Minkowski-r power metrics (see Figure 1 for the family of curves), 1 .... )' E = - L I (Yi - Yi I r . • (6) ~ .. ~ I : : C'f 0 § .eo • ·20 0 ao ... 10 NfIII Figure 1: Minkowski-r Family Using equations 24 above with equation 6 we can easily find an expression for the gradient in the general Minkowski-r case, dE I .... I ,-1 1) (y ..... ) ~ = ( Yi - Yi) Yi( -Yi ",.sgn i - Yi aw,.; (7) This gradient is used in the weight update rule proposed by Rumelhart, Hinton and Williams [1], 351 dE whi(n+l) = (X- + wAi(n) dWAi (8) Since the gradient computed for the hidden layer is a function of the gradient for the output, the hidden layer weight updating proceeds in the same way as in the Euclidian case [1], simply substituting this new Minkowski-r gradient. It is also possible to define a gradient over r such that a minimum in error would be sought. Such a gradient was suggested by White [3, see also 4] for maximum likelihood estimation of r, and can be shown to be, dIO£E) = (1-1Ir)(1Ir) + (llr)2/og (r) + (lIr) 2",(1lr) + (1/r) 21Yi-Yi 1 -(1/r)(IYi -Yil)'/og(IYi -Yi I) (9) An approximation of this gradient (using the last term of equation 9) has been implemented and investigated for simple problems and shown to be fairly robust in recovering similar r values. However, it is important that the r update rule changes slower than the weight update rule. In the simulations we ran r was changed once for every 10 times the weight values were changed. This rate might be expected to vary with the problem and rate of convergence. Local minima may be expected in larger problems while seeking an optimal r. It may be more infonnative for the moment to examine different classes of problems with fixed r and consider the specific rationale for those classes of problems. 3. Variations in r Various r values may be useful for various aspects of representing infonnation in the feature domain. Changing r basically results in a reweighting of errors from output bitsl . Small r's give less weight for large deviations and tend to reduce the influence of outlier points in the feature space during learning. In fact, it can be shown that if the distributions of feature vectors are non-gaussian, then the r=2 case 1. It is possible to entcltain r values that are negative, which would give largest weight to small errors close to zero and smallest weight to very large emn. Values of r lea than 1 generally are non-metric. i.e. they viola1e 81ieast one of the meuic axioms. For example. r<O violates the triangle inequality. Fa' aome problems this may make sense and the need for a metric em:r weighting may be unnecessary. These issues are not explored in this paper. 352 will not be a maximum likelihood estimator of the weights [5]. The city block case, r=1, in fact, arises if the underlying conditional probability distributions are Laplace [5]. More generally. r's less than two will tend to model non~gaussian distributions where the tails of the distributions are more pronounced than in the gaussian. Better estimators can be shown to exist for general noise reduction and have been studied in the area of robust estimation procedures [5] of which the Minkowski-r metric is only one possible case to consider. r<2. It is generally recommended that 1'=1.5 may be optimal for many noise reduction problems [6]. However, noise reduction may also be expected to vary with the problem and nature of the noise. One example we have looked at involves the recovery of an arbitrary 3 dimensional smooth surface as shown in Figure 2a, after the addition of random noise. This surface was generated from a gaussian curve in the 2 dimensions. Uniform random noise equal to the width (standard deviation) of the surface shape was added point-wise to the surface producing the noise plus surface shape shown in Figure 2b. b Figure 2: Shape surface (2a), Shape plus noise surface (2b) and recovered Shape sUrface (2c) The shape in Figure 2a was used as target points for Minkowski-r back~propagation2 and recovered with some distortion of the slope of the shape near the peak of the 2. All simulation runs, unless otherwise stated, used the same learning rate (.05) and smoothing value (.9) and stopping critmon defined in tenns of absolute mean deviation. The number of iterations to meet the stopping criterion varied considerably as r was changed (see below). 353 surface (see Fiaure 2c). Next the noise plus shape surface was used as target points for the learning procedure with r=2. The shape shown in Figure 3a was recovered, however. with considerable distortion iaround the base and peak. The value of r was reduced to 1.5 (Figure 3b) and then finally to 1.2 (Figure 3c) before shape distortions were eliminated. Although, the major properties of the shape of the surface were recovered. the scale seems distorted (however, easily restored with renormalization into the 0.1 range). Figure 3: Shape surface recovered with r=2 (3a), r=1.5 (3b) and r=1.2 (3c) r>2. Large r's tend to weight large deviations. When noise is not possible in the feature space (as in an arbitrary boolean problem) or where the token clusters are compact and isolated tllen simpler (in the sense of the number and placement of partition planes) genenuization surfaces may be created with larger r values. For example, in the simple XOR problem, the main effect of increasing r is to pull the decision boundaries closer into the non-zero targets (compare high activation regions in Figure 4a and 4b). In this particular problem clearly such compression of the target regions does not constitute simpler decision surfaces. However, if more hidden units are used than are needed for pattern class separation, then increasing r during training will tend to reduce the number of cuts in the space to the minimum needed. This seems to be primarily due to the sensitivity of the hyper-plane placement in the feature space to the geometry of the targets. A more complex case illustrating the same idea comes from an example suggested by Minsky & Papen [7] called "the mesh". This type of pattern recognition problem is also. like XOR, a non-linearly separable problem. An optimal 354 Figure 4: XOR solved with r=2 (4a) and r=4 (4b) solution involves only three cuts in feature space to separate the two "meshed" cluSten (see Figure Sa). f14W'" 1 b Figure 5: Mesh problem with minimwn cut solution (5a) and Performance Surface(5b) Typical solutions for r=2 in this case tend to use a large number of hidden units to separate the two sets of exemplars (see Figure 5b for a perfonnance surface). For examplet in Figure 6a notice that a typical (based on several runs) Euclidian backprop starting with 16 hidden units has found a solution involving five decision boundaries (lines shown in the plane also representing hidden units) while the r=3 case used primarily three decision boundaries and placed a number of other 355 boundaries redundantly near the center of the meshed region (see Figure 6b) where there is maximum uncertainty about the cluster identification. ~ ~ ~ ID 0 0 .. ID 0 0 • • 0 0 C'lI C'lI 0 0 0 ~ 0 0 0.0 0.2 G.4 0.8 0.8 1.0 b 0.0 0.2 0.4 0.8 0.8 1.0 Figure 6: Mesh solved with r=2 (6a) and r=3 (6b) Speech Recognition. A final case in which large r's may be appropriate is data that has been previously processed with a transformation that produced compact regions requiring separation in the feature space. One example we have looked at involves spoken digit recognition. The first 10 cepstral coefficients of spoken digits ("one" through "ten") were used for input to a network. In this case an advantage is shown for larger r's with smaller training set sizes. Shown in Figure 7 are transfer data for 50 spoken digits replicated in ten different runs per point (bars show standard error of the mean). Transfer shows a training set size effect for both r=2 and r=3, however for the larger r value at smaller training set sizes (10 and 20) note that transfer is enhanced. We speculate that this may be due to the larger r backprop creating discrimination regions that are better able to capture the compactness of the clusters inherent in a small number of training points. 4. Conver&ence Properties It should be generally noted that as r increases. convergence time tends to grow roughly linearly (although this may be problem dependent). Consequently, decreasing r can significantly improve convergence, without much change to the nature of solution. Further, if noise is present decreasing r may reduce it dramatically. Note finally that the gradient for the Minkowski-r back-propagation is nonlinear and therefore more complex for implementing learning procedures. 356 8 ~ 0 co c ! ... 0 ~ co ·c :J 0 ~ 0 ~ ~ ~ 0 u ~ 0 0 C\I 0 ... ---.......... ----.---.. -- - -- 1..~ '~- '. ~ I ~ .Q i t------l--:::~+-·/'- ". . ! / i1 : : R=2 ,.1 ! I t· :' ! ! T.:' 10 replications of 50 transfer POints : i' I: I / ..... ·.C/ L ____ U.____ ----.. o 10 20 30 40 50 TRAINING SET SIZE Figure 7: Digit Recognition Set Size Effect 5. Summary and Conclusion A new procedure which is a variation on the Back-propagation algorithm is derived and simulated in a number of different problem domains. Noise in the target domain may be reduced by using power values less than 2 and the sensitivity of partition planes to the geometry of the problem may be increased with increasing power values. Other types of objective functions should be explored for their potential consequences on network resources and ensuing pattern recognition capabilities. References 1. Rumelhart D. E., Hinton G. E., Williams R., Learning Internal Representations by error propagation. Nature. 1986. 2. Burr D. I. and Hanson S. I .• Knowledge Representation in Connectionist Networks. Bellcore. Technical Report, 3. White. H. Personal Communication. 1987. 4. White, H. Some Asymptotic Results for Learning in Single Hidden Layer Feedforward Network Models. Unpublished Manuscript. 1987. 357 S. Mosteller, F. & Tukey, 1. Robust Estimation Procedures, Addison Wesley, 1980. 6. Tukey, 1. Personal Communication, 1987. 7. Minsky, M. & Papert, S., Perceptrons: An Introduction to Computational Geometry, MIT Press, 1969.
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804 INTRODUCTION TO A SYSTEM FOR IMPLEMENTING NEURAL NET CONNECTIONS ON SIMD ARCHITECTURES Sherryl Tomboulian Institute for Computer Applications in Science and Engineering NASA Langley Research Center, Hampton VA 23665 ABSTRACT Neural networks have attracted much interest recently, and using parallel architectures to simulate neural networks is a natural and necessary application. The SIMD model of parallel computation is chosen, because systems of this type can be built with large numbers of processing elements. However, such systems are not naturally suited to generalized communication. A method is proposed that allows an implementation of neural network connections on massively parallel SIMD architectures. The key to this system is an algorithm that allows the formation of arbitrary connections between the "neurons". A feature is the ability to add new connections quickly. It also has error recovery ability and is robust over a variety of network topologies. Simulations of the general connection system, and its implementation on the Connection Machine, indicate that the time and space requirements are proportional to the product of the average number of connections per neuron and the diameter of the interconnection network. INTRODUCTION Neural Networks hold great promise for biological research, artificial intelligence, and even as general computational devices. However, to study systems in a realistic manner, it is highly desirable to be able to simulate a network with tens of thousands or hundreds of thousands of neurons. This suggests the use of parallel hardware. The most natural method of exploiting parallelism would have each processor simulating a single neuron. Consider the requirements of such a system. There should be a very large number of processing elements which can work in parallel. The computation that occurs at these elements is simple and based on local data. The processing elements must be able to have connections to other elements. All connections in the system must be able to be traversed in parallel. Connections must be added and deleted dynamically. Given current technology, the only type of parallel model that can be constructed with tens of thousands or hundreds of thousands of processors is an SIMD architecture. In exchange for being able to build a system with so many processors, there are some inherent limitations. SIMD stands for single instruction multiple datal which means that all processors can work in parallel, but they must do exactly the same thing at the same time. This machine model is sufficient for the computation required within a neuron, however in such a system it is difficult to implement arbitrary connections between neurons. The Connection Machine2 provides such a model, but uses a device called the router This work was supported by the National Aeronautics and Space Administration under NASA Constract No. NASl-18010-7 while the author was in residence at ICASE. © American Institute of Physics 1988 805 to deliver messages. The router is a complex piece of hardware that uses significant chip area, and without the additional hardware for the router, a machine could be built with significantly more processors. Since one of the objectives is to maximize the number of "neurons" it is desirable to eliminate the extra cost of a hardware router and instead use a software method. Existing software algorithms for forming connections on SIMD machines are not sufficient for the requirements of a neural networks. They restrict the form of graph (neural network) that can be embedded to permutations!·· or sorts5.6combinedwith7, the methods are network specific, and adding a new connection is highly time consuming. The software routing method presented here is a unique algorithm which allows arbitrary neural networks to be embedded in machines with a wide variety of network topologies. The advantages of such an approach are numerous: A new connection can be added dynamically in the same amount of time that it takes to perform a parallel traversal of all connections. The method has error recovery ability in case of network failures. This method has relationships with natural neural models. When a new connection is to be formed, the two neurons being connected are activated, and then the system forms the connection without any knowledge of the "address" of the neuron-processors and without any instruction as to the method of forming the connecting path. The connections are entirely distributed; a processor only knows that connections pass through it - it doesn't know a connection's origin or final destination. Some neural network applications have been implemented on massively parallel architectures, but they have run into restrictions due to communication. An implementation on the Connection Machines discovered that it was more desirable to cluster processors in groups, and have each processor in a group represent one connection, rather than having one processor per neuron, because the router is designed to deliver one message at a time from each processor. This approach is contrary with the more natural paradigm of having one processor represent a neuron. The MPP 9, a massively parallel architecture with processors arranged in a mesh, has been used to implement neural nets10, but because of a lack of generalized communication software, the method for edge connections is a regular communication pattern with all neurons within a specified distance. This is not an unreasonable approach, since within the brain neurons are usually locally connected, but there is also a need for longer connections between groups of neurons. The algorithms presented here can be used on both machines to facilitate arbitrary connections with an irregular number of connections at each processor. MACHINE MODEL As mentioned previously, since we desire to build a system with an large number of processing elements, the only technology currently available for building such large systems is the SIMD architecture model. In the SIMD model there is a single control unit and a very large number of slave processors that can execute the same instruction stream simultaneously. It is possible to disable some processors so that only some execute an instruction, but it is not possible to have two processor performing different instructions at the same time. The processors have exclusively local memory which is small (only a few thousand bits), and they have no facilities for local indirect addressing. In this scheme an Instruction involves both a particular operation code and the local memory 806 address. All processors must do this same thing to the same areas of their local memory at the same time. The basic model of computation is bit-serial - each instruction operates on a bit at a time. To perform multiple bit operations, such as integer addition, requires several instructions. This model is chosen because it requires less hardware logic, and so would allow a machine to be built with a larger number of processors than could otherwise be achieved with a standard word-oriented approach. Of course, the algorithms presented here will also work for machines with more complex instruction abilities; the machine model described satisfies the minimal requirements. An important requirement for connection formation is that the processors are connected in some topology. For instance, the processors might be connected in a grid so that each processor has a North, South, East, and West neighbor. The methods presented here work for a wide variety of network topologies. The requirements are: (1) there must be some path between any two proeessors; (2) every neighbor )ink must be bi-directional, i.e. if A is a neighbor of B, then B must be a neighbor of A; (3) the neighbor relations between processors must have a consistent invertible labeling. A more precise definition of the labeling requirements can be found in 11. It suffices that most networks 12, including grid, hypercube, cube connected cycles1S, shuffle exchange14, and mesh of trees15 are admissible under the scheme. Additional requirements are that the processors be able to read from or write to their neighbors' memories, and that at least one of the processors acts as a serial port between the processors and the controller. COMPUTATIONAL REQUIREMENTS The machine model described here is sufficient for the computational requirements of a neuron. Adopt the paradigm that each processor represents one neuron. While several different models of neural networks exist with slightly different features, they are all fairly well characterized by computing a sum or product of the neighbors values, and if a certain threshold is exceeded, then the processor neuron will fire, Le. activate other neurons. The machine model described here is more efficient at boolean computation, such as described by McCulloch and Pitts16, since it is bit serial. Neural net models using integers and floating point arithmetic 17,18 will also work but will be somewhat slower since the time for computation is proportional to the number of bits of the operands. The only computational difficulty lies in the fact that the system is SIMD, which means that the processes are synchronous. For some neural net models this is sufficient18 however others require asynchronous behavior 17. This can easily be achieved simply by turning the processors on and off based on a specified probability distribution. (For a survey of some different neural networks see 19). CONNECTION ASSUMPTIONS Many models of neural networks assume fully connected systems. This model is considered unrealistic, and the method presented here will work better for models that contain more sparsely connected systems. While the method will work for dense connections, the time and space required is proportional to 807 the number of edges, and becomes prohibitively expensive. Other than the sparse assumptions, there are no restrictions to the topological form of the network being simulated. For example, multiple layered systems, slightly irregular structures, and completely random connections are all handled easily. The system does function better if there is locality in the neural network. These assumptions seem to fit the biological model of neurons. THE CONNECTION FORMATION METHOD A fundamental part of a neural network implementation is the realization of the connections between neurons. This is done using a software scheme first presented in 11,20. The original method was intended for realizing directed graphs in SIMD architectures. Since a neural network is a graph with the neurons being vertices and the connections being arcs, the method maps perfectly to this system. Henceforth the terms neuron and vertex and the terms arc and connection will be used interchangeably. The software system presented here for implementing the connections has several parts. Each processor will be assigned exactly one neuron. (Of course some processors may be "free" or unallocated, but even "free" processor participate in the routing process.) Each connection will be realized as a path in the topology of processors. A labeling of these paths in time and space is introduced which allows efficient routing algorithms and a set-up strategy is introduced that allows new connections to be added quickly. The standard computer science approach to forming the connection would be to store the addresses of the processors to which a given neuron is connected. Then, using a routing algorithm, messages could be passed to the processors with the specified destination. However, the SIMD architecture does not lend itself to standard message passing schemes because processors cannot do indirect addressing, so buffering of values is difficult and costly. Instead, a scheme is introduced which is closer to the natural neuron-synapse structures. Instead of having an address for each connection, the connection is actually represented as a fixed path between the processors, using time as a virtual dimension. The path a connection takes through the network of processors is statically encoded in the local memories of the neurons that it passes through. To achieve this, the following data structures will be resident at each processor. ALLOCATED ---- boolean flag indicating whether this processor is assigned a vertex (neuron) in the graph VERTEX LABEL --- label of graph vertex (neuron) HAS_NEIGHBOR[l .. neighbor_limit] flag indicating the existence of neighbors SLOTS[l .. T] OF arc path information START----------new arc starts here DIRECTION------direction to send {l .. neighbor_limit.FREE} END-----------arc ends here ARC LABEL-----label of arc 808 The ALLOCATED and VERTEX LABEL field indicates that the processor has been assigned a vertex in the graph (neuron). The HAS NEIGHBOR field is used to indicate whether a physical wire exists in the particular direction; it allows irregular network topologies and boundary conditions to be supported. The SLOTS data structure is the key to realizing the connections. It is used to instruct the processor where to send a message and to insure that paths are constructed in such a way that no collisions will occur. SLOTS is an array with T elements. The value T is called the time quantum. Traversing all the edges of the embedded graph in parallel will take a certain amount of time since messages must be passed along through a sequence of neighboring processors. Forming these parallel connections will be considered an uninterruptable operation which will take T steps. The SLOTS array is used to tell the processors what they should do on each relative time position within the time quantum. One of the characteristics of this algorithm is that a fixed path is chosen to represent the connection between two processors, and once chosen it is never changed. For example, consider the grid below. I I I I I --A--B--C--D--E-I I I I I --F--G--H--I--J-I I I I I Fig. 1. Grid Example If there is an arc between A and H, there are several possible paths: EastEast-South, East-South-East, and South-East-East. Only one of these paths will be chosen between A and H, and that same path will always be used. Besides being invariant in space, paths are also invariant in time. As stated above, traversal is done within a time quantum T. Paths do no have to start on time 1, but can be scheduled to start at some relative offset within the time quantum. Once the starting time for the path has been fixed, it is never changed. Another requirement is that a message can not be buffered, it must proceed along the specified directions without interruption. For example, if the path is of length 3 and it starts at time 1, then it will arrive at time 4. Alternatively, if it starts at time 2 it will arrive at time 5. Further, it is necessary to place the paths so that no collisions occur; that is, no two paths can be at the same processor at the same instant in time. Essentially time adds an extra dimension to the topology of the network, and within this spacetime network all data paths must be non-conflicting. The rules for constructing paths that fulfill these requirements are listed below . • At most one connection can enter a processor at a given time, and at most one connection can leave a processor at a given time. It is possible to have both one coming and one going at the same time. Note that this does not mean that a processor can have only one connection; it means that it can have only one connection during anyone of the T time steps. It can have as many as T connections going through it . • Any path between two processors (u,v) repr('senting a connection must consist of steps at contiguous times. For example, if the path from processor u to processor v is u,f,g,h,v, then if the arc from u-f is assigned time 1, f-g must have time 2, g-h time 3, and h-v time 4. Likewise if u-f occurs at time 5, then arc h-v will occur time 8. 809 When these rules are used when forming paths, the SLOTS structure can be used to mark the paths. Each path goes through neighboring processors at successive time steps. For each of these time steps the DffiECTION field of the SLOTS structure is marked, telling the processor which direction it should pass a message if it receives it on that time. SLOTS serves both to instruct the processors how to send messages, and to indicate that a processor is busy at a certain time slot so that when new paths are constructed it can be guaranteed that they won't conflict with current paths. Consider the following example. Suppose we are given the directed graph with vertices A,B,C,D and edges A - > C, B - > C,B - > D, and D - > A. This is to be done where A,B,C, and D have been assigned to successive elements of a linear array. (A linear array in not a good network for this scheme, but is a convenient source of examples.) Lo~ical Connections Faa. 2. GIapb Example A.B.C.D are successive members in a linear array 1---2---3---4 A---B---C---D First. A ->C can be completed with the map East-East. so Slots[A][1].direction = E. Slots[B][2].direction=E. Slots[C][2].end = 1 . B->C can be done with the map East. it can start at time 1. since Slots[B] [1] . direction and Slots[C] [1].end are free. B->D goes through C then to D. its map is East-East. B is occupied at time 1 and 2. It is free at time 3. so Slots[B] [3].direction = E. Slots[C] [4].direction = E. Slots[D] [4].end = 1. D->A must go through C.B.A. using map West-West-West. D is free on time 1. C is free on time 2. but B is occupied on time 3. D is free on time 2. but C is occupied on time 3. It can start from D at time 3. Slots[D] [3].direction = W. Slots[C] [4] .direction = W. Slots[B] [5].direction = W. Slots [A] [5].end=1 810 Every processor acts as a conduit for its neighbors messages. No processor knows where any message is going to or coming from, but each processor knows what it must do to establish the local connections. The use of contiguous time slots is vital to the correct operation of the system. If all edge-paths are established according to the above rules, there is a simple method for making the connections. The paths have been restricted so that there will be no collisions, and paths' directions use consecutive time slots. Hence if all arcs at time i send a message to their neighbors, then each processor is guaranteed no more than 1 message coming to it. The end of a path is specified by setting a separate bit that is tested after each message is received. A separate start bit indicates when a path starts. The start bit is needed because the SLOTS array just tells the processors where to send a message, regardless of how that message arrived. The start array indicates when a message originates, as opposed to arriving from a neighbor. The following algorithm is basic to the routing system. for i = time 1 to T FORALL processors /* if an arc starts or is passing through at this time*/ if SLOT[i] . START = 1 or active = 1 for j=1 to neighbor-limit if SLOT[i].direction= j write message bit to in-box of neighbor j: set active = 0: FORALL processor that just received a message if end[i] move in-box to message-destination; else move in-box to out-box: set active bit = 1: This code follows the method mentioned above. The time slots are looped through and the messages are passed in the appropriate directions as specified in the SLOTS array. Two bits, in-box and out-box, are used for message passing so that an out-going message won't be overwritten by an in-coming message before it gets transferred. The inner loop lor j = 1 to neighbor limit checks each of the possible neighbor directions and sends the message to the correct neighbor. For instance, in a grid the neighbor limit is 4, for North, South, East, and West neighbors. The time complexity of data movement is O(T times neighbor-limi t) . SETTING UP CONNECTIONS One of the goals in developing this system was to have a method for adding new connections quickly. Paths are added so that they don't conflict with any previously constructed path. Once a path is placed it will not be re-routed 811 by the basic placement algorithm; it will always start at the same spot at the same time. The basic idea of the method for placing a connection is to start from the source processor and in parallel examine all possible paths outward from it that do not conflict with pre-established paths and which adhere to the sequential time constraint. As the trial paths are flooding the system, they are recorded in temporary storage. At the end of this deluge of trial paths all possible paths will have been examined. If the destination processor has been reached, then a path exists under the current time-space restrictions. Using the stored information a path can be backtraced and recorded in the SLOTS structure. This is similar to the Lee-Moore routing algorithm21•22 for finding a path in a system, but with the sequential time restriction. For example, suppose that the connection (u,v) is to be added. First it is assumed that processors for u and v have already been determined, otherwise (as a simplification) assume a random allocation from a pool of free processors. A parallel breadth-first search will be performed starting from the source processor. During the propagation phase a processor which receives a message checks its SLOTS array to see if they are busy on that time step, if not it will propagate to its neighbors on the next time step. For instance, suppose a trial path starts at time 1 and moves to a neighboring processor, but that neighbor is already busy at time 1 (as can be seen by examining the DIRECTION-SLOT.) Since a path that would go through this neighbor at this time is not legal, the trial path would commit suicide, that is, it stops propagating itself. If the processor slot for time 2 was free, the trial path would attempt to propagate to all of its' neighbors at time 3. Using this technique paths can be constructed with essentially no knowledge of the relative locations of the "neurons" being connected or the underlying topology. Variations on the outlined method, such as choosing the shortest path, can improve the choice of paths with very little overhead. If the entire network were known ahead of time, an off-line method could be used to construct the paths more efficiently; work on off-line methods is underway. However, the simple elegance of this basic method holds great appeal for systems that change slowly over time in unpredictable ways. PERFORMANCE Adding an edge (assuming one can be added), deleting any set of edges, or traversing all the edges in parallel, all have time complexity O(T x neighborlimit). If it is assumed that neighbor limit is a small constant then the complexity is O(T). Since T is related both to the time and space needed, it is a crucial factor in determining the value of the algorithms presented. Some analytic bounds on T were presented inll, but it is difficult to get a tight bound on T for general interconnection networks and dynamically changing graphs. A simulator was constructed to examine the behavior of the algorithms. Besides the simulated data, the algorithms mentioned were actually implemented for the Connection Machine. The data produced by the simulator is consistent with that produced by the real machine. The major result is that the size of T appears proportional to the average degree of the graph times the diameter of the interconnection network20• 812 FURTHER RESEARCH This paper has been largely concerned with a system that can realize the connections in a neural network when the two neurons to be joined have been activated. The tests conducted have been concerned with the validity of the method for implementing connections, rather than with a full simulation of a neural network. Clearly this is the next step. A natural extension of this method is a system which can form its .own connections based solely on the activity of certain neurons, without having to explicitly activate the source and destination neurons. This is an exciting avenue, and further results should be forthcoming. Another area of research involves the formation of branching paths. The current method takes an arc in the neural network and realizes it as a unique path in space-time. A variation that has similarities to dendritic structure would allow a path coming from a neuron to branch and go to several target neurons. This extension would allow for a much more economical embedding system. Simulations are currently underway. CONCLUSIONS A method has been outlined which allows the implementation of neural nets connections on a class of parallel architectures which can be constructed with very large numbers of processing elements. To economize on hardware so as to maximize the number of processing element buildable, it was assumed that the processors only have local connections; no hardware is provided for communication. Some simple algorithms have been presented which allow neural nets with arbitrary connections to be embedded in SIMD architectures having a variety of topologies. The time for performing a parallel traversal and for adding a new connection appears to be proportional to the diameter of the topology times the average number of arcs in the graph being embedded. In a system where the topology has diameter O(logN), and where the degree of the graph being embedded is bounded by a constant, the time is apparently O(logN). This makes it competitive with existing methods for SIMD routing, with the advantages that there are no apriori requirements for the form of the data, and the topological requirements are extremely general. Also, with our approach new arcs can be added without reconfiguring the entire system. The simplicity of the implementation and the flexibility of the method suggest that it could be an important tool for using SIMD architectures for neural network simulation. BIBLIOGRAPHY 1. M.J. Flynn, "Some computer organizations and their effectiveness", IEEE Trans Comput., vol C-21, no.9, pp. 948-960. 2. W. Hillis, "The Connection Machine", MIT Press, Cambridge, Mass, 1985. 3. D. Nassimi, S. Sahni, "Parallel Algorithms to Set-up the Benes Permutation Network", Proc. Workshop on Interconnection Networks for Parallel and Distributed Processing, April 1980. 4. D. Nassimi, S. Sahni, "Benes Network and Parallel Permutation Algorithms", IEEE Transactions on Computers, Vol C-30, No 5, May 1981. 5. D. Nassimi, S. Sahni, "Parallel Permutation and Sorting Algorithms and a 813 New Generalized Connection Network" , JACM, Vol. 29, No.3, July 1982 pp. 642-667 6. K.E. Batcher, "Sorting Networks and their Applications", The Proceedings of AFIPS 1968 SJCC, 1968, pp. 307-314. 7. C. Thompson, "Generalized connection networks for parallel processor intercommunication", IEEE Tran. Computers, Vol C, No 27, Dec 78, pp. 1119-1125. 8. Nathan H. Brown, Jr., "Neural Network Implementation Approaches for the Connection Machine", presented at the 1987 conference on Neural Information Processing Systems - Natural and Synthetic. 9. K.E. Batcher, "Design of a massively parallel processor", IEEE Trans on Computers, Sept 1980, pp. 836-840. 10. H.M. Hastings, S. Waner, "Neural Nets on the MPP" , Frontiers of Massively Parallel Scientific Computation, NASA Conference Publication 2478, NASA Goddard Space Flight Center, Greenbelt Maryland, 1986. 11. S. Tomboulian, "A System for Routing Arbitrary Communication Graphs on SIMD Architectures", Doctoral Dissertation, Dept of Computer Science, Duke University, Durham NC. 12. T. Feng, "A Survey of Interconnection Networks", Computer, Dec 1981, pp.12-27. 13. F. Preparata and J. Vuillemin, "The Cube Connected Cycles: a Versatile Network for Parallel Computation", Comm. ACM, Vol 24, No 5 May 1981, pp. 300-309. 14. H. Stone, "Parallel processing with the perfect shuffle", IEEE Trans. Computers, Vol C, No 20, Feb 1971, pp. 153-161. 15. T. Leighton, "Parallel Computation Using Meshes of Trees", Proc. International Workshop on Graph Theory Concepts in Computer Science, 1983. 16. W.S. McCulloch, and W. Pitts, "A Logical Calculus of the Ideas Imminent in Nervous Activity," Bulletin of Mathematical Biophysics, Vol 5, 1943, pp.115133. 17. J.J. Hopfield, "Neural networks and physical systems with emergent collective computational abilities", Prot!. Natl. Aca. Sci., Vol 79, April 1982, pp. 2554-2558. 18. T. Kohonen, "Self-Organization and Associative Memory, Springer-Verlag, Berlin, 1984. 19. R.P. Lippmann, "An Introduction to Computing with Neural Nets", IEEE AASP, Apri11987, pp. 4-22. 20. S. Tomboulian, "A System for Routing Directed Graphs on SIMD Architectures", ICASE Report No. 87-14, NASA Langley Research Center, Hampton, VA. 21. C.Y. Lee, "An algorithm for path connections and its applications", IRE Trans Elec Comput, Vol. EC-I0, Sept. 1961, pp. 346-365. 22. E. F. Moore, "Shortest path through a maze", A nnals of Computation Laboratory, vol. 30. Cambridge, MA: Harvard Univ. Press, 1959, pp.285-292.
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72 ANALYSIS AND COMPARISON OF DIFFERENT LEARNING ALGORITHMS FOR PATTERN ASSOCIATION PROBLEMS J. Bernasconi Brown Boveri Research Center CH-S40S Baden, Switzerland ABSTRACT We investigate the behavior of different learning algorithms for networks of neuron-like units. As test cases we use simple pattern association problems, such as the XOR-problem and symmetry detection problems. The algorithms considered are either versions of the Boltzmann machine learning rule or based on the backpropagation of errors. We also propose and analyze a generalized delta rule for linear threshold units. We find that the performance of a given learning algorithm depends strongly on the type of units used. In particular, we observe that networks with ±1 units quite generally exhibit a significantly better learning behavior than the corresponding 0,1 versions. We also demonstrate that an adaption of the weight-structure to the symmetries of the problem can lead to a drastic increase in learning speed. INTRODUCTION In the past few years, a number of learning procedures for neural network models with hidden units have been proposed 1 ,2. They can all be considered as strategies to minimize a suitably chosen error measure. Most of these strategies represent local optimization procedures (e.g. gradient descent) and therefore suffer from all the problems with local m1n1ma or cycles. The corresponding learning rates, moreover, are usually very slow. The performance of a given learning scheme may depend criticallyon a number of parameters and implementation details. General analytical results concerning these dependences, however, are practically non-existent. As a first step, we have therefore attempted to study empirically the influence of some factors that could have a significant effect on the learning behavior of neural network systems. Our preliminary investigations are restricted to very small networks and to a few simple examples. Nevertheless, we have made some interesting observations which appear to be rather general and which can thus be expected to remain valid also for much larger and more complex systems. NEURAL NETWORK MODELS FOR PATTERN ASSOCIATION An artificial neural network consists of a set of interconnected units (formal neurons). The state of the i-th unit is described by a variable S. which can be discrete (e.g. S. = 0,1 or S. = ±1) or continuous (e.l. 0 < S. < 1 or -1 < S. < +ll, and each ~onnection j-7i carries a weight- W.1. which can be 1positive, zero, or negative. 1J © American Institute of Physics 1988 73 The dynamics of the network is determined by a local update rule, S.(t+l) 1 = HI W . . S . (t)) j 1J J (1) where f is a nonlinear activation function, specifically a threshold function in the case of discrete units and a sigmoid-type function, e.g. (2) or (3) respectively, in the case of continuous units. The individual units can be given different thresholds by introducing an extra unit which always has a value of 1. If the network is supposed to perform a pattern association task, it is convenient to divide its units into input units, output units, and hidden units. Learning then consists in adjusting the weights in such a way that, for a given input pattern, the network relaxes (under the prescribed dynamics) to a state in which the output units represent the desired output pattern. Neural networks learn from examples (input/output pairs) which are presented many times, and a typical learning procedure can be viewed as a strategy to minimize a suitably defined error function F. In most cases, this strategy is a (stochastic) gradient descent method: To a clamped input pattern, randomly chosen from the learning examples, the network produces an output pattern {O . }. This is compared with the desired output, say {T . }, and the erfor F( {O. }, {T . }) is calculated . Subsequently, each 1weight is changed by ~an am~unt proportional to the respective gradient of F, b.W .. ~J of = -r} -oW .. ~J (4) and the procedure is repeated for a new learning example until F is minimized to a satisfactory level. In our investigations, we shall consider two different types of learning schemes. The first is a deterministic version of the Boltzmann machine learning rule! and has been proposed by Yann Le Cun2 • It applies to networks with symmetric weights, W .. = W .. , so that an ~J J ~ energy E(~) == - I W .. S. S . (i ,j) ~J ~ J (5) can be associated with each state S = {S.}. If X refers to the net1 work state when only the input units are clamped and Y to the state when both the input and output units are clamped, the error function 74 is defined as F = E c:~) E QO and the gradients are simply given by of - -- = Y. Y. oW. . 1 J 1J x. X. 1 J (6) (7) The second scheme, called backpropagation or generalized delta rule 1 ,3, probably represents the most widely used learning algorithm. In its original form, it applies to networks with feedforward connections only, and it uses gradient descent to minimize the mean squared error of the output signal, F = -21 L (T . - 0.)2 .11 1 (8) For a weight W .. from an (input or hidden) unit j to an output unit i, we simply ha~ (9 ) where f' is the derivative of the nonlinear activation function introduced in Eq. (1), and for weights which do not connect to an output unit, the gradients can successively be determined by applying the chain rule of differentiation. In the case of discrete units, f is a threshold function, so that the backpropagation algorithm described above cannot be applied. We remark, however, that the perceptron learning rUle 4 , ~W .. = £(T. - O.)S. 1J 1 1 J (10) is nothing else than Eq. (9) with f' replaced by a constant £. Therefore, we propose that a generalized delta rule for linear threshold units can be obtained if f' is replaced by a constant £ in all the backpropagation expressions for of/oW ... This generalization of the perceptron rule is, of course, not u1dque. In layered networks, e.g., the value of the constant which replaces f' need not be the same for the different layers. ANALYSIS OF LEARNING ALGORITHMS The proposed learning algorithms suffer from all the problems of gradient descent on a complicated landscape. If we use small weight changes, learning becomes prohibitively slow, while large weight changes inevitably lead to oscillations which prevent the algorithm from converging to a good solution. The error surface, moreover, may contain many local minima, so that gradient descent is not guaranteed to find a global minimum. 75 There are several ways to improve a stochastic gradient descent procedure. The weight changes may, e.g., be accumulated over a number of learning examples before the weights are actually changed. Another often used method consists in smoothing the weight changes by overrelaxation, ~W .. (k+1) 1J of = -~ ~W + a ~W .. (k) a .. 1J 1J (11) where ~W .. (k) refers to the weight change after the presentation of the k-th11earning example (or group of learning examples, respectively). The use of a weight decay term, ~W .. 1J of = -11 ~W BW .. a .. 1J 1J (12) prevents the algorithm from generating very large weights which may create such high barriers that a solution cannot be found in reasonable time. Such smoothing methods suppress the occurrence of oscillations, at least to a certain extent, and thus allow us to use higher learning rates. They cannot prevent, however, that the algorithm may become trapped in bad local minimum. An obvious way to deal with the problem of local minima is to restart the algorithm with different initial weights or, equivalently, to randomize the weights with a certain probability p during the learning procedure. More sophisticated approaches involve, e.g., the use of hill-climbing methods. The properties of the error-surface over the weight space not only depend on the choice of the error function F, but also on the network architecture, on the type of units used, and on possible restrictions concerning the values which the weights are allowed to assume. The performance of a learning algorithm thus depends on many factors and parameters. These dependences are conveniently analyzed in terms of the behavior of an appropriately defined learning curve. For our small examples, where the learning set always consists of all input/output cases, we have chosen to represent the performance of a learning procedure by the fraction of networks that are "perfect" after the presentation of N input patterns. (Perfect networks are networks which for every input pattern produce the correct output). Such learning curves give us much more detailed information about the behavior of the system than, e. g., averaged quantities like the mean learning time. RESULTS In the following, we shall present and discuss some representative results of our empirical study. All learning curves refer to a set of 100 networks that have been exposed to the same learning procedure, where we have varied the initial weights, or the sequence 76 of learning examples, or both. With one exception (Figure 4), the sequences of learning examples are always random. A prototype pattern association problem is the exclusive-or (XOR) problem. Corresponding networks have two input units and one output unit. Let us first consider an XOR-network with only one hidden unit, but in which the input units also have direct connections to the output unit. The weights are symmetric, and we use the deterministic version of the Boltzmann learning rule (see Eqs. (5) to (7)). Figure 1 shows results for the case of tabula rasa initial conditions, i.e. the initial weights are all set equal to zero. If the weights are changed after every learning example, about 2/3 of the networks learn the problem with less than 25 presentations per pattern (which corresponds to a total number of 4 x 25 = 100 presentations). The remaining networks (about 1/3), however, never learn to solve the XOR-problem, no matter how many input/output cases are presented. This can be understood by analyzing the corresponding evolution-tree in weight-space which contains an attractor consisting of 14 "non-perfect" weight-configurations. The probability to become trapped by this attractor is exactly 1/3. If the weight changes are accumulated over 4 learning examples, no such attractor 100 I I I I en 80 ~ a:: 0 0 0 0 0 0 0 ij 0 0 0 ~ ••• • • • i • • • • • • • • • .60 I000 w 0 z • 0 0 .• 0 u • 00 w 40 0 0 lL. • a:: • 0 W Q. 0 20 ..... ·0 ~ 0 0 ~ 0 .o.~. I I I I 0 20 40 60 80 100 #: PRESENTATIONS /PATTERN Fig. 1. Learning curves for an XOR-network with one hidden unit (deterministic Boltzmann learning, discrete ±I units, initial weights zero). Full circles: weights changed after every learning example; open circles: weight changes accumulated over 4 learning examples. 77 seems to exist (see Fig. 1), but for some networks learning at least takes an extremely long time . The same saturation effect is observed with random initial weights (uniformly distributed between -1 and +1), see Fig. 2. Figure 2 also exhibits the difference in learning behavior between networks with ±1 units and such with 0,1 units. In both cases, weight randomization leads to a considerably improved learning behavior. A weight decay term, by the way, has the same effect. The most striking observation, however, is that ±1 networks learn much faster than 0,1 networks (the respective average learning times differ by about a factor of 5). In this connection, we should mention that ~ = 0.1 is about optimal for 0,1 units and that for ±1 networks the learning behavior is practically independent of the value of ~. It therefore seems that ±1 units lead to a much more well-behaved error-surface than 0,1 units. One can argue, of course, that a discrete 0,1 model can always be translated into a ±1 model, but this would lead to an energy function which has a considerably more complicated weight dependence than Eq. (5). 100 en 80 ~ a::: 0 3: .... 60 w z .... u 40 w lL. a::: w a.. ~ 20 0 0 2 5 10 20 50 100 200 1000 # PRESENTATIONS / PATTERN Fig. 2. Learning curves for an XOR-network with one hidden unit (deterministic Boltzmann learning, initial weights random, weight changes accumulated over 5 learning examples). Circles: discrete ±1 units, ~ = 1; triangles: discrete 0,1 units, ~ = 0.1; broken curves: without weight randomization; solid curves: with weight randomization (p = 0.025). 78 Figures 3 and 4 refer to a feedforward XOR-network with 3 hidden units, and to backpropagation or generalized delta rule learning. In all cases we have included an overrelaxation (or momentum) term with a = 0.9 (see Eq. (11». For the networks with continuous units we have used the activation functions given by Eqs. (2) and (3), respectively, and a network was considered "perfect" if for all input/output cases the error was smaller than 0.1 in the 0,1 case, or smaller than 0.2 in the ±1 case, respectively. In Figure 3, the weights have been changed after every learning example, and all curves refer to an optimal choice of the only remaining parameter, £. or ", respectively. For discrete as well as for continuous units, the ±1 networks again perform much better than their 0,1 counterparts. In the continuous case, the average learning times differ by about a factor of 7, and in the discrete case the discrepancy is even more pronounced. In addition, we observe that in ±1 networks learning with the generalized delta rule for discrete units is about twice as fast as with the backpropagation algorithm for continuous units. 100~--~--~----~----~~~~--~--~ en 80 ~ a:: 0 ~ I60 w Z I0 40 w lL. a:: w a. ~ 20 0 O~----~--~------~----~----~~~~--~ 5 10 20 50 100 200 500 1000 :# PRESENTATIONS / PATTERN Fig. 3. Learning curves for an XOR-network with three hidden units (backpropagation/generalized delta rule, initial weights random, weights changed after every learning example). Open circles: discrete ±1 units, £. = 0.05; open triangles: discrete 0,1 units, £. = 0.025; full circles: continuous ±1 units, " = 0.125; full triangles; continuous 0,1 units, " = 0.25. 79 In Figure 4, the weight changes are accumulated over all 4 input/output cases, and only networks with continuous units are considered. Also in this case, the ±1 units lead to an improved learning behavior (the optimal Il-values are about 2.5 and 5.0, respectively). They not only lead to significantly smaller learning times, but ±1 networks also appear to be less sensitive with respect to a variation of 11 than the corresponding 0,1 versions. The better performance of the ±1 models with continuous units can partly be attributed to the steeper slope of the chosen activation function, Eq. (3). A comparison with activation functions that have the same slope, however, shows that the networks with ±1 units still perform significantly better than those with 0,1 units. If the weights are updated after every learning example, e.g., the reduction in learning time remains as large as a factor of 5. In the case of backpropagation learning, the main reason for the better performance of ±1 units thus seems to be related to the fact that the algorithm does not modify weights which emerge from a unit with value zero. Similar observations have been made by Stornetta and Huberman,s who further find that the discrepancies become even more pronounced if the network size is increased. 100 "1 = 5.0 CI) 80 ~ a: 0 ~ I60 w z I-u 40 w lL. a: w a.. ~ 20 0 0 0 50 100 150 200 250 # PRESENTATIONS I PATTERN Fig. 4. Learning curves for an XOR-network with three hidden units (backpropagation, initial weights random, weight changes accumulated over all 4 input/output cases). Circles: continuous ±1 units; triangles: continuous 0,1 units. 80 In Figure 5, finally, we present results for a network that learns to detect mirror symmetry in the input pattern. The network consists of one output, one hidden, and four input units which are ' also directly connected to the output unit. We use the deterministic version of Boltzmann learning and change the weights after every presentation of a learning pattern . If the weights are allowed to assume arbitrary values, learning is rather slow and on average requires almost 700 presentations per pattern. We have observed, however, that the algorithm preferably seems to converge to solutions in which geometrically symmetric weights are opposite in sign and almost equal in magnitude (see also Ref. 3). This means that the symmetric input patterns are automatically treated as equivalent, as their net input to the hidden as well as to the output unit is zero. We have therefore investigated what happens if the weights are forced to be antisymmetric from the beginning. (The learning procedure, of course, has to be adjusted such that it preserves this antisymmetry). Figure 5 shows that such a problem-adapted weightstructure leads to a dramatic decrease in learning time. 100 • 0 • 0 • 0 en 80 • 0 ~ • 0 a:: • 0 3: • 0 • 0 I60 w 0 z • • 0 l• 0 (,) w • LL. 40 • a:: • 0 lLI 0 a.. 0 ~ • 0 0 20 0 • 0 • 0 0 2 5 10 20 50 100 200 500 2000 # PRESENTATIONS I PATTERN Fig. 5. Learning curves for a symmetry detection network with 4 input units and one hidden unit (deterministic Boltzmann learning, 11 = 1, discrete ±1 units, initial weights random, weights changed after every learning example). Full circles: symmetry-adapted weights; open circles: arbitrary weights, weight randomization (p = 0.015). 81 CONCLUSIONS The main results of our empirical study can be summarized as follows: - Networks with ±1 units quite generally exhibit a significantly faster learning than the corresponding 0,1 versions. - In addition, ±1 networks are often less sensitive to parameter variations than 0,1 networks. - An adaptation of the weight-structure to the symmetries of the problem can lead to a drastic improvement of the learning behavior. Our qualitative interpretations seem to indicate that the observed effects should not be restricted to the small examples considered in this paper. It would be very valuable, however, to have corresponding analytical results. REFERENCES 1. "Parallel Distributed Processing: Explorations in the Microstructure of Cognition", vol. 1: "Foundations", ed. by D.E. Rumelhart and J.L. McClelland (MIT Press, Cambridge), 1986, Chapters 7 & 8. 2. Y. Ie Cun, in "Disordered Systems and Biological Organization", ed . by E. Bienenstock, F. Fogelman Soulie, and G. Weisbuch (Springer, Berlin), 1986, pp. 233-240. 3. D.E. Rumelhart, G.E. Hinton, and R.J. Williams, Nature 323, 533 (1986). 4. M.L. Minsky and S. Papert, "Perceptrons" (MIT Press, Cambridge), 1969. 5. W.S. Stornetta and B.A. Huberman, IEEE Conference on "Neural Networks", San Diego, California, 21-24 June 1987.
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410 NEURAL CONTROL OF SENSORY ACQUISITION: THE VESTIBULO-OCULAR REFLEX. Michael G. Paulin, Mark E. Nelson and James M. Bower Division of Biology California Institute of Technology Pasadena, CA 91125 ABSTRACT We present a new hypothesis that the cerebellum plays a key role in actively controlling the acquisition of sensory infonnation by the nervous system. In this paper we explore this idea by examining the function of a simple cerebellar-related behavior, the vestibula-ocular reflex or VOR, in which eye movements are generated to minimize image slip on the retina during rapid head movements. Considering this system from the point of view of statistical estimation theory, our results suggest that the transfer function of the VOR, often regarded as a static or slowly modifiable feature of the system, should actually be continuously and rapidly changed during head movements. We further suggest that these changes are under the direct control of the cerebellar cortex and propose experiments to test this hypothesis. 1. INTRODUCTION A major thrust of research in our laboratory involves exploring the way in which the nervous system actively controls the acquisition of infonnation about the outside world. This emphasis is founded on our suspicion that the principal role of the cerebellum, through its influence on motor systems, is to monitor and optimize the quality of sensory information entering the brain. To explore this question, we have undertaken an investigation of the simplest example of a cerebellar-related motor activity that results in improved sensory inputs, the vestibulo-ocular reflex (VOR). This reflex is responsible for moving the eyes to compensate for rapid head movements to prevent retinal image slip which would otherwise significantly degrade visual acuity (Carpenter, 1977). 2. VESTIBULO-OCULAR REFLEX (VOR) The VOR relies on the vestibular apparatus of the inner ear which is an inertial sensor that detects movements of the head. Vestibular output caused by head movements give rise to compensatory eye movements through an anatomically well described neural pathway in the brain stem (for a review see Ito, 1984). Visual feedback also makes an important contribution to compensatory eye movements during slow head movements, Neural Control of Sensory Acquisition 411 but during rapid head movements with frequency components greater than about 1Hz, the vestibular component dominates (Carpenter, 1977). A simple analysis of the image stabilization problem indicates that during head rotation in a single plane, the eyes should be made to rotate at equal velocity in the opposite direction. This implies that, in a simple feedforward control model, the VOR transfer function should have unity gain and a 1800 phase shift. This would assure stabilized retinal images of distant objects. It turns out, however, that actual measurements reveal the situation is not this simple. Furman, O'Leary and Wolfe (1982), for example, found that the monkey VOR has approximately unity gain and 1800 phase shift only in a narrow frequency band around 2Hz. At 4Hz the gain is too high by a factor of about 30% (fig. 1). 1.2 ~ -< C) 1.0 0.8 ,...., I til f f1d1t H f~ff 1\ j ~ 5 0 I Hf lIJ (f) ~O n.. -5 2 3 4 5 2 3 4 5 FREQUENCY (Hz) FREQUENCY (Hz) Figure 1: Bode gain and phase plots for the transfer function of the horizontal component of the VOR of the alert Rhesus monkey at high frequencies (Data from Furman et al. (1982». Given the expectation of unity gain, one might be tempted to conclude from the monkey data that the VOR simply does not perform well at high frequencies. But 4Hz is not a very high frequency for head movements, and perhaps it is not the VOR which is performing poorly, but the simplified analysis using classical control theory. In this paper, we argue that the VOR uses a more sophisticated strategy and that the "excessive" gain in the system seen at higher frequencies actually improves VOR performance. 3. OPTIMAL ESTIMATION In order to understand the discrepancy between the predictions of simple control theory models and measured VOR dynamics, we believe it is necessary to take into account more of the real world conditions under which the VOR operates. Examples include noisy head velocity measurements, conduction delays and multiple, possibly conflicting, measurements of head velocity, acceleration, muscle contractions, etc., generated by different sensory modalities. The mathematical framework that is appropriate for analyz412 Paulin, Nelson and Bower ing problems of this kind is stochastic state-space dynamical systems theory (Davis and Vinter, 1985). This framework is an extension of classical linear dynamical systems theory that accommodates multiple inputs and outputs, nonlinearities, time-varying dynamics, noise and delays. One area of application of the state space theory has been in target tracking, where the basic principle involves using knowledge of the dynamics of a target to estimate its most probable trajectory given imprecise data. The VOR can be viewed as a target tracking system whose target is the "world", which moves in head coordinates. We have reexamined the VOR from this point of view. The Basic VOR. To begin our analysis of the VOR we have modeled the eye-head-neck system as a damped inverted pendulum with linear restoring forces (fig. 2) where the model system is driven by random (Gaussian white) torque. Within this model, we want to predict the correct compensatory "eye" movements during "head" movements to stabilize the direction in which the eye is pointing. Figure 2 shows the amplitude spectrum of head velocity for this model. In this case, the parameters of the model result in a system that has a natural resonance in the range of 1 to 2 Hz and attenuates higher frequencies. 20 • • ••• SIS z, s. u • '. • •• i. I 0.1 1.0 FREQUENCY Figure 2: Amplitude spectrum of model head velocity. We provide noisy measurements of "head" velocity and then ask what transfer function, or filter, will give the most accurate "eye" movement compensation? This is an estimation problem and, for Gaussian measurement error, the solution was discovered by Kalman and Bucy (1961). The optimal fIlter or estimator is often called the KalmanBucy filter. The gain and phase plots of the optimal filter for tracking movements of the inverted pendulum model are shown in figure 3. It can be seen that the gain of the optimal estimator for this system peaks near the maximum in the spectrum of "head-neck" velocity (fig. 2). This is a general feature of optimal filters. Accordingly, to accurately compensate for head movement in this system, the VOR would need to have a frequency dependent gain. ~ 20 ~ 0 0 5 ~ -20 ~ Neural Control of Sensory Acquisition 413 bO 0 So j ~ tI) ~90 0.1 1.0 10.0 .0 Figure 3: Bode gain plot (left) and phase plot (right) of an optimal estimator for tracking the inverted pendulum using noisy data. Time Varying dynamics and the VOR So far we have considered our model for VOR optimization only in the simple case of a constant head-neck velocity power spectrum. Under natural conditions, however, this spectrum would be expected to change. For example, when gait changes from walking to running, corresponding changes in the VOR transfer function would be necessary to maintain optimal performance. To explore this, we added a second inverted pendulum to our model to simulate body dynamics. We simulated changes in gait by changing the resonant frequency of the trunk. Figure 4 compares the spectra of head-neck velocity with two different trunk parameters. As in the previous example, we then computed transfer functions of the optimal filters for estimating head velocity from noisy measurements in these two cases. The gain and phase characteristics of these filters are also shown in Figure 5. These plots demonstrate that significant changes in the transfer function of the VOR would be necessary to maintain visual acuity in our model system under these different conditions. Of course, in the real situation head-neck dynamics will change rapidly and continuously with changes in gait, posture, substrate, etc. requiring rapid continuous changes in VOR dynamics rather than the simple switch implied here. HEAD ~ 20 ~ § 0 ....... ----~~-~ 5 ~ ~ -20 0.1 1.0 10.0 FREQUENCY Figure 4: Head velocity spectrum during "walking" (light) and "running" (heavy). 414 Paulin, Nelson and Bower ~ 0 .8 ~ 1----""'::3I~ § 5 ~ -20 • • •• , Ci. D •• .1 1.0 10.0 .1 1.0 10.0 Figure 5: Bode gain plots (left) and phase plots (right) for optimal estimators of head angular velocity during "walking" (light) and "running" (heavy). 4. SIGNIFICANCE TO THE REAL VOR Our results show that the optimal VOR transfer function requires a frequency dependent gain to accurately adjust to a wide range of head movements under real world conditions. Thus, the deviations from unity gain seen in actual measurements of the VOR may not represent poor, but rather optimal, performance. Our modeling similarly suggests that several other experimental results can be reinterpreted. For example, localized peaks or valleys in the VOR gain function can be induced experimentally through prolonged sinusoidal oscillations of subjects wearing magnifying or reducing lenses. However, this "frequency selectivity" is not thought to occur naturally and has been interpreted to imply the existence of frequency selective channels in the VOR control network (Lisberger, Miles and Optican, 1983). In our view there is no real distinction between this phenomenon and the "excessive" gain in normal monkey VOR; in each case the VOR optimizes its response for the particular task which it has to solve. This is testable. If we are correct, then frequency selective gain changes will occur following prolonged narrow-band rotation in the light without wearing lenses. In the classical framework there is no reason for any gain changes to occur in this situation. Another phenomenon which has been observed experimentally and that the current modeling sheds new light on is referred to as "pattern storage". After single-frequency sinusoidal oscillation on a turntable in the light for several hours, rabbits will continue to produce oscillatory eye movements when the lights are extinguished and the turntable stops. Trained rabbits also produce eye oscillations at the training frequency when oscillated in the dark at a different frequency (Collewijn, 1985). In this case the sinusoidal pattern seems to be "stored" in the nervous system. However, the effect is naturally accounted for by our optimal estimator hypothesis without relying on an explicit "pattern storage mechanism". An optimal estimator works by matching its dynamics to the dynamics of the signal generator, and in effect it tries to force an internal model to mimic the signal generator by comparing actual and expected patterns of sensory inputs. When Neural Control of Sensory Acquisition 415 no data is available, or the data is thought to be very unreliable, an optimal estimator relies completely, or almost completely, on the model. In cases where the signal is patterned the estimator will behave as though it had memorized the pattern. Thus, if we hypothesize that the VOR is an optimal estimator we do not need an extra hypothesis to explain pattern storage. Again, our hypothesis is testable. If we are correct, then repeating the pattern storage experiments using rotational velocity waveforms obtained by driving a frequency-tuned oscillator with Gaussian white noise will produce identical dynamical effects in the VOR. There is no sinusoidal pattern in the stimulus, but we predict that the rabbits can be induced to generate sinusoidal eye movements in the dark after this training. The modeling results shown in figures 4 and 5 represent an extension of our ideas into the area of gait (or more generally "context") dependent changes in VOR which has not been considered very much in VOR research. In fact, VOR experimental paradigms, in general, are explicitly set up to produce the most stable VOR dynamics possible. Accordingly, little work has been done to quantify the short term changes in VOR dynamics that must occur in response to changes in effective head-neck dynamics. Experiments of this type would be valuable and are no more difficult technically than experiments which have already been done. For example, training an animal on a turntable which can be driven randomly with two distinct velocity power spectra, i.e. two "gaits", and providing the animal with external cues to indicate the gait would, we predict, result in an animal that could use the cues to switch its VOR dynamics. A more difficult but also more compelling demonstration would be to test VOR dynamics with impulsive head accelerations in different natural situations, using an unrestrained animal. s. SENSOR FUSION AND PREDICTION To this point, we have discussed compensatory eye movements by treating the VOR as a single input, single output system. This allowed us to concentrate on a particular aspect of VOR control: tracking a time-varying dynamical system (the head) using noisy data. In reality there are a number of other factors which make control of compensatory eye movements a somewhat more complex task than it appears to be when it is modeled using classical control theory. For example, a variety of vestibular as well as non-vestibular signals (e.g. visual, proprioceptive) relating to head movements are transmitted to the compensatory eye movement control network (Ito, 1984). This gives rise to a "sensor fusion" problem where data from different sources must be combined. The optimal solution to this problem for a multiple input - multiple output, time-varying linear, stochastic system is also given by the Kalman-Bucy filter (Davis and Vinter, 1985). Borah, Young and Curry (1988) have demonstrated that a Kalman-Bucy filter model of visualvestibular sensor fusion is able to account for visual-vestibular interactions in motion perception. Oman (1982) has also developed a Kalman-Bucy filter model of visualvestibular interactions. Their results show that the optimal estimation approach is useful 416 Paulin, Nelson and Bower for analyzing multivariate aspects of compensatory eye movement control, and complement our analysis of dynamical aspects. Another set of problems arises in the VOR because of small time delays in neural transmission and muscle activation. To optimize its response, the mammalian VOR needs to make up for these delays by predicting head movements about lOmsec in advance (ret). Once the dynamics of the signal generator have been identified, prediction can be performed using model-based estimation (Davis and Vinter, 1985). A neural analog of a Taylor series expansion has also been proposed as a model of prediction in the VOR (pellionisz and LUnas, 1979), but this mec.hanism is extremely sensitive to noise in the data and was abandoned as a practical technique for general signal prediction several decades ago in favor of model-based techniques (Wiener, 1948). The later approach may be more appropriate for analyzing neural mechanisms of prediction (Arbib and Amari, 1985). An elementary description of optimal estimation theory for target tracking, and its possible relation to cerebellar function, is given by Paulin (1988). 6. ROLE OF CEREBELLAR CORTEX IN VOR CONTROL To this point we have presented a novel characterization of the problem of compensatory eye movement control without considering the physical circuitry which implements the behavior. However, there are two parts to the optimal estimation problem. At each instant it is necessary to (a) filter the data using the optimal transfer function to drive the desired response and (b) determine what transfer function is optimal at that instant and adjust the filtering network accordingly. The first problem is fairly straightforward, and existing models of VOR demonstrate how a network of neurons based on known brains tern circuitry can implement a particular transfer function (Cannon and Robinson, 1985). The second problem is more difficult because requires continuous monitoring of the context in which head movements occur using a variety of sources of relevant data to tune the optimal filter for that context. We speculate that the cerebellar cortex performs this task. First, the cortex of the vestibulo-cerebellum is in a position to mflke the required compu, tation, since it receives detailed information from multiple sensory modalities that provide information on the state of the motor system (Ito, 1985). Second, the cerebellum projects to and appears to modulate the brain stem compensatory eye movement control network (Mackay and Murphy, 1979). We predict that the cerebellar cortex is necessary to produce rapid, context-dependent optimal state dependent changes in VOR transfer function which we have discussed. This speculation can be tested with turntable experiments similar to those described in section 4 above in the presence and absence of the cerebellar cortex. Neural Control of Sensory Acquisition 417 7. THE GENERAL FUNCTION OF CEREBELLAR CORTEX According to our hypothesis, the cerebellar cortex is required for making optimal compensatory eye movements during head movements. This is accomplished by continuously modifying the dynamics of the underlying control network in the brainstem, based on current sensory information. The function of the cerebellar cortex in this case can then be seen in a larger context as using primary sensory information (vestibular, visual) to coordinate the use of a motor system (the extraoccular eye muscles) to position a sensory array (the retina) to optimize the quality of sensory information available to the brain. We believe that this is the role played by the rest of the cerebellum for other sensory systems. Thus, we suspect that the hemispheres of the rat cerebellum, with their peri-oral tactile input (Bower et al., 1983), are involved in controlling the optimal use of these tactile surfaces in sensory exploration through the control of facial musculature. Similarly, the hemispheres of the primate cerebellum, which have hand and finger tactile inputs (Ito, 1984), may be involved in an analogous exploratory task in primates. These tactile sensory-motor systems are difficult to analyze, and we are currently studying a functionally analogous but more accessible model system, the electric sense of weakly electric fish (cf Rasnow et al., this volume). 8.CONCLUSION Our view of the cerebellum assigns it an important dynamic role which contrasts markedly with the more limited role it was assumed to have in the past as a learning device (Marr, 1969; Albus, 1971; Robinson, 1976). There is evidence that cerebellar cortex has some learning abilities (Ito, 1984), but it is recognized that cerebellar cortex has an important dynamic role in motor control. However, there are widely differing opinions as to the nature of that role (Ito, 1985; Miles and Lisberger, 1981; Pellionisz and Llinas, 1979). Our proposal, that the VOR is a neural analog of an optimal estimator and that the cerebellar cortex monitors context and sets reflex dynamics accordingly, should not be interpreted as a claim that the nervous system actually implements the computations which are involved in applied optimal estimation, such as the KalmanBucy filter. Understanding the neural basis of cerebellar function will require the combined power of a number of experimental, theoretical and modeling approaches (cf Wilson et al., this volume). We believe that analyses of the kind presented here have an important role in characterizing behaviors controlled by the cerebellum. Acknowledgments This work was supported by the NIH (BNS 22205), the NSF (EET-8700064), and the Joseph Drown Foundation. References Arbib M.A. and Amari S. 1985. Sensori-moto Transformations in the Brain (with a critique of the tensor theory of the cerebellum). J. Theor. BioI. 112:123-155 418 Paulin, Nelson and Bower Borah J., Young L.R. and Curry, R.E. 1988. Optimal Estimator Model for Human Spatial Orientation. In: Proc. N.Y. Acad. Sci. B. Cohen and V. Henn (eds.). In Press. Bower lM. and Woolston D.C. 1983. The Vertical Organization of Cerebellar Cortex. J. Nemophysiol. 49: 745-766. Carpenter R.H.S. 1977. Movements of the Eyes. Pion, London. Davis M.B.A. and Vinter R.B. 1985. Stochastic Modelling and Control. Chapman and Hall, NY. Funnan J.M., O'Leary D.P. and Wolfe lW. 1982. Dynamic Range of the Frequency Response of the Horizontal Vestibulo-Ocular Reflex of the Alert Rhesus Monkey. Acta Otolaryngol. 93: 81 Ito, M. 1984. The Cerebellum and Neural Control. Raven Press, NY. Kalman R.E. 1960. A New Approach to Linear Filtering and Prediction Problems. l Basic Eng., March 1960. Kalman R.E. and Bucy R.S. 1961. New Results in Linear Filtering and Prediction Theory. 1. Basic Eng., March 1961. Lisberger, S.G. 1988. The Nemal Basis for Learning of Simple Motor Skills. Science, 242:728735. Lisberger S.G. , Miles F.A. and Optican L.M. 1983. Frequency Selective Adaptation: Evidence for Channels in the Vestibulo-Ocular Reflex. J. Neurosci. 3:1234-1244 Mackay W.A. and Murphy J.T. 1979. Cerebellar Modulation of reflex Gain. Prog. Neurobiol. 13:361-417. Oman C.M. 1982. A heuristic mathematical Model for the Dynamics of Sensory Conflict and Motion Sickness. Acta Oto-Laryngol. S392. Paulin M.G. 1988. A Kalman Filter Model of the Cerebellum. In: Dynamic Interactions in Nemal Networks: Models and Data. M. Arbib and S. Amari (eds). Springer-Verlag, NY. pp239-261. Pellionisz A. and Llinas R. 1979. Brain Modelling by Tensor Network Theory and Computer Simulation. The Cerebellum: Distributed Processor for Predictive Coordination. Nemoscience 4:323-348. Robinson D.A. 1976. Adaptive Control of the Vestibulo-Ocular Reflex by the Cerebellum. J. Nemophys.36:954-969. Robinson D.A. 1981. The Use of Control Systems Analysis in the Neurophysiology of Eye Movements. Ann. Rev. Neurosci. 4:463-503. Wiener, N. 1948. Cybernetics: Communication and Control in the Animal and the Machine. MIT Press, Boston.
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232 SPEECH PRODUCTION USING A NEURAL NETWORK WITH A COOPERATIVE LEARNING MECHANISM Mitsuo Komura Akio Tanaka International Institute for Advanced Study of Social Information Science, Fujitsu Limited 140 Miyamoto, Numazu-shi Shizuoka, 410-03 Japan ABSTRACT We propose a new neural network model and its learning algorithm. The proposed neural network consists of four layers - input, hidden, output and final output layers. The hidden and output layers are multiple. Using the proposed SICL(Spread Pattern Information and Cooperative Learning) algorithm, it is possible to learn analog data accurately and to obtain smooth outputs. Using this neural network, we have developed a speech production system consisting of a phonemic symbol production subsystem and a speech parameter production subsystem. We have succeeded in producing natural speech waves with high accuracy. INTRODUCTION Our purpose is to produce natural speech waves. In general, speech synthesis by rule is used for producing speech waves. However, there are some difficulties in speech synthesis by rule. First, the rules are very complicated. Second, extracting a generalized rule is difficult. Therefore, it is hard to synthesize a natural speech wave by using rules. We use a neural network for producing speech waves. Using a neural network, it is possible to learn speech parameters without rules. (Instead of describing rules explicitly, selecting a training data set becomes an important subject.) In this paper, we propose a new neural network model and its learning algorithm. Using the proposed neural network, it is possible to learn and produce analog data accurately. We apply the network to a speech production system and examine the system performance. PROPOSED NEURAL NETWORK AND ITS LEARNING ALGORITHM We use an analog neuron-like element in a neural network. The element has a logistic activation function presented by equation (3). As a learning algorithm, Speech Production Using A Neural Network 233 the BP(Back Propagation) method is widely used. By using this method it is possible to learn the weighting coefficients of the units whose target values are not given directly. However, there are disadvantages. First, there are singular points at 0 and 1 (outputs of the neuron-like element). Second, finding the optimum values of learning constants is not easy. We have proposed a new neural network model and its learning algorithm to solve this problem. The proposed SICL(Spread Pattern Information and Cooperative Learning) method has the following features. (a)The singular points of the BP method are removed. (Outputs are not simply o or 1.) This improves the convergence rate. (b)A spread pattern information(SI) learning algorithm is proposed. In the SI learning algorithm, the weighting coefficients from the hidden layers to the output layers are fixed to random values. Pattern information is spread over the memory space of the weighting coefficients. As a result, the network can learn analog data accurately. (c)A cooperative learning(CL) algorithm is proposed. This algorithm makes it possible to obtain smooth and stable output. The CL system is shown in Fig.1 where D(L) is a delay line which delays L time units. In the following sections, we define a three-layer network, introduce the BP method, and propose the SICL method . . ' .. ' ", ..... , ..•.•. Input layar .... ". '. -'. ,:yi ,/ Hlddan layar. ~,;" I';; Output layar. Figure 1. Cooperative Learning System k=K Final output layar (Speech Parameter I Phonemic Symbol Production Subsystem) 234 Komura and Tanaka THREE·LA YER NETWORK We define a three-layer network that has an input layer, a hidden layer, and an output layer. The propagation of a signal from the input layer to the hidden layer is represented as follows. Uj = 'EiwIHijXi, Yj = f(Uj -OJ) , (1) where i= 1,2, ... ,1; j= 1,2, ... ,J and Xi is an input, Yj the output of the hidden layer, and OJ a threshold value. The propagation of a signal from the hidden layer to the output layer is represented as follows. (2) where Zk is the output of the output layer and k= 1,2, ... ,K. The activation function f(u) is defined by f(u) = (1+exp(-u+O»-l. (3) Settingy = f(u), the derivation off(u) is then given by f' (u) = y(1-y). BACK PROPAGATION (BP) METHOD The three-layer BP algorithm is shown as follows. The back propagation error, 80k(n) , for an output unit is given by (4) where n is an iteration number. The back propagation error, 8Hin), for a hidden unit is given by 8Hin) = ('Ek80k(n) WHOjk) f' (u)<n» . (5) The change to the weight from the i-th to thej-th unit, flwIH;,in) is given by flwIHi/n) =a8H/n)xi(n) + jJflwIH;,in-l) . (6) The change to the weight from thej-th to the k-th unit, flwH0jk(n) is given by flwHOjk(n) = a80k(n) Yin) + jJflwHOjk(n-l) , (7) where a and jJ are learning constants, and have positive values. SPREAD PATTERN INFORMATION AND COOPERATIVE LEARNING (SICL) METHOD The proposed learning algorithm - SICL method is shown as follows. The propagation of a signal from the input layer to the hidden layer is given by u/l,n) = 'EiwlHij(l,n-l) xi(n) , y/l,n) =f(u/l,n) -oin) , (8) where l is a stage number (l = -LI2, ... ,LI2). The propagation of a signal from the hidden layer to the output layer is given by vk(l,n) = 'EjwH0jk(n Yin) , zk(l,n) = f(vk(l,n) -Ok(n) . (9) The back propagation error, 80k(n), for an output unit is given by Speech Production Using A Neural Network 235 (10) where y is a constant for removing singular points. The back propagation error, OHj(l,n), for a hidden unit is given by 8HjCI,n) = (EkoOk(l,n) wH0jk(l) (Yil,n)(1-yjCl,n» + y) . (11) The change to the weight from the i-th to thej-th unit, ll.wIHi/l,n) is given by AwIHij(l,n) =aOHjCl,n)xln) + jJAwIHij(l,n-l) . (12) The weight from thej-th to the k-th unit,wHOjk(l) is given by wH°jk(l) = Cjk(l), (13) where Cjk(l) is a fixed value and a random number with normal distribution. A final output is a weighting sum of outputs zk(l,n) ,and is given by ZFk( n) = E,Wl zk(l,n). (14) A SPEECH PRODUCTION SYSTEM USING THE PROPOSED NEURAL NETWORK The block diagram of a speech production system is shown in Fig.2. The system consists of a phonemic symbol production subsystem and a speech parameter production subsystem using the proposed neural network. Phonemic Symbols Speech ParameterslAnalog detal Input String ! ® Phonemic SYlnbol = Speech Parameter I'Iaduclim SiRy.1n _ Production Subsystern @ Automatic Input Speech S~lg...Lnll~I-------li Speech Psremeter ExtlClIon S.mystem 1 """"®~-=0--1 Synthesized Speech Figure 2. The Speech Production System Using the Proposed Neural Network In the automatic speech parameter extraction subsystem@, speech parameters are extracted automatically from an input speech signal. Speech parameters are composed of source parameters( voiced/unvoiced ratio, pitch and source power) and a vocal tract area function(PARCOR coefficients). The extracted speech parameters are used as training data of the speech parameter production subsystem. In the speech parameter production subsystem®, input is the string of phonemic symbols. (In the training stage, phonemic symbols are decided by the teacher using input speech data. After training, phonemic symbols are given by phonemic symbol production subsystem.) Targets are speech parameters extracted in @ . The phonemic symbol production subsystem CD consists of a preprocessor and a learning system using the proposed neural network. In the preprocessor, a 236 Komura and Tanaka string ofinput characters is converted to a string of phonemic symbols with the mean length of utterance. The input is the string of phonemic symbols converted by the preprocessor. In the training stage, the targets are actual phonemic symbols, namely, the inputs of subsystem @ which are decided by the teacher. The output speech parameters are converted into the synthesized speech wave using the PARCOR synthesizer circuit~. EXPERIMENTS We performed two separate experiments. Experiment I Automatic Speech Parameter Extraction Subsystem@ Sampling frequency: 8 KHz Frame length : 20 ms Frame period: 10 ms Speech Parameter Production Subsystem @ Input layer: 1,044(36 X 29) units Hidden layers: 80 units X 9 stages Output layers: 13 units X 9 stages Final output layer: 13 units For source parameters, 3(VIUV, pitch, source power) units are assigned. For a vocal tract area function, 10(PARCOR coefficients) units are assigned. This system has an input layer, 9 hidden layers, 9 output layers, and a final output layer. For each output layer, different target data are assigned. In the final output layer, these outputs are summed with weighting coefficients. Phonemic Symbol Production SubsystemCD Input layer: 1,044(36 X 29) units Hidden layers: 80 units X 9 stages Output layers: 36 units X 9 stages Final output layer: 36 units Input Speech Signal No.1 The input speech signal No.1 is r ASAHA YAKU BANGARONI DENPOGA TODOITA J, that is, Japanese sentence which means that "A telegram was sent to the bungalow early in the morning.". This signal is a 408 frame (4.08s) sequence. Experimental Results In Fig.3, the targets and the outputs of the speech parameter production subsystem are shown. The dotted line is the sequence of target values. The solid line is the sequence of output values after the training stage. In this case, the learning constant a is equal to 0.7 and jJ to 0.2. After training, the actual outputs produced by SICL system agree well with the targets. Fig.4 (a) shows the learning behaviors of the speech parameter production subsystem. In this case, the input speech signal is No.1. The learning constant a is equal to 0.7 and P to 0.2. The learning curve based on the SICL and SI 1 Speech Production Using A Neural Network 237 methods converged. However, the learning curve based on the BP method did not converge. Fig.4 (b) also shows the learning behaviors. The learning constant a is equal to 0.07 and jJ to 0.02. In this case, all of the learning curves converged. These results show that if we use the SICL or SI method instead of the BP method, we can obtain better results. 1.0 0.5 0.0 ~. If '. . \ I I I o 1.0 2.0 3.0 4.0 s Source Power a = 0.7,/1 = 0.2 Input speech signal : No.1 Figure 3. Targets and Outputs of Speech Parameter Production Subsystem MSE(log) MSE(log) 1 10-1 10-1 10-2 10-2 10-3 10-4 10-5 10-3 10-4 10.5 1 10 102 103 (NNI) 1 10 102 103 (a) a = 0.7 ,jJ = 0.2 (b) a = 0.07 ,p = 0.02 MSE: Mean square error NNI : Normalized number of iteration G-£J : SICL method (9stage) G-B: SI method (l = 0) * *: BP method Figure 4. Learning Behavior of Speech Parameter Production Subsystem Table 1 shows mean square errors of the system using SICL method for a, p. It should be noted that the domain of convergence is very wide. From these 238 Komura and Tanaka experiments, it is seen that the SICL method almost always allows stable and smooth output to be obtained. We also examined the system performance for another input speech signal which is a 1,60S frame (16.0S s) sequence. The learning curves converged and were similar to those for the input speech signal No.1 0.07 0.01 7.9S X 10-5 0.02 7.97X10-5 0.1 7.8SX 10-5 0.2 7.85 X 10-5 0.35 7.82x 10-5 0.7 7.80X 10-5 0.20 7.S2X 10-5 7.S2X10-5 7.S1 X 10-5 7.S0X 10-5 7.S1 X 10-5 8_26X 10-5 Input speech signal: No.1. 0.7 1.03 X 10-4 1.09X 10-4 1.26 X 10-4 1.72 X 10-4 2.S6X 10-4 1.72x 10-3 TABLE 1. Mean Square Errors for a, p Experiment II Speech Parameter Production Subsystem @ Input layer: 665 (35X 19)units Hidden layers: 400 units X 9 stages Output layers: 14 (12 + 2)units X 9 stages For the first target data(VIUV, power ratio, 10 PARCOR coefficients), 12 units are . assigned. For the second target data(pitch, source power), 2 units are assigned. . Input Speech Signal No.2 This signal is a 1:),700 frame (137s) sequence shown as follows. IARAARN,IARAIRN, .... ,1 ARAKARN, I ARAKIRN, ..... ,1ARASARN, ..... . I ARARAN,I!RARA!/, ..... ,1KARARAKN ,1KIRARAKI/, ... ,1SARARASN, .... . In experiment II , a learning test for producing an arbitrary combination of the input phonemes (ai) was done. The learning behavior of subsystem® is shown in Fig.5. In this case, the fIrst 12 target data items are trained. The other 2 target data it~ms should be trained using another network. The upper line is a learning curve for the SI method, and the lower line for the SICL method. Some of the learning curves for the SI method didn't converge. However, the learning curve for the SICL method did. We can thus say that the SICL method is very powerful for actual use. (In this case, 400 hidden units is not sufficient for the size of input data. If more hidden units are used, the MSE will be small.) CONCLUSION Speech Production Using A Neural Network 239 MSE (log) ~---------..., 10-1 G-B : SICL method (9stage) G-E) : SI method (l= 0) 10-2 10-3 a = 0.2, 11 = 0.035 Input speech signal: No.2 10-4 1 10 102 NNI Figure 5. Learning Behavior of Speech Parameter Production Subsystem From the experiments shown in previous section, it should be noted that using the SICL (Spread Pattern Information and Cooperative Learning) method makes it possible to learn speech parameters or phonemic symbols stably and to produce more natural speech waves than those synthesized by rules. If the input is the combination of a word and postpositional particle, it is easy to produce sound for unknown input data using the proposed speech production system. However, the number of the hidden units and training data will be great. Therefore we have to make the system learn phonemes among phoneme sequence. Using this training data makes it possible to produce an arbitrary sequence of phonemes. In experiment II , phonemic information (V IUV I power ratio, PARCOR coefficients) is trained. Then the range of input window was set to be 190 ms. For prosodic information (pitch, source power), we must use another network. Because, if we want to make the system learn prosodic information, we must set the range of input data wider than that of words. U sing these strategies, it is possible to produce arbitrary natural speech. Acknowledgments The authors thank Dr. Tosio Kitagawa, the president of liAS-SIS for his encouragement. References D. E. Rumelhart, J. L. McClelland and PDP Research Group Parallel Distributed Processing. VOL.l The MIT Press (1987). T.J.Sejnowski and C.R.Rosenberg Parallel Networks that Learn to Pronounce English Text. Complex Systems, 1, pp.145-168 (1987). M.Komura and A.Tanaka Speech Synthesis Using a Neural Network with a Cooperative Learning Mechanism. IEICE Tech. Rep. MBE88-8 (1988) (in Japanese).
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20 ASSOCIATIVE LEARNING VIA INHIBITORY SEARCH David H. Ackley Bell Communications Research Cognitive Science Research Group ABSTRACT ALVIS is a reinforcement-based connectionist architecture that learns associative maps in continuous multidimensional environments. The discovered locations of positive and negative reinforcements are recorded in "do be" and "don't be" subnetworks, respectively. The outputs of the subnetworks relevant to the current goal are combined and compared with the current location to produce an error vector. This vector is backpropagated through a motor-perceptual mapping network. to produce an action vector that leads the system towards do-be locations and away from don 't-be locations. AL VIS is demonstrated with a simulated robot posed a target-seeking task. INTRODUCTION The "backpropagation algorithm" or generalized delta rule (Rumelhart, Hinton, & Williams, 1986) is sometimes criticized on the grounds that it is a "supervised" learning algorithm, which requires a "teacher" to provide correct outputs, and apparently leaves open the question of how the teacher learned the right answers. However, work. by Rumelhart (personal communication, 1987) and Miyata (1988) has shown how the environment that a system is embedded in can serve as the "teacher." If, as in this paper, a backpropagation network is posed the task of mapping from a vector 9 of robot arm joint angles to the resulting vector X of arm coordinates in space (the "forward kinematics problem"), then input-output training data can be obtained by supplying sets of joint angles to the arm and observing the resulting configurations. Although this "environment as teacher" strategy shows how a "teacher" can come to possess useful information without an infinite regress learning it, it is not a complete solution. There are problems for which the "laws of physics" of an environment do not suffice to determine the solution. Suppose, for example, that a robot is posed the problem of learning to reach for different positions in space depending on which of a set of signals is currently presented, and that the only feedback available from the environment is success or failure information about the current arm configuration. Associative Learning via Inhibitory Search 21 d Figure 1. A "trunk" robot. What is needed in such a case is a mechanism to search through the space of possible arm configurations, recording the successful configurations associated with the various inputs. ALVIS Associative Learning Via Inhibitory Search provides one such mechanism. The next section applies backpropagation to the 9 --+ X mapping and shows how the resulting network can sometimes be used to solve X --+ 9 problems. The third section, "Self-supervision and inhibitory search," integrates that network into the overall ALVIS algorithm. The final section contains some discussion and conclusions. An expanded version of this paper may be found in Ackley (1988). FORWARD AND INVERSE KINEMATICS The in verse kinematics problem in controlling an arm is the problem of determining what joint angles are needed to produce a specific position and orientation of a hand. In the general case it is a difficult problem. An itch on your back suggests the kinds of questions that arise. Which hand should you use? Should you go up from around your waist, or down from over your shoulders? Can you be sure you know what will work without actually trying it? From a computational standpoint, forward kinematics deciding where your limbs will end up given a set of joint angles is an easier problem. Figure 1 depicts the planar "robot" that was used in this work. I call it the "trunk" robot. (The work discussed in Ackley (1988) also used a two-handed "pincer" robot.) Of course, the trunk is a far cry from a real robot, and the only significant constraint is that the possible joint angles are limited, but this suffices to pose non-trivial kinematics problems. The trunk has five joints, and each joint angle is limited to a range of 40 to 1760 with respect to the previous limb. I simulated a backpropagation network with five real-valued input units (the joint angles), sixty hidden units in a single layer, and twelve linear output units (the Cartesian joint positions). Joint angles were expressed in radians, so the range of input unit values was from about 0.07 to about 3.07. The configuration vector X was represented by twelve output units corresponding to six pairs of (x, y) coordi22 Ackley Figure 2. A "logarithmic strobe" display of the trunk's asymptotic convergence on a specified position and orientation. The arm position is displayed after iterations 1,2,4,8, ... ,256. nates, one pair for each joint a through f. With the trunk robot (though not with the "pincer") f:e and fy have constant values, since they end up being part of the "anchor." The state of an output unit equals the sum of its inputs, and the error propagated out of an output unit equals the error propagated into it. Errors were defined by the difference between the predicted configuration and the actual configuration, and after extensive training on the trunk robot forward kinematics problem, the network achieved high accuracy over most of the joint ranges. In typical backpropagation applications, once the desired mapping has been learned, the backward "error channels" in the network are no longer used. However, suppose some other error computation, different from that used to train the weights, was then incorporated. Those errors can be propagated from the outputs of the trained network all the way back to the inputs. The goal is no longer to change weights in the network since they already represent a useful mapping but to use the trained network to translate output-space errors, however defined, into errors at the inputs to the network. Figure 2 illustrates one use of this process, showing how a trained forward kinematics network can be used to perform a cheap kind of inverse kinematics. The figure shows superimposed outputs of the trained network under the influence of a task-specific error computationj in this case the trunk is trying to reduce the distances between the front and back of a "target arrow" and the front and back of its first arm section. The target arrow is defined by a head (h:e, hy) and a tail (t:e, ty). The errors for output units a:e and ay are defined by e(a:e} = h:e - a:e and e(ay) = hy-ay, the errors for output units b:e and by are defined by e(b:e) = t:e-b:e and e(by) = ty - by, and the errors at all other outputs are set to zero. The algorithm used to generate this behavior has the following steps: 1. Compute errors for one or more output units based on the current positions of the joints and the desired positioning and orienting information. If "close Associative Learning via Inhibitory Search 23 o Figure 3. The trunk kinking itself. enough" to the target, exit, otherwise, store these errors on the selected output units, and set the other error terms to zero. 2. Backpropagate the errors all the way through the network to the input units. This produces an error term e(9i) for each joint angle 9i. Produce new joint angles: 9~ = 9i + ke(9i) where k is a scaling constant. Clip the joint angles against their minimum and maximum values. 3. Forward propagate through the network based on the new joint angles, to produce new current positions for the joints. Go to step 1. Whereas the training phase has forward propagation of activations (states) followed by back propagation of errors, this usage reverses the order. Backpropagation of errors is followed by changes in inputs followed by forward propagation of activations. This is a general gradient descent technique usable when a backpropagation network can learn to map from a control space 9 to an error or evaluation space X. Figure 3 illustrates how gradient descent's familiar limitation can manifest itself: The target is reachable but the robot fails to reach it. The initial configuration was such that while approaching the target, the trunk kinked itself too short to reach. If the robot had "thought" to open 93 instead of closing it, it could have succeeded. In that sense, the problem arises because the error computation only specified errors for the tip of the trunk, and not for the rest of the arm. If, instead, there were indications where all of the joints were to be placed, failures due to local minima could be greatly reduced. SELF-SUPERVISION AND INHIBITORY SEARCH The feedback control network of the previous section locally minimizes joint position errors however they are generated by translating them into joint angle space and moving downhill. AL VIS uses the feedback control network for arm control; this section shows how ALVIS learns to generate appropriate joint position space errors given only a reinforcement signal. There are two key points. The first is this: Once an action producing a positive reinforcement has somehow been found, 24 Ackley the problem reduces to associative mapping between the input and the discovered correct output. In ALVIS, "do-be units" are used to record such successes. The second point is this: When negative reinforcement occurs, the current configuration can be associated with the input in a behavior-reversed fashion as a place to avoid in the future. In ALVIS, "don't-be units" are used to record such failures. The overall idea, then, is to perform inhibitory search by remembering failures as they occur and avoiding them in the future, and to perform associative learning by remembering successful configurations as the search process uncovers them and recreating them in the future. In effect, ALVIS constructs input-dependent "attractors" at arm configurations associated with success and "repellors" at configurations associated with failure. Figure 4 summarizes the algorithm. A few points to note are these: • The do-be and don't-be units use the spherical non-linear function explored by Burr & Hanson (1987). The response of a spherical unit is maximal and equal to one when the input vector and the weight vector are identical. The response of the unit decreases monotonically with the Euclidean distance between the two vectors, and the radius r governs the rate of decay. • The don't-be units of each subnetwork (i.e., relevant to one goal) are in a competitive network (see, e.g., Feldman 1982). The don't-be unit with the largest activation value (which is a function of both the distance from the current position and the radius) is the only don't-be unit that has effects on the rest of the system. In the simulations reported here, I used m = 4 don't-be units per goal. • In addition to the parameters associated with spherical units, each do-be and don't-be unit has a strength parameter a that specifies how much influence the unit has over the behavior of the arm. Do-be strength (at) grows logarithmically with positive reinforcement and shrinks linearly with negative; don't-be strength (a;i) grows logarithmically with negative reinforcement and shrinks linearly with positive. Figure 5 illustrates a situation from early in a run of the system. From left to right, the three displays show the state of the relevant do-be unit, the relevant don't-be subnetwork, and the current configuration of the arm. Since this particular goal has never been achieved before, the do-be map provides no useful information its weight vector contains small random values (as it happens, the origin is below and right of the display) and its strength is zero. The display of the don't-be map shows the positions of all four relevant don't-be units, with the currently selected don't-be (unit number 3) drawn somewhat darker. The don't-be units are spread around configuration space, creating "hills" that push away the arm if it comes too close. As the arm moves about without reaching the target, different don't-be units win the competition and take control. Negative reinforcement accrues, and the winning don't-be consequently moves toward the various current configurations and gets stronger, until the arm is pushed elsewhere. Figure 6 illustrates the behavior of the system after more extensive learning. The Associative Learning via Inhibitory Search 25 Figure 4. AL VIS O. (Initialize) Given: a space X of h dimensions, a backpropagation network trained on 8 -+ X, a set G of goals and a mapping from G to regions of space. Create an AL VIS network with n = IGI goal units 91, ... ,9n, n dobe/don't-be subnetworks consisting of one do-be unit dt and m don't-be units dii, ... , dfu., and h current position units ZI,'" Zh. Create modifiable connections Wzi from z's to d's, Wiz from d's to z's, and a modifiable strength s for each d. Set all do-be strengths st and all don't-be strengths sti to zero. Set all weights Wzi and Wiz to small random values. Set 8 to a random legal vector and produce a current configuration X. 1. (New stimulus) Choose t at random from 1 ... n. 2. (Do's/Don'ts) Compute activations for do-be's and don't-be's using the spherical function: d = 1/ (1 + ~JL./:=1 (Zi - wzil2). In subnetwork t, let dt: be the unit with the largest activation. 3. (Errors) Let wt denote the weights from dt and w~ denote the weights from dt:, and similarly for strengths st and st:. Compute errors for each component of X: e(zi) = si(wt - Zi) + Si:.(Zi w~). 4. (Move) Backpropagate to produce e(8). Generate angle changes: A8i = min(q, max( -q, k,.e(8i ))), with parameters q and k,.. Generate new angles respecting the maximum vt and minimum vi possible joint angles: 8~ = min( vt, max(vi ,8i + A8i )). Forward propagate to produce a new configuration X'. 5. (Positive reinforcement) Determine whether X' satisfies goal t. If it does not, go to step 6. Otherwise, 5 1 t;\ • 1 hIt +, I d +, I . ror 1 , ••• , , e wiz - zi an wzi - zi' 5.2 Let st' = min (5, si + p+ / (1 + si)), for positive reinf p+ > O. 5.3 Let ri = k,/ max(.I, st'), with parameter k,. 5.4 For i = 1, ... , m, let Sti' = max(O, sti-p+), and rti = k,f max(.I, sti'l. 5.5 Go to step 1. 6. (Negative reinforcement) Perform the following: 6.1 For i = 1, ... , h, let wi: = w~ + 77(zi w~) and w;/ = w;i + e + 77(Z; w~) with parameter 77, where e is a uniform random variable between ±O.O1. 6.2 Let S;:' = min (5, s;: + p- / (1 + s;:)), for negative reinf p- > O. 6.3 Let rt = k,/ max ( .1, S;:'). 6.4 Let st' = max(O, si - p-), and ri = k,/ max(O.I, Si'l. 6.5 Go to step 2. 26 Ackley ~~~Be~:~0~.~00~O~OO~0 ___________ I_D_on_'_t_~~:_3~1~.~29~4~70~3 ________ I_B~e ____________________ 1 Figure 5. A display of the internal state of the trunk robot in the process of learning to associate a set of twelve arbitrary stimuli with specified positions in space. The current signal (though the system has not discovered this yet) means "touch 5". Do Be: 4.998667 Don't Be: 3 0.007976 Be ~~~~~~----------I~~~~~~--------Figure 6. A display of the internal state of the same trunk robot later in the learning process. The previous goal was "touch *" and the current goal is "touch 3." \ do-be map now contains an accurate image of a successful configuration for the "touch 3" goal, and its strength is high. The strength of the selected don't-be unit is low. The current configuration map in Figure 6 shows each iteration of the algorithm between the time it achieved its previous goal and the time it achieved the current goal. Finally, Figure 7 displays the average time-per-goal as a function of the number of goals achieved. For 75 repetitions, the trunk network was initialized and run until 500 goals had been achieved, and the resulting time-per-goal data was averaged to produce the graph. The average time-per-goal declines rapidly as goals are presented, then seems to rise slightly, and then stabilizes around an average value of about 300. To have some kind of standard of comparison, albeit unsophisticated, if the joint angles are simply changed by uniform random values between q and -q (see Figure 4) on each iteration, the average time-per-goal is observed to be about 490. Associative Learning via Inhibitory Search 27 :nOD 24DO 2100 liDO 15DD 1200 900 600 300 Figure 7. A graph of the average time taken per goal as a function of the number of goals achieved. The horizontal line shows the average performance of random joint changes. DISCUSSION ALVIS is a preliminary, exploratory system. Of course, the ALVIS environment is but a pale shadow of the real world, but even granting the limited scope of the problem formulation, several aspects of AL VIS are incompletely satisfying and in need of improvement. To my eye, the biggest drawback of the current implementation is the local goal representation which essentially requires that the set of goals be enumerable at network definition time. Related problems include the inability to share information between one goal and another and the inability to pursue more than one goal simultaneously. To determine behavior, the constraints of goals must be integrated with the possibilities of the current situation. In AL VIS this is done as a strictly two-step process: the goal selects a subnetwork, and the current situation selects units within the subnetwork. Work such as Jordan (1986) and Miyata (1988) shows how goal information and context information can be integrated by supplying both as inputs to a single network. ALVIS is a pure feedback control model, and can suffer from the traditional problem of that approach: when the errors are small, the resulting joint angle changes are small, and the arm converges only slowly. If the gain at the joints is increased to speed con vergence, overshoot and oscillation become more likely. However, in ALVIS oscillations gradually die out, as don't-be units shift positions under the negative reinforcement, and sometimes such temporary oscillations actually help with the search, causing the tip of the arm to explore a variety of different points. 28 Ackley The aspect of AL VIS behavior that I find most irritating reveals something about the approach in general. In some cases usually on more "peripheral" targets AL VIS learns to hit the very edge of the target region. While approaching such targets, ALVIS experiences negative reinforcement, and the don't-be units, consequently, gain a little strength. The resulting interference occasionally causes a very long search for a goal that had previously been rapidly achieved. AL VIS has a representation only for its own body; a better system would also be able to represent other objects in the world, and useful relations on the expanded set. The "mostly motor" emphasis evident in the present system needs to be balanced by more sophistication on the perceptual side. Though limited in scope, ALVIS demonstrates three ideas I think worth highlightmg: • The reuse of the error channel of a backpropagation network, after training, for translating arbitrary output-space gradients into input-space gradients. • The recording of previous actual outputs to be used as future desired outputs. • The use of "repellors" (don 't-be units) as well as attractors in defining errors, and the resulting process of search-by-inhibition generated by negative reinforcement. Characterizing the behavior of a machine in terms of attractor dynamics is a familiar notion, but "repellor dynamics" seems to be largely unknown territory. Indeed, in ALVIS there is an ephemeral quality to the don't-be units: When all answers have been discovered, all strength accrues to the do-be's, good performances become routine, and AL VIS behavior is essentially attractor-based. In watching such a "grown-up" AL VIS, it is easy to forget how it was in the beginning, when the world was big and answers were scarce, and ALVIS was doing well just to discover a new mistake. References Ackley, D.H. (1988). Associative learning via inhibitory search. Teclmical memorandum TM ARH-012509. Morristown. NJ: Bell Communications Research. Burr. D.J .• & Hanson. S.J. (1987). Knowledge representation in connectionist networks. Technical memorandum TM-ARH-008733. Morristown. NJ: Bell Communications Research. Feldman. J.A. (1982). Dynamic connections in neural networks. Biological Cybernetics, 36, 193202. Jordan. M.1. (1986). Serial order: A parallel. distributed processing approach. Technical report ICS-86M. La Jolla, CA: University of California. Institute for Cognitive Science. Miyata, Y. (1988). The learning and planning of actions. Unpublished doctoral dissertation in psychology, University of California San Diego. Rumelhart, D.E .• Hinton. G.E .• & Williams. R.J. (1986). Learning representations by backpropagating errors. Nature, 3Z3. 533-536. Rumelhart. D.E. (personal communication. 1987). Also cited as personal communication in Miyata (1988).
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748 Performance of a Stochastic Learning Microchip • Joshua Alspector, Bhusan Gupta, and Robert B. Allen Bellcore, Morristown, NJ 07960 We have fabricated a test chip in 2 micron CMOS that can perform supervised learning in a manner similar to the Boltzmann machine. Patterns can be presented to it at 100,000 per second. The chip learns to solve the XOR problem in a few milliseconds. We also have demonstrated the capability to do unsupervised competitive learning with it. The functions of the chip components are examined and the performance is assessed. 1. INTRODUCTION In previous work,(l] (2] we have pointed out the importance of a local learning rule, feedback connections, and stochastic elements(3] for making learning models that are electronically implementable. We have fabricated a test chip in 2 micron CMOS technology that embodies these ideas and we report our evaluation of the microchip and our plans for improvements. Knowledge is encoded in the test chip by presenting digital patterns to it that are examples of a desired input-output Boolean mapping. This knowledge is learned and stored entirely on chip in a digitally controlled synapse-like element in the form of connection strengths between neuron-like elements. The only portion of this learning system which is off chip is the VLSI test equipment used to present the patterns. This learning system uses a modified Boltzmann machine algorithm[3] which, if simulated on a serial digital computer, takes enormous amounts of computer time. Our physical implementation is about 100,000 times faster. The test chip, if expanded to a board-level system of thousands of neurons, would be an appropriate architecture for solving artificial intelligence problems whose solutions are hard to specify using a conventional rule-based approach. Examples include speech and pattern recognition and encoding some types of expert knowledge. 2. CIllP COMPONENTS I Fig. 1 is a photograph of the silicon chip. It contains various test structures, the largest of which. in the lower left, is a neural-style learning network composed of 6 neurons, each with its own noise amplifier, and 15 bidirectional synapses which potentially allow the network to be fully connected. In order to study these components separately, there is a also a noise amplifier in the upper left comer of the chip, a neuron in the upper right, and 2 synapses in the lower right. • Pennanent address: University of California, Berkeley; EE Dep't, Cory Hall; Berkeley, CA 94720 • ,-- .. Performance of a Stochastic Learning Microchip 749 - -H .. • I _____ ~_J Figure 1. Photograph of Test Chip Containing a Learning Network in Lower Left. 2.1 Neuron The electronic neuron perfonns the physical computation: activation=/ (LWjj sj+noise )=/ (gain*netj) where / is a monotonic non-linear function such as tanh. In some of our computer simulations this is a step function corresponding to a high value of gain. The signal from other neurons to neuron i is the sum of neural states Sj giving input weighted by the connection strengths Wjj, while the noise simulates a temperature in a physical thermodynamic system. Their sum is the effective net input netj . The model neuron is a double differential amplifier as shown in Fig. 2. Noise and signal have separate differential inputs and are summed at low gain. The differential outputs of this summing stage are converted to a single output by a high gain stage before being fed into a switching arrangement. This selects either the net input or an external clamping signal which forces the neuron into a desired state. The output of the switch is then 750 Alspector. Gupta and Allen Sdeelred Figure 2. Circuitry of Electronic Analog Neuron. further amplified before driving the network. The final output approximates a two-state binary neuron. 2.2 Noise amplifier anneal I Figure 3. Block Diagram of Noise Amplifier. Vnot .. >-........ Performance of a Stochastic Learning Microchip 751 Fig. 3 is a block diagram of the noise amplifier. The original idea was to amplify the thermal noise in the channel of a transistor with a gain of nearly a million but to stabilize the de output using low pass negative feedback in 3 stages. By controlling the feedback. one could control both the bandpass of the noise signal as well as the gain to provide for annealin$ the temperature (amount of noise) as required by the Boltzmann machine algorithm. (3) Unfortunately this amplifier proved unstable at high gain values leading to oscillations of a few MHz which were highly correlated among all the noise amplifiers in the network. In spite of this undesirable correlation in the noise signals. the network was still able to learn (see section 3). Rather than a slow "annealing". we used a rapid "heating" and "flash freezing" of the network to randomize· it. This was done by momentarily -opening a "noise on" switch during the time allotted for annealing. Learning was also demonstrated by clamping the free running neurons momentarily to a pseudo-random state and then releasing them to allow the network to settle. 2.3 Synapse Fig. 4 is a block diagram of the digitally controlled electronic synapse. The weights are stored as a sign and four bits of magnitude in five flip-flops arranged as an up-down counter. The correlation logic tests whether the two neurons that the synapse connects have the same binary state (correlated) or not at the end of the anneal cycle. If the neurons are correlated in the "teacher" phase (when the teacher is clamping the output neurons in the correct state) and not in the "student" phase (when the output neurons are running free). then a signal to the counter increments the weight by one. If the reverse is true. the counter is decremented. If the "teacher" and "student" phase have the same correlation. no change is made. SJ or' SJ or' In,or UP. son down, 0 S, correlation ement &set WIJ or II logic logic SJ 2 3 1n 1or Figure 4. Block Diagram of Synapse. 752 Alspector, Gupta and Allen The digital weight is converted to an analog conductance by a set of pass transistors with graduated binary conductance ratios. Measurements confirmed that the synapse conductance increased monotonically from a value of -15 though +15 as the counter was incremented. The -0 value, when loaded into the synapse, disconnected that link. We usually initialized all the weights to +0 before learning. 3. PERFORMANCE EVALUATION OF NETWORK 3.1 XOR tests The most difficult test for our 6 neuron network was to have it learn the exclusive-OR function. The network was arranged with 2 input neurons, 2 hidden neurons, and 1 output neuron as shown in Fig. 5. There is also a so-called 'true' neuron which is always clamped on. The negative of the weights from that neuron provide the threshold for the other neurons. The exclusive-OR function is of historical interest because the neural models of the 1960's could not learn it.[4] [5] This is because those learning algorithms did not work when there was a layer of hidden neurons. Networks with only a single layer of modifiable weights could learn the logical OR function but not the exclusive-OR (XOR). The truth table in Fig. 5 shows that the XOR is 1 (on or true) when either one of the two inputs is 1, but not when both are 1. However, recent algorithms such as the Boltzmann machine are able to learn with a hidden layer and hence can solve the XOR. out hidden In out 1 2 XOR In hidden 00 01 10 11 ? o 1 1 o Learn 'rules' to solve problem Figure 5. 2-2-1 Network to Learn XOR. To teach a network to be an XOR, we start with a blank slate where all the weights are zero and then present the patterns of 1 's and O's in the figure with the teacher~dtemately clamping the output to the correct state and letting it run free. On each presentation, the network is jittered by noise and correlations are counted by each synapse. At the end of each teacher-student cycle, weights are adjusted. Performance of a Stochastic Learning Microchip 753 Tests of the chip were conducted using an HP 8180A data generator to present digital patterns to the chip, an HP 8182 data analyzer to capture the chip's digital outputs, and an HP 54112A digitizing oscilloscope to capture wavefonns. Analog wavefonns were generated using an HP 8770A arbitrary wavefonn synthesizer feeding a Comlinear E20 1 I amplifier. These instruments were controlled by an HP 9836 computer running UNIX with test programs written in C. A pattern presentation phase consisted of five subphases and hence five clock cycles of the data generator. The input and/or output pattern to be presented to the clamped neurons is present during all five cycles. The first cycle presents noise or an annealing wavefonn to the network. The second cycle sends a signal to each synapse to count correlations. The fourth cycle can be used to send a signal to each synapse to adjust weights. This is usually done only after two 5 cycle phases, one for the "teacher" phase and one for the "student" phase. Thus, during learning, ten digital words were used in the data generator for each pattern presentation. In addition to presenting patterns, digital weights can also be read into the chip with a similar 5 cycle phase. This uses the flip-flop storage arranged as a shift register for weight storage and readout. Because the memory of the data generator was only 1024 bits deep, we would present only 66 patterns (660 words) each time the data generator was loaded by the control computer. The remaining memory was used to initialize the network to its previous value after the destructive readout of weights. In this way, perfonnance of the network was monitored after sets of 66 pseudo-randomly selected patterns. 100 test patterns could also be presented, without learning, to see what perfonnance the network achieved at that point. For the XOR, we organized the connectivity as in Fig. 5. For example, the connections between input and output neurons were fixed at zero. In order to test the settling of the network, we loaded a set of synapse weights that were learned in one of the computer simulations. We then checked the settling times of the network for various transitions of input states. These varied from 130 to 1700 nanoseconds, with most transitions in the 250 to 600 nanosecond range. The shortest time is a simple settling of the neuron amplifier while the longest time represents several loops of settling of the network before a stable state is found. For the learning trials, we initialized all weights to zero. Fig. 6 shows three learning curves for a 2-2-1 XOR network (Fig. 5). At first the network performs at chance but it soon learns all the patterns. The values of the weights (which have an accuracy of 4 bits plus a sign) after learning are also shown for one of the trials. The chip had an easier time learning the XOR function in a network with only one hidden unit provided there were also direct connections from input to output as shown in the inset of Fig. 7. This also demonstrates the flexibility of the connectivity on the chip which would not be possible if we organized it as a strictly layered network. The figure shows the learning curves at various speeds of pattern presentation from 500 to 256,000 patterns per second. The clock rate of the data generator at the highest speed was 2.56 MHz so that the time during which noise was applied was only 400 nanoseconds. The noise amplifier often did not produce an excursion of neural states at these frequencies 754 Alspector. Gupta and Allen f r • c t I c o· r r • c · t I. f r • C t I 0 .8 n c o. r r • c t 132 198 264 330 3911 482 numlMt of training pattern. 528 594 Figure 6. Proportion Correct for On-chip Learning vs. Patterns Presented. ... 1 Z 528 Figure 7. Learning Curves for 2-1-1 XOR at Various Speeds. effectively limiting learning above this rate. We could have increased the rate by compressing the five cycle phase to three or by random clamping of free running neurons, but probably not by an order of magnitude. Note that noise is necessary for learning by this system as shown by the curve at 500 Hz without noise. Performance of a Stochastic Learning Microchip 755 Fig. 8 is an oscilloscope trace of the 4 neural states as a function of time during the pattern presentations. 44.a400 _ lM.a400 _ 1M ..... _ output unit hidden unit Input unit A f 1'* ........ +-4--. ..... I , I , I , .. I I I I , ....... =$ .... ~ ..... ,...,...,. ...... -c Input unit B ..--------. .pply nol •• • dJu.t weight. ph .... t •• ch.r « . student PI .m ..... ~_.a... • • • I •• __ ~ ••• ~ ... t •••• ~a.....a--' I ••••• '__....a._L.a...-......,. e_, •• ...a.r t '" s "'r t '" s "'r t '" s "'r t '" s "'r t "'. "'r t '" s "'r t "'. "'r t '" s "'r t A. A 11 11 10 10 00 01 11 10 00 Figure 8. Neural States during Learning. The time during which noise is applied is apparent from the rapid changes of state in the hidden neuron and also in the output neuron when it is not clamped. Since each pattern presentation can take as little as 5 microseconds, the XOR function can be learned in a few milliseconds. A pattern presentation on a 1 MIP serial computer such as a VAX 11n80 takes about 0.5 seconds with our simulation software. 3.2 Unsupervised Learning So far, we have described only supervised learning procedures, but the chip can also do unsupervised learning which has no teacher. Nevertheless, the network can learn to classify input patterns according to their similarity to one another. We set the chip connectivity as in Fig. 9 with 4 input neurons and 2 output neurons arranged so that they strongly inhibit each other to form a 'competitive' layer. With noise, this output layer performs a 'winner-take-all' function in that the output neuron which has the strongest net input is on and the other is off. This is because they inhibit each other strongly (are connected to each other with a large negative weight) so that only one can be on. The usual supervised learning rule was effectively simplified by removing the teacher requirement so that correlations always increment weights. Specifically, we stored a comparison pattern in the student phase which consisted of the 'on' state for the two competitive neurons and 'off' for all the input neurons. We then presented patterns to the chip with the "teacher" phase signal on. This has the effect of always decrementing the competitive connections which therefore remain at the lower limit of -15 since it is not possible to have more correlations than the stored "student" phase correlation. On the other hand, the stored "student" phase correlation for the weights leading from the input 756 Alspector, Gupta and Allen Input Figure 9. A Competitive Learning Network. to the competitive layer is zero. Then, the winning output neuron will always be correlated with those input neurons which are ton' and hence these weights will be incremented. A decay signal decremented weights occasionally to keep them from growing too large. The net effect of such a procedure is for the output neurons to classify the input space among themselves, such that each responds to a particular neighborhood of similar patterns. (2] To demonstrate competitive learning, an input set was prepared such that the four input bits were not quite random. We picked two input neurons to represent 'left' and the other two to represent 'right". Patterns were never used with an equal number of left and right neurons on. Eventually one of the two output neurons responded to left weighted patterns and the other to right weighted patterns. Fig. 9 shows one set of weights which were obtained. Therefore the chip learned left from right although nothing in its wiring predisposed it in any way. 3.3 Computer Simulations or Chip Test Conditions Computer tests were conducted which simulated limitations of the operating chip such as correlated noise. Table 1 presents summaries of 10 replications of 2000 pattern presentations across 5 testing conditions. The Table reports the mean percent correct on the last 100 patterns and, in parentheses, the number of networks which reached 100% performance during at least one block of 100 pattern presentations. The first line of the table shows the performance of the network with no noise. In the next four lines, two parameters of the noise were varied yielding 4 conditions. Specifically, noise was either correlated or uncorrelated across neurons and it was either presented as a single pulse in a "flash freeze" schedule or following a broad annealing schedule. Performance of a Stochastic Learning Microchip 757 The 2-1-1 XOR. in which the inputs are directly connected to the outputs. demonstrated very good performance across conditions. Indeed. additional tests of the 2-1-1 in the nonoise condition showed that within 10k patterns all networks reached 100%. This suggests there are deterministic solutions for the 2-1-1. TABLE 1. Results of Computer Simulations. nOIse schedule 2-1-1 XOR 2-2-1 XOR 4-4-1 parity no noise 92(9) 67(0) 72(0) correlated flash freeze 95(9) 83(5) 71(0) correlated anneal temperature 99(10) 78(2) 74(0) uncorrelated flash freeze 99(10) 84(4) 67(0) uncorrelated anneal temperature 99(10) 85(5) 79(0) no noise anneal gain 99(9) 81(4) 85(2) The 2-2-1 networks learned to only 67% correct without noise. Learning with correlated noise degraded performance compared to learning with uncorrelated noise. While the chip contained only 6 neurons it was of interest to consider how limitations such as those studied here might affect solutions to larger problems. Thus. the solution to parity problems were considered and are included in the table. It is worth noting that the full complexity of the chip's settling and noise distribution is not captured in the discrete time simulations on the computer. The fact that we do not use a circuit simulation may account for some of the differences between the simulations and chip performance. It is interesting to note that learning by the chip was generally faster than learning by the simulation program and that the chip seemed to require noise for learning more than the simulator. We also considered a system without random noise in which we annealed the inverse gain of the neurons like a temperature through a broad annealing schedule covering the values previously exam ined [2] • As shown in the last line of the Table this performed comparably to temperature annealing reported above. 10 runs of a 2-2-1 XOR gave a mean performance of 81 % with 4 networks reaching 100%. On the 4-4-1 parity problem the mean performance was better than the results of annealing temperature. The mean performance was 85% and 2 networks reached 100%. For still larger problems. such as 6-8-1 parity. performance was comparable to annealing with noise. 4. FUTURE DIRECTIONS 4.1 Applications of Learning Systems Learning systems give us a way to encode knowledge as a set of training examples rather than as a set of rules. Learned behavior emerges from the training set in ways that depend on the input representation. the network architecture. and the learning procedure. This technique is suitable for problem domains where there are too many rules or where the rules are not known. Two general categories of problems suitable for learning 758 Alspector, Gupta and Allen systems are pattern recognition and some types of expert systems. Pattern recognition of something like an oak leaf is· difficult because of the many variations a rule-based system would have to consider even when variations of scale, rotation, and translation are accounted for. Yet, it is quite easy to give a learning system many training examples of oak leaves. Scale, rotation, and translation invariance can be built into the network structure. Similarly, recognition of speech sounds is difficult, but many training examples exist. Here also, pre-processing of the auditory data is important to obtain a useful representation. Another pattern learning task useful in telecommunications is learning the codebook for vector quantization in a real-time visual data compression system. [61 Expert knowledge is often easier to encode by training examples as well. Experts often do not know the rules they use to troubleshoot equipment or give advice. Again, it is quite easy, by taking a history of such advice, to build a large database of training examples. As knowledge changes, training is a more graceful way of Updating a knowledge base than changing the rules. In telephone networks, fault handling or traffic routing are examples of problems for which training is a suitable way of encoding knowledge. 4.2 Future Large-Scale Learning Systems Because training takes too much computer time in a simulation, physical implementations of learning systems such as ours are necessary for speed. It takes several hours to train a network to recognize a few milliseconds of speech. [7] If we could expand our system to the thousand-neuron level, it would be possible to learn simple speech recognition in real time. Because the chip uses Ohm's law to multiply, charge conservation to add, device physics to create a threshold step, and a physical noise mechanism for random number generation, we can present training patterns to this chip about 100,000 times faster than the computer simulator. This factor, mostly due to the physical analog computation at this small network size, will increase with the size of the system due to its inherently parallel nature. It would also be possible to build fast special-purpose digital hardware to perform the multiply-accumulate calculations and do fast compares in parallel. Such hardware would take up considerably more silicon area but may be a good way to integrate neural network calculations into existing computer systems. If we could build a large VLSI learning system of, say, 10,000 neurons and 1,000,000 synapses, it would be about a billion times faster than a simulator on a 1 MIP machine. Presumably, such a system will be able to learn things beyond the capability of simulations even if they are run on supercomputers. However, there are several challenges to building these systems. An algorithmic problem divorced from implementation is the effect of scaling to large size in highly connected networks. The learning time of such a system scales exponentially with the size of the problem. [8] The traditional way of handling complexity in large problems is to break them into smaller subpieces. An effective algorithm is yet to be discovered for doing learning in the modular, hierarchical networks which would be required to handle large problems. Perfonnance of a Stochastic Learning Microchip 759 Even from a technological viewpoint, modularity is necessary to manage the connectivity in a typical multiple chip system. A highly connected system, even if it could be built, would take too long to settle even considering the technology and parallel speedups available. Constraints such as power dissipation, capacitive loading across chips, and interchip communication are difficult to solve. If we succeed in these challenges, we will have the problem of presenting data to the system at extremely high rates amounting to several thousand (or more) bits every few microseconds. Biology solves these problems in the visual system, for example, by highly parallel communication via the optic nerve. It is unlikely that we will be able to use a million bit wide bus in our electronic system, however. Can one take the weights learned by a learning system and simply load them onto a much simpler system with programmable rather than adaptive synapses? This is perhaps possible for smaller systems where analog inaccuracies and defects can be controlled. Modular networks provide a way of handling inaccuracies. However, for large analog systems, adaptation mechanisms are needed to maintain accuracy. Even if the accuracy were a few percent, a system of only a hundred neurons would be inaccurate across chips. In biological systems, if one were to place the connection strengths found in brain A onto the structures of brain B, the result would be chaos rather than a brain transplant The robustness of neural systems depends on having the neurons and synapses adapt to the particular environment they find themselves in. Nevertheless, some amount of hardwiring is probably possible in modular systems if it is modifiable by a trainable portion of the network. A speech recognition system may, for example, adapt in real time to the accents and timbre of a particular speaker. It is also likely that the system would require at least partial training beforehand for robustness. We plan to design a larger version of our test chip containing both neurons and synapses which can form part of a still larger multiple chip network with the addition of chips containing only synapses. This next chip will have self-powered synapses so that each neuron need only signal its state rather than drive an unknown number of neurons from other chips. In addition, the noise generator will be improved so that true annealing is possible. We may also go further toward a fully analog chip[2] by having a variable gain neuron. Analog charge domain storage of weights and transport of states would further reduce the silicon area necessary but the technology required is not standard. There are many challenges in scaling learning networks up to the 1 ()4 neuron and 1 ()6 synapse range although these large electronic learning networks will have on the order of a billionfold speed advantage over simulations based on serial computers. Thus they may be able to address many longstanding problems in artificial intelligence which have resisted attack by more conventional methods. 760 Alspector, Gupta and Allen References 1. J. Alspector & R.B. Allen, "A neuromorphic VLSI learning system", in Advanced Research in VLSI: Proceedings of the 1987 Stanford Conference. edited by P. Losleben (MIT Press, Cambridge, MA, 1987) pp. 313-349. 2. J. Alspector, R.B. Allen, V. Hu, & S. Satyanarayana, "Stochastic learning networks and their electronic implementation", Neural Information Processing Systems (Denver, Nov. 1987) pp. 9-21. 3. D.H. Ackley, G.E. Hinton, & T J. Sejnowski, "A learning algorithm for Boltzmann machines", Cognitive Science 9 (1985) pp. 147-169. 4. B. Widrow & M.E. Hoff, "Adaptive switching circuits". IRE WESCON Convention Record Part 4, (1960) pp. 96-104. 5. F. Rosenblatt. Principles of neurodynamics: Perceptrons and the theory of brain mechanisms. Spartan Books, Washington. D.C. (1961). 6. J. Alspector, "A VLSI approach to neural-style information processing", in VLSI Signal Processing III. edited by R.W. Brodersen and H.S. Moscovitz (IEEE Press, New York, 1988) pp. 232-243. 7. T.K. Landauer, C. Kamm, & S. Singhal, "Teaching a minimally structured backpropagation network to recognize speech sounds", Proceedings of the Cognitive Science Society (Seattle, Aug. 1987) pp. 531-536. 8. G. Tesauro & B. Janssens, "Scaling relationships in back-propagation learning". Complex Systems 2 (1988) pp. 39-44.
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SONG LEARNING IN BIRDS M. Konishi Division of Biology California Institute of Technology Birds sing to communicate. Male birds use song to advertise their territories and attract females. Each bird species has a unique song or set of songs. Song conveys both species and individual identity. In most species, young birds learn some features of adult song. Song develops gradually from amorphous to fixed patterns of vocalization as if crystals form out of liquid. Learning of a song proceeds in two steps; birds commit the song to memory in the first stage and then they vocally reproduce it in the second stage. The two stages overlap each other in some species, while they are separated by several months in other species. The ability of a bird to commit a song to memory is restricted to a period known as the sensitive phase. Vocal reproduction of the memorized song requires auditory feedback. Birds deafened before the second stage cannot reproduce the memorized song. Birds change vocal output until it matches with the memorized song, which thus serves as a template. Birds use a built-in template when a tutor model is not available. Exposure to a tutor model modifies this innate template. A series of brain nuclei controls song production and patterning. Recording multiand single neurons from this nuclei in the singing bird is possible. The learned temporal pattern of song is recognizable in the neural discharge of these nuclei. The need for auditory feedback for song learning suggests the presence of links between the auditory and vocal control systems. One such link is found in the HV c, one of the forebrain song nuclei. This nucleus contains neurons sensitive to sound in addition to those which control song production. In the white-crowned sparrow, the HVc contains neurons selective for the bird's own individual song. The stimulus selectivity of these neurons are thus shaped by the bird's hearing of its own voice during song development. [1] Konishi, M. (1985) Birdson: from behavior to neuron. Ann. Rev. Neurosci. 8:125-170. [2] Konishi, M. (1985) The role of auditory feedback in the control of vocalization in the white-crowned sparrow. Z. Tierpsychol. 22:770-783. [3] McCasland, J. S. (1987) Neuronal control of bird song production. J. Neurosci., 723-739. [4] Margoliash, D. (1983) Acoustic parameters underlying the responses of songspecific neurons in the white-crowned sparrow. J. Neurosci. 3:10389-1057. [5] Nottebohm, F. T., Stokes, M., & Leonard, C. M. (1976) Central control of song in the canary Serinus canarius. J. Compo Neurol. 165:457-486. 795
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384 MODELING SMALL OSCILLATING BIOLOGICAL NETWORKS IN ANALOG VLSI Sylvie Ryckebusch, James M. Bower, and Carver Mead California Instit ute of Technology Pasadena, CA 91125 ABSTRACT We have used analog VLSI technology to model a class of small oscillating biological neural circuits known as central pattern generators (CPG). These circuits generate rhythmic patterns of activity which drive locomotor behaviour in the animal. We have designed, fabricated, and tested a model neuron circuit which relies on many of the same mechanisms as a biological central pattern generator neuron, such as delays and internal feedback. We show that this neuron can be used to build several small circuits based on known biological CPG circuits, and that these circuits produce patterns of output which are very similar to the observed biological patterns. To date, researchers in applied neural networks have tended to focus on mammalian systems as the primary source of potentially useful biological information. However, invertebrate systems may represent a source of ideas in many ways more appropriate, given current levels of engineering sophistication in building neural-like systems, and given the state of biological understanding of mammalian circuits. Invertebrate systems are based on orders of magnitude smaller numbers of neurons than are mammalian systems. The networks we will consider here, for example, are composed of about a dozen neurons, which is well within the demonstrated capabilities of current hardware fabrication techniques. Furthermore, since much more detailed structural information is available about these systems than for most systems in higher animals, insights can be guided by real information rather than by guesswork. Finally, even though they are constructed of small numbers of neurons, these networks have numerous interesting and potentially even useful properties. CENTRAL PATTERN GENERATORS Of all the invertebrate neural networks currently being investigated by neurobiologists, the class of networks known as central pattern generators (CPGs) may be especially worthy of attention. A CPG is responsible for generating oscillatory neural activity that governs specific patterns of motor output, and can generate its pattern of activity when isolated from its normal neuronal inputs. This propModeling Small Oscillating Biological Networks 385 erty, which greatly facilitates experiments, has enabled biologists to describe several CPGs in detail at the cellular and synaptic level. These networks have been found in all animals, but have been extensively studied in invertebrates [Selverston, 1985]. We chose to model several small CPG networks using analog VLSI technology. Our model differs from most computer simulation models of biological networks [Wilson and Bower, in press] in that we did not attempt to model the details of the individual ionic currents, nor did we attempt to model each known connection in the networks. Rather, our aim was to determine the basic functionality of a set of CPG networks by modeling them as the minimum set of connections required to reproduce output qualitatively similar to that produced by the real network under certain conditions. MODELING CPG NEURONS The basic building block for our model is a general purpose CPG neuron circuit. This circuit, shown in Figure 1, is our model for a typical neuron found in central pattern generators, and contains some of the essential elements of real biological neurons. Like real neurons, this model integrates current and uses positive feedback to output a train of pulses, or action potentials, whose frequency depends on the magnitude of the current input. The part of the circuit which generates these pulses is shown in Figure 2a [Mead, 19891. The second element in the CPG neuron circuit is the synapse. In Figure 1, each pair of transistors functions as a synapse. The p-well transistors are. excitatory synapses, whereas the n-well transistors are inhibitory synapses. One of the transistors in the pair sets the strength of the synapse, while the other transistor is the input of the synapse. Each CPG neuron has four different synapses. The third element of our model CPG neuron involves temporal delays. Delays are an essential element in the function of CPGs, and biology has evolved many different mechanisms to introduce delays into neural networks. The membrane capacitance of the cell body, different rates of chemical reactions, and axonal transmission are just a few of the mechanisms which have time constants associated with them. In our model we have included synaptic delay as the principle source of delay in the network. This is modeled as an RC delay, implemented by the follower-integrator circuit shown in Figure 2b [Mead, 19891. The time constant of the delay is a function of the conductance of the amplifier, set by the bias G. A multiple time constant delay line is formed by cascading several of these elements. Our neuron circuit uses a delay line with three time constants. The synapses which are before the delay element are slow synapses, whereas the undelayed synapses are fa.st synapses. We fabricated the circuit shown in Figure 1 using CMOS, VLSI technology. Several of these circuits were put on each chip, with all of the inputs and controls going out to pads, so that these cells could be externally connected to form the network of interest. 386 Ryckebusch, Bower, and Mead slow excitation -t G Figure 1. The CPG neuron circuit. r I- Pulse Length ~ (a) Yout. pulse length Yout. -111 0 -QJ(b) Figure 2. (a). The neuron spike-generating circuit. (b). The follower-integrater circuit. Each delay box 0 contains a delay line formed by three follower-integrater circuits. The Endogenous Bursting Neuron One type of cell which has been found to play an important role in many oscillatory circuits is the endogenous bursting neuron. This type of cell has an intrinsic oscillatory membrane potential, enabling it to produce bursts of action potentials at rhythmic intervals. These cells have been shown to act both as external "pacemakers" which set the rhythm for the CPG, or as an integral part of a central pattern generator. Figure 3a shows the output from a biological endogenous bursting neuron. Figure 3b demonstrates how we can configure our CPG neuron to be an endogenous bursting neuron. The delay element in the cell must have three time constants in order for this circuit to oscillate stably. Note that in the circuit, the Modeling Small Oscillating Biological Networks 387 cell has internal negative feedback. Since real neurons don't actually make synaptic connections onto themselves, this connection should be thought of as representing an internal molecular or ionic mechanism which results in feedback within the cell. 4 mV AS 1 sec (a) (b) (c) Figure 3. (a). The output from the AB cell in the lobster stomatogastric ganglion CPG [Eisen and Marder, 1982]. This cell is known to burst endogenously. (b). The CPG neuron circuit configured as an endogenous bursting neuron and (c) the output from this circuit. Postinhibitory Rebound A neuron configured to be an endogenous burster also exhibits another property common to many neurons, including many CPG neurons. This property, illustrated in Figures 4a and 4b, is known as postinhibitory rebound (PIR). Neurons with this property display increased excitation for a certain period of time following the release of an inhibitory influence. This property is a useful one for central pattern generator neurons to have, because it enables patterns of oscillations to be reset following the release of inhibition. 388 Ryckebusch, Bower and Mead -(a) (b) I N' " ... .. I .. • ' LA 4 t •••• I ........ (c) Figure 4. (a) The output of a ganglion cell of the mudpuppy retina exhibiting postinhibitory rebound [Miller and Dacheux, 19761. The bar under the trace indicates the duration of the inhibition. (b) To exhibit PIR in the CPG neuron circuit, we inhibit ,the cell with the square pulse shown in (c). When the inhibition is released, the circuit outputs a brief burst of pulses. MODELING CENTRAL PATTERN GENERATORS The Lobster Stomatogastric Ganglion The stomatogastric ganglion is a CPG which controls the movement of the teeth in the lobster's stomach. This network is relatively complex, and we have only modeled the relationships between two of the neurons in the CPG (the PD and LP cells) which have a kind of interaction found in many CPGs known as reciprocal inhibition (Figure Sa). In this case, each cell inhibits the other, which produces a pattern of output in which the cells fire alternatively (Figure Sb). Note that in the absence of external input, a mechanism such as postinhibitory rebound must exist in order for a cell to begin firing again once it has been released from inhibition. (b) Modeling Small Oscillating Biological Networks 389 (a) (d) (c) _ 120 ~rrN Figure 5. (a) Output from the PD and LP cells in the lobster stomatogastric ganglion [Miller and Selverston, 1985]. (c) and (d) demonstrate reciprocal inhibition with two CPG neuron circuits. The Locust Flight CPG A CPG has been shown to play an important role in producing the motor pattern for flight in the locust [Robertson and Pearson, 19851. Two of the cells in the CPG, the 301 and 501 cells, fire bursts of action potentials as shown in Figure 6a. The 301 cell is active when the wings of the locust are elevated, whereas the 501 cell is active when the wings are depressed. The phase relationship between the two cells is very similar to the reciprocal inhibition pattern just discussed, but the circuit that produces this pattern is quite different. The connections between these two cells are shown in Figure 6b. The 301 cell makes a delayed excitatory connection onto the 501 cell, and the 501 cell makes fast inhibitory contact with the 301 cell. Therefore, the 301 cell begins to fire, and after some delay, the 501 cell is activated. When the 501 cell begins to fire, it immediately shuts off the 301 cell. Since the 501 cell is no longer receiving excitatory input, it will eventually stop firing, releasing the 301 cell from inhibition. The cycle then repeats. This same circuit has been reproduced with our model in Figures 6c and 6d. 390 Ryckebusch, Bower and Mead • ---.J50mv ~ 100ms \5'~'1 I--~ 501 301~~~~=-~~~~~~ __ ~ __ ~~ Dl (a) (b) 301 301 CELL SOl CELL (e) (d) Figure 6. (a) The 301 and 501 cells in the locust flight CPG [Robertson and Pearson, 19851. (b) Simultaneous intracellular recordings of 301 and 501 during flight. (c) The model circuit and (d) its output. The Tritonia Swim CPG One of the best studied central pattern generators is the CPG which controls the swimming in the small marine mollusc Tritonia. This CPG was studied in great detail by Peter Getting and his colleagues at the University of Iowa, and it is one of the few biological neural networks for which most of the connections and the synaptic parameters are known in detail. Tritonia swims by making alternating dorsal and ventral flexions. The dorsal and ventral motor neurons are innervated by the DSI and VSI cells, respectively. Shown in Figure 7a and 7b is a simplified schematic diagram for the network and the corresponding output. The DSI and VSI cells fire out of phase, which is consistent with the alternating nature of the animal's swimming motion. The basic circuit consists of reciprocal inhibition between DSI and VSI paralleled by delayed excitation via the C2 cell. The DSI and VSI cells fire out of phase, and the DSI and C2 cells fire in phase. Swimming is initiated by sensory stimuli which feed into DSI and cause it to begin to fire a burst of impulses. DSI inhibits VSI, and at the same time excites C2. C2 has excitatory synapses on VSIj however, the initial response of VSI neurons is delayed. VSI then fires, during which there is inhibition by VSI of C2 and DSI. During this period, VSI no longer receives excitatory input from C2, and hence the VSI firing rate declines; DSI is therefore released from inhibition, and is ready to fire again to initiate a new cycle. Figure 7c and 8 show the model circuit which is identical to the circuit Modeling Small Oscillating Biological Networks 391 shown in Figure 7a, and the output from this circuit. Note that although the model output closely resembles the biological data, there are small differences in the phase relationships between the cells which can be accounted for by taking into account other connections and delays in the circuit not currently incorporated in our model. (a) 1-0 OSI : I VSI B C2 .' .~ .' .' ........... . - - --I \ \ .. \ ..... \ .. .. \ ..... \ ..... , ...... \ DSI t' ... , ...... ' , , , \ ,---------(c) (b) C2 VSI 50 mV J 5 sec Figure 1. (a) Simplified schematic diagram of the Tritonia CPG (which actually has 14 cells) and (b) output from the three types of cells in the circuit.(c) The model circuit. 392 Ryckebusch, Bower and Mead VSI C2 DSI Figure 8. Output from the circuit shown in Figure 7c. CONCLUSIONS One may ask why it is interesting to model these systems in analog VLSI, or, for that matter, why it is interesting to model invertebrate networks altogether. Analog VLSI is a very nice medium for this type of modeling, because in addition to being compact, it runs in real time, eliminating the need to wait hours to get the results of a simulation. In addition, the electronic circuits rely on the same physical principles as neural processes {including gain, delays, and feedback}, allowing us to exploit the inherent properties of the medium in which we work rather than having to explicitly model them as in a digital simulation. Like all models, we hope that this work will help us learn something about the systems we are studying. But in addition, although invertebrate neural networks are relatively simple and have small numbers of cells, the behaviours of these networks and animals can be fairly complex. At the same time, their small size allows us to understand how they are engineered in detail. Accordingly, modeling these networks allows us to study a well engineered system at the component level-a level of modeling not yet possible for more complex mammalian systems, for which detailed structural information is scarce. Modeling Small Oscillating Biological Networks 393 Acknowledgments This work relies on information supplied by the hard work of many experimentalists. We would especially like to acknowledge the effort and dedication of Peter Getting who devoted 12 years to understanding the organization of the Tritonia network of 14 neurons. We also thank Hewlett-Packard for computing support, and DARPA and MOSIS for chip fabrication. This work was sponsored by the Office of Naval Research, the System Development Foundation, and the NSF (EET-8700064 to J.B.). Reference8 Eisen, Judith S. and Marder, Eve (1982). Mechanisms underlying pattern generation in lobster stomatogastric ganglion as determined by selective inactivation of identified neurons. III. Synaptic connections of electrically coupled pyloric neurons. J. Neurophysiol. 48:1392-1415. Getting, Peter A. and Dekin, Michael S. (1985). Tritonia swimming: A model system for integration within rhythmic motor systems. In Allen I. Selverston (Ed.), Model Neural Networks and Behavior, New York, NY: Plenum Press. Mead, Carver A. (in press). Analog VLSI and Neural Systems. Reading, MA: Addison-Wesley. Miller, John P. and Selverston, Allen I. (1985). Neural Mechanisms for the production of the lobster pyloric motor pattern. In Allen I. Selverston (Ed.), Model Neural Networks and Behavior, New York, NY: Plenum Press. Miller, R. F. and Dacheux, R. F. (1976). Synaptic organization and ionic basis of on and off channels in mudpuppy retina. J. Gen. Physiol. 67:639-690. Robertson, R. M. and Pearson, K. G. (1985). Neural circuits in the ft.ight system of the locust. J. Neurophysiol. 53:110-128. Selverston, Allen I. and Moulins, Maurice (1985). Oscillatory neural networks. Ann. Rev. Physiol. 47:29-48. Wilson, M. and Bower, J. M. (in press). Simulation oflarge scale neuronal networks. In C. Koch and I. Segev (Eds.), Methods in Neuronal Modeling: From Synapses to Networks, Cambridge j MA: MIT Press.
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COMPARING BIASES FOR MINIMAL NETWORK CONSTRUCTION WITH BACK-PROPAGATION Lorien Y. Pratt Rutgers University Stephen Jo~ Hansont Bell Communications Research Morristown. New Jersey 07960 New Brunswick. New Jersey 08903 ABSTRACT Rumelhart (1987). has proposed a method for choosing minimal or "simple" representations during learning in Back-propagation networks. This approach can be used to (a) dynamically select the number of hidden units. (b) construct a representation that is appropriate for the problem and (c) thus improve the generalization ability of Back-propagation networks. The method Rumelhart suggests involves adding penalty terms to the usual error function. In this paper we introduce Rumelhart·s minimal networks idea and compare two possible biases on the weight search space. These biases are compared in both simple counting problems and a speech recognition problem. In general. the constrained search does seem to minimize the number of hidden units required with an expected increase in local minima. INTRODUCTION Many supervised connectionist models use gradient descent in error to solve various kinds of tasks (Rumelhart. Hinton & Williams. 1986). However. such gradient descent methods tend to be ".opportunistic" and can solve problems in an arbitrary way dependent on starting point in weight space and peculiarities of the training set. For example. in Figure 1 we show a "mesh" problem which consists of a random distribution of exemplars from two categories. The spatial geometry of the categories impose a meshed or overlapping subset of the exemplars in the two dimensional feature space. As the meshed part of the categories increase the problem becomes more complex and must involve the combination of more linear cuts in feature space and consequently more nonlinear cuts for category separation. In the top left corner of Figure l(a). we show a mesh geometry requiring only three cuts for category separation. In the bottom center t Also member of Cognitive Science Laboratory, 221 Nassau Street, Princeton University, Princeton, New lersey,08S42 177 178 Hanson and Pratt I (b) is the projection of the three cut solution of the mesh in output space. In the top right of this Figure I(c) is a typical solution provided by back-propagation starting with 16 hidden units. This Figure shows the two dimensional featme space in which 9 of the lines cuts are projected (the other 7 are outside the [0.1] unit plane). ~ r-------------------------~ 0.0 0.2 U 0.' 0.8 '.0 Figure 1: Mesh problem (a). output space (b) and typical back-propagation solution (c) Examining the weights in the next layer of the network indicates that in fact. 7 of these 9 line segments are used in order to construct the output surface shown in Figure l(b). Consequently. the underlying feature relations determining the output surface and category separation are arbitrary. more complex then necessary and may result in anomalous generalizations. Rumelhart (1987). has proposed a way to increase the generalization capabilities of learning networks which use gradient descent methods and to automatically control the resources learning networks use-for example. in tenns of "hidden" units. His hypothesis concerns the nature of the 'representation in the network: " ... the simplest most robust network which accounts/or a data set will, on awrage,lead to the best generalization to the population from which the training set has been drawn". The basic approach involves adding penalty terms to the usual error function in order to constrain the search and cause weights to differentially decay. This is similar to many proposals in statistical regression where a "simplicity" measure is minimized along with the error term and is sometimes referred to as "biased" regression (Rawlings. 1988). Basically. the statistical concept of biased regression derives from parameter estimation approaches that attempt to achieve a best linear unbiased estimator ("BLUE"). By definition an unbiased estimator is one with the lowest possible variance and theoretically. unless there is significant collinearityl or nonlinearity amongst the 1. For example, Ridge regreuiolt is a special case of biased regression which attempts to make a singular correlation matrix non-lingular by adding a small arbitrary coostant to the diagonal of the matrix. This increase in the diagonal may lower the impact of the off-diagonal elements and thus reduce the effects of collinearity • Comparing Biases for Minimal Network Construction 179 variables. a least squares estimator (LSE) can be also shown to be a BLUE. If on the other hand. input variables are correlated or nonlinear with the output variables (as is the case in back-propagation) then there is no guarantee that the LSE will also be unbiased. Consequently. introducing a bias may actually reduce the variance of the estimator of that below the theoretically unbiased estimator. Since back-propagation is a special case of multivariate nonlinear regression methods we must immediately give up on achieving a BLUE. Worse ye4 the input variables are also very likely to be collinear in that input data are typically expected to be used for feature extraction. Consequently. the neural network framework leads naturally to the exploration of biased regression techniques. unfortunately. it is not obvious what sorts of biases ought to be introduced and whether they may be problem dependent Furthennore. the choice of particular biases probably determines the particular representation that is chosen and its nature in tenns of size. structure and "simplicity". This representation bias may in turn induce generalization behavior which is greater in accuracy with larger coverage over the domain. Nonetheless. since there is no particular motivation for minimizing a least squares estimator it is important to begin exploring possible biases that would lead to lower variance and more robust estimators. In this paper we explore two general type of bias which introduce explicit constraints on the hidden units. First we discuss the standard back-propagation method. various past methods of biasing which have been called "weight decay". the properties of our biases. and finally some simple benchmark tests using parity and a speech recognition task. BACK·PROPAGATION The Back-propagation method [2] is a supervised learning technique using a gradient descent in an error variable. The error is established by comparing an output value to a desired or expected value. These errors can be accumulated over the sample: E = LL (yu - ;ir)2 (1) • i Assuming the output function is differentiable then a gradient of the error can be found, and we require that this derivative be decreasing. dE --=0 dWij • (2) Over multiple layers we pass back a weighted sum of each derivative from units above. WEIGHT DECAY Past wo~ using biases have generally been based on ad hoc arguments that weights should differentially decay allowing large weights to persist and small weights to tend 180 Hanson and Pratt towards zero sooner. Apparently. this would tend to concentrate more of the input into a smaller number of weights. Generally. the intuitive notion is to somehow reduce the complexity of the network as defined by the nmnber of connections and number of hidden units. A simple but inefficient way of doing this is to include a weight decay tenn in the usual delta updating rule causing all weights to decay on each learning step (where W = Wjj throughout): (3) Solving this difference equation shows that for P < 1.0 weights are decaying exponentially over steps towards zero. " . aE w" = a 1: P"'" (--)j + P" Wo (4) ;=1 Ow This approach introduces the decay tenn in the derivative itself causing error tenns to also decrease over learning steps which may not be desirable. BIAS The sort of weight decay just discussed can also be derived from genezal consideration of "costs" on weights. For example it is possible to consider E with a bias tenn which in the simple decay case is quadratic with weight value (Le. w2). We now combine this bias with E producing an objective function that includes both the error term and this bias function: O=E+B (5) where. we now want to minimize ao = aE + aB (6) o~·· ow·· ow·· 'I 'I 'I In the quadratic case the updating rule becomes. aE W,,+1 = a (- :\.... - 2w,,) + w" (7) ClW;J Solving this difference equation derives the updating rule from equation 4. " . oE w" = a I:(1-2a)""'(- Ow ); + (l-2a)"wo (8) lal ij In this case. however without introduction of other parameters. a is both the learning rate 2. MOlt tX the wort discussed here has not been previously publilhed but nonetheless has entered into general use in many cormectionisl models and wu recently summarized on the COlIMctionist Bw/~tin Board by John Kruschke. Comparing Biases for Minimal Network Construction 181 and related to the decay tenn and must be strictly < ~ for weight decay. Unifonn weight decay has a disadvantage in that large weights are decaying at the same rate as small weights. It is possible to design biases that influence weights only when they are relatively small or even in a particular range of values. For example. Rumelhart has entertained a number of biases. one fonn in particular that we will also explore is based on a rectangular hyperbolic function. w1 B:: (1+w2) (9) It is infonnative to examine the derivative associated with this function in order to understand its effect on the weight updates. dB 2w dwij ::- (1+w2)1 (10) This derivative is plotted in Figure 2 (indicated as Rumelhart) and is non-monotonic showing a strong differential effect on small weights (+ or -) pushing them towards zero. while near zero and large weight values are not significantly affected. BIAS PER UNIT It is possible to consider bias on each hidden unit weight group. This has the potentially desirable effect of isolating weight changes to hidden unit weight groups and could effectively eliminate hidden units. Consequently. the hidden units are directly determining the bias. In order to do this. first define w·::~lw··1 I If..I 'I' (11) j where i is the ith hidden unit. Hyperbolic Bias Now consider a function similar to Rumelhart's but this time with Wi, the ith hidden group as the variable. W· B' - 1 +AWi· The new gradient includes the term from the bias which is. aB Asgn(wij) - dWij = (1 +Wi)2 Exponential Bias A similar kind of bias would be to consider the negative exponential: (12) (13) 182 Hanson and Pratt (14) This bias is similar to the hyperbolic bias tenn as above but involves the exponential which potentially produce more unifonn and gradual rate changes towards zero, dB sgn(wij) --= (15) dWij ( e ~Wi) . The behavior of these two biases (hyperbolic, exponential) are shown as function of weight magnitudes in Figure 2. Notice that the exponential bias term is more similar in slope change to RumelharCs (even though his is non-monotonic) than the hyperbolic as weight magnitude to a hidden unit increases. q II) i d 0 ~ 0 'iI d 'i 'tJ an 9 q -. -3 -2 -1 o weightvalu8 1 2 3 Figure 2: Bias function behavior of Rumelharfs, Hyperbolic and Exponential Obviously there are many more kinds of bias that one can consider. These two were chosen in order to provide a systematic test of varying biases and exploring their differential effectiveness in minimizing network complexity. SOME COMPARISONS Parity These biased Back-propagation methods were applied to several counting problems and to a speech (digit) recognition problem. In the following graphs for example, we show the results of 100 runs of XOR and 4-bit parity at 11 =.1 (learning rate) and ex=.8 (moving average) starting with 10 hidden units. The parameter A. was optimized for the bias runs. Comparing Biases for Minimal Network Construction 183 II I , I: ......... ., .... ---...... oJ~~~~~~~=-~. • • \I 12 _._J: J: ...-------.--a 4 12 _.-~-----r....... ---r-rrt-~ , , , , , . • I •• .'1 _._Figure 3: Exclusive OR runs for standard, hyperbolic and exponential biasing Shown are runs for the standard case without biases, the hyperbolic bias and the exponential bias. Once a solution was reached all hidden Wlits were tested individually by removing each of them one at a time from the network and then testing on the training set Any hidden unit which was unnecessary was removed for data analysis. Only the number of these "functional units" are reported in the histograms. Notice the number of hidden units decrease with bias runs. An analysis of variance (statistical test) verified this improvement for both the hyperbolic and exponential over the standard. Also note that the exponential is significandy better than the hyperbolic. This is also confinned for the 4-bit parity case as shown in Figure 4. - -_ .... _... --..... -I' • II' -_. __ .... _........ -----. . --..........-..-.. _. , -_. Figure 4: four-bit parity runs for standard. hyperbolic and exponential biasing 184 Hanson and Pratt Speech Recognition Samples of 10 spoken digits (0-9) each were collected (same speaker throughout--DJ. BUlT kindly supplied data). Samples were then preprocessed using FFTs retaining the first 12 Cepstral coefficients. To avoid ceiling effects only two tokens each of the 10 digits were used for training ("0", "0","1","1", .... "9",.,9., .. ) each network. Eight such 2 token samples were used for replications. Another set of 50 spoken digits (5 samples of each of the 10 digits) were collected for transfer. All runs were matched across methods for number ofleaming sweeps «300),11=.05, a=.2, and A=.01 which were optimized for the exponential bias. Shown in the following table is the results of the 8 replications for the standard and the exponential bias. doll COIII&I'IiIlecl{up 1 IIIIIDIe 1'rInIrer , HIdden Units TrlDafer 'HWcnUnill rl 5K II 64~ 10 r2 6K 11 76~ 13 r3 62~ 18 64" 14 1'4 6A 14 74" 14 d 62~ 16 56 .. 11 16 c569& 19 68 .. 14 t7 58" 18 54" 11 IS sa" 18 64 .. 9 17%.56 65~ 12.%.71 Table 1: Eight replications with transfer for standard and exponential bias. In this case there appears to both an improvement in the average number of hidden units (functional ones) and transfer. A typical correlation of the improved transfer and reduced hidden unit usage for a single replication is plotted in the next graph. J! ~ y- -1.21+ 71.7. ,- -.trT l ~ I 2 I :I i I/) or ~ 10 12 14 18 11 I1IMnber of hidden unIIs Figure 5: Transfer as a function of hidden unit usage for a single replication We note that introduction of biases decrease the probability of convergence relative to the standard case (as many as 75% of the parity runs did not converge within criteria Comparing Biases for Minimal Network Construction 185 number of sweeps.) Since the search problem is made more difficult by introducing biases it now becomes even more important to explore methods for improving convergence similar for example. to simulated annealing (Kirkpatrick. Gelatt & Vecchi. 1983) CONCLUSIONS Minimal networks were defined and two types of bias were compared in a simple counting problem and a speech recognition problem. In the counting problems under biasing conditions the number hidden units tended to decrease towards the minimum required for the problem although with a concomitant decrease in convergence rate. In the speech problem also under biasing conditions the number of hidden units tended to decrease as the transfer rate tended to improve. Acknowledgements We thank Dave Rumelhart for discussions concerning the minimal network concept. the Bellcore connectionist group and members of the Princeton Cognitive Science Lab for a lively environment for the development of these ideas. References Kirkpalrick. S .• Gelatt. C. D .• & Vecchi. M. P .• Optimization by simulated annealing. Science. 220. 671-680. (1983). Rawlings. I. 0 •• Applied Regression Analysis. Wadsworth & Brooks/Cole, (1988). Rumelhart D. E .• Personal Communication, Princeton. (1987). Rumelhart D. E., Hinton G. E .• & Williams R .• Learning Internal Representations by error propagation. Nature. (1986).
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WHAT SIZE NET GIVES VALID GENERALIZATION?* Eric B. Baum Department of Physics Princeton University Princeton NJ 08540 David Haussler Computer and Information Science University of California Santa Cruz, CA 95064 ABSTRACT We address the question of when a network can be expected to generalize from m random training examples chosen from some arbitrary probability distribution, assuming that future test examples are drawn from the same distribution. Among our results are the following bounds on appropriate sample vs. network size. Assume o < £ $ 1/8. We show that if m > O( ~log~) random examples can be loaded on a feedforward network of linear threshold functions with N nodes and W weights, so that at least a fraction 1 - t of the examples are correctly classified, then one has confidence approaching certainty that the network will correctly classify a fraction 1 £ of future test examples drawn from the same distribution. Conversely, for fully-connected feedforward nets with one hidden layer, any learning algorithm using fewer than O( '!') random training examples will, for some distributions of examples consistent with an appropriate weight choice, fail at least some fixed fraction of the time to find a weight choice that will correctly classify more than a 1 £ fraction of the future test examples. INTRODUCTION In the last few years, many diverse real-world problems have been attacked by back propagation. For example "expert systems" have been produced for mapping text to phonemes [sr87], for determining the secondary structure of proteins [qs88], and for playing backgammon [ts88]. In such problems, one starts with a training database, chooses (by making an educated guess) a network, and then uses back propagation to load as many of the training examples as possible onto the network. The hope is that the network so designed will generalize to predict correctly on future examples of the same problem. This hope is not always realized. * This paper will appear in the January 1989 issue of Neural Computation. For completeness, we reprint this full version here, with the kind permission of MIT Press. © 1989, MIT Press 81 82 Baum and Haussler We address the question of when valid generalization can be expected. Given a training database of m examples, what size net should we attempt to load these on? We will assume that the examples are drawn from some fixed but arbitrary probability distribution, that the learner is given some accuracy parameter E, and that his goal is to produce with high probability a feedforward neural network that predicts correctly at least a fraction 1 E of future examples drawn from the same distribution. These reasonable assumptions are suggested by the protocol proposed by Valiant for learning from examples [val84]. However, here we do not assume the existence of any "target function"; indeed the underlying process generating the examples may classify them in a stochastic manner, as in e.g. [dh73]. Our treatment of the problem of valid generalization will be quite general in that the results we give will hold for arbitrary learning algorithms and not just for back propagation. The results are based on the notion of capacity introduced by Cover [cov65] and developed by Vapnik and Chervonenkis [vc7l], [vap82]. Recent overviews of this theory are given in [dev88], [behw87b] and [poI84], from the various perspectives of pattern recognition, Valiant's computational learning theory, and pure probability theory, respectively. This theory generalizes the simpler counting arguments based on cardinality and entropy used in [behw87a] and [dswshhj87], in the latter case specifically to study the question of generalization in feedforward nets (see [vap82] or [behw87b]). The particular measures of capacity we use here are the maximum number of dichotomies that can be induced on m inputs, and the Vapnik-CheMlonenki. (Ve) Dimen.ion, defined below. We give upper and lower bounds on these measures for classes of networks obtained by varying the weights in a fixed feedforward architecture. These results show that the VC dimension is closely related to the number of weights in the architecture, in analogy with the number of coefficients or "degrees of freedom" in regression models. One particular result, of some interest independent of its implications for learning, is a construction of a near minimal size net architecture capable of implementing all dichotomies on a randomly chosen set of points on the n-hypercube with high probability. Applying these results, we address the question of when a network can be expected to generalize from m random training examples chosen from some arbitrary probability distribution, assuming that future test examples are drawn from the same distribution. Assume 0 < E < 1/8. We show that ifm ~ O(~log.r:) random examples can be loaded on a feedforward network of linear threshold functions with N nodes and W weights, so that at least a fraction 1 - j of the examples are correctly classified, then one has confidence approaching certainty that the network will correctly classify a fraction 1 E of future test examples drawn from the same distribution. Conversely, for fully-connected feedforward nets with one hidden layer, any learning algorithm using fewer than O( ~) random training examples will, for some distributions of examples consistent with an appropriate weight choice, fail at least some fixed fraction of the time to find a weight choice that will correctly classify more than a 1 E fraction of the future test examples. What Size Net Gives Valid Generalization? 83 Ignoring the constant and logarithmic factors, these results suggest that the appropriate number of training examples is approximately the number of weights times the inversel of the accuracy parameter E. Thus, for example, if we desire an accuracy level of 90%, corresponding to E = 0.1, we might guess that we would need about 10 times as many training examples as we have weights in the network. This is in fact the rule of thumb suggested by Widrow [wid87], and appears to work fairly well in practice. At the end of Section 3, we briefly discuss why learning algorithms that try to minimize the number of non-zero weights in the network [rum87] [hin87] may need fewer training examples. DEFINITIONS We use In to denote the natural logarithm and log to denote the logarithm base 2. We define an ezample as a pair (i, a), i E ~n, a E {-I, +1}. We define a random sample as a sequence of examples drawn independently at random from some distribution D on ~n X {-1, +1}. Let I be a function from ~n into {-1, +1}. We define the error of I, with respect to D, as the probability a;/; I(i) for (i,a) a random example. Let F be a class of {-1, +l}-valued functions on ~n and let S be a set of m points in ~n . A dichotomy of S induced by I E F is a partition of S into two disjoint subsets S+ and S- such that I(i) = +1 for i E S+ and I(i) = -1 for i E S-. By .6.F (S) we denote the number of distinct dichotomies of S induced by functions I E F, and by .6.F(m) we denote the maximum of .6.F(S) over all S C ~n of cardinality m. We say S is shattered by F if .6.F(S) = 2151, i.e. all dichotomies of S can be induced by functions in F. The Vapnik-CheMlonenkis (VC) dimension of F, denoted VCdim(F), is the cardinality of the largest S C ~n that is shattered by F, i.e. the largest m such that .6.F ( m) = 2m • A feedforward net with input from ~n is a directed acyclic graph G with an ordered sequence ofn source nodes (called inputs) and one sink (called the output). Nodes of G that are not source nodes are called computation nodes, nodes that are neither source nor sink nodes are called hidden nodes. With each computation node n. there is associated a function" : ~inde't'ee(n,) ~ {-I, +1}, where indeg7'ee(n.) is the number of incoming edges for node n,. With the net itself there is associated a function I : ~n ~ {-I, +1} defined by composing the I,'s in the obvious way, assuming that component i of the input i is placed at the it" input node. A Jeedlorward architecture is a class of feedforward nets all of which share the same underlying graph. Given a graph G we define a feedforward architecture by associating to each computation node n, a class of functions F, from ~'nde't'ee(n,) 1 It should be noted that our bounds differ significantly from those given in [dev88] in that the latter exhibit a dependence on the inverse of e2• This is because we derive our results from Vapnik's theorem on the uniform relative deviation of frequencies from their probabilities ([vap82], see Appendix A3 of [behw87b]), giving sharper bounds as E approaches o. 84 Baum and Haussler to {-I, +1}. The resulting architecture consists of all feedforward nets obtained by choosing a particular function" from F, for each computation node ft,. We will identify an architecture with the class offunctions computed by the individual nets within the architecture when no confusion will arise. CONDITIONS SUFFICIENT FOR VALID GENERALIZATION Theorem 1: Let F be a feedforward architecture generated by an underlying graph G with N > 2 computation nodes and F, be the class of functions associated with computation node ft, of G, 1 < i < N. Let d = E~l VCdim(Fl). Then AF(m) < n~lAF,(m)::; (Nem/d)d for m > d, where e is the base of the natural logarithm. Proof: Assume G has n input nodes and that the computation nodes of G are ordered so that node n, receives inputs only from input nodes and from computation nodes nj, 1 < j ::; i-I. Let S be a set of m points in ~n. The dichotomy induced on S by the function in node nl can be chosen in at most AFI (m) ways. This choice determines the input to node nz for each of the m points in S. The dichotomy induced on these m inputs by the function in node nz can be chosen in at most AF:a(m) ways, etc. Any dichotomy of S induced by the whole network can be obtained by choosing dichotomies for each of the ni's in this manner, hence AF(m) < nf:l AF,(m). By a theorem of Sauer [sau72], whenever VCdim(F) = Ie < 00, AF(m) < (em/Ie)l for all m > Ie (see also [behw87b]). Let ~ = VCdim(Fi), 1 < i < N. Thus d = Ef:l~. Then n~l AF,(m) < n~l(em/~)'" for m > d. Using the fact that E~l -ailogai < logN whenever a. > 0, 1 < i < N, and E~l ai = I, and setting ai = ~/d, it is easily verified that n~l ~d. > (d/N)d. Hence n~l(em/di)d. < (Nem/d)d. Corollary 2: Let F be the class of all functions computed by feedforward nets defined on a fixed underlying graph G with E edges and N > 2 computation nodes, each of which computes a linear threshold function. Let W = E + N (the total number of weights in the network, including one weight per edge and one threshold per computation node). Then AF(m) < (Nem/W)W for all m > Wand VCdim(F) < 2Wlog(eN). Proof: The first inequality follows from directly from Theorem 1 using the fact that VCdim(F) = Ie + 1 when F is the class of all linear threshold functions on ~l (see e.g. [wd81]). For the second inequality, it is easily verified that for N > 2 and m = 2Wlog(eN), (N em/W)W < 2m. Hence this is an upper bound on VCdim(F). Using VC dimension bounds given in [wd81], related corollaries can be obtained for nets that use spherical and other types of polynomial threshold functions. These bounds can be used in the following. What Size Net Gives Valid Generalization? 85 Theorem 3 [vapS2} (see [behw87b), Theorem A3.3): Let F be a class offunctions2 on ~n, 0 < l' < 1,0 < £,6 < 1. Let S be a random sequence of m examples drawn independently according to the distribution D. The probability that there exists a function in F that disagrees with at most a fraction (1 - 1')£ of the examples in S and yet has error greater than £ (w.r.t. D) is less than From Corollary 2 and Theorem 3, we get: Corollary 4: Given a fixed graph G with E edges and N linear threshold units (i.e. W = E + N weights), fixed 0 < £ < 1/2, and m random training examples, where 32W1 32N m>-n-, £ € if one can find a choice of weights so that at least a fraction 1-£/2 of the m training examples are correctly loaded, then one has confidence at least 1 - Se- 1•5W that the net will correctly classify all but a fraction € of future examples drawn from the same distribution. For 64W I 64N m > --;- n--;-, the confidence is at least 1 - Se-em/S2. Proof: Let l' = 1/2 and apply Theorem 3, using the bound on aF(m) given in Corollary 2. This shows that the probability that there exists a choice of the weights that defines a function with error greater than £ that is consistent with at least a fraction 1 - £/2 of the training examples is at most When m = !ll!.ln!!K this is S(2e E In!!K)W which is less than Se-1.5W for N > e e' 3fN' E ' 2 and £ < 1/2. When m > 84EW In 8~N, (2N em/W) W < eEm/S2, so S(2N em/W) W e- Em/16 < Se-em/S2. The constant 32 is undoubtably an overestimate. No serious attempt has been made to minimize it. Further, we do not know if the log term is unavoidable. Nevertheless, even without these terms, for nets with many weights this may represent a considerable number of examples. Such nets are common in cases where the complexity of the rule being learned is not known in advance, so a large architecture is chosen 2 We assume some measurability conditions on the class F. See [poI84], [behwS7b1 for details. 86 Baum and Haussler in order to increase the chances that the rule can be represented. To counteract the concomitant increase in the size of the training sample needed, one method that has been explored is the use of learning algorithms that try to use as little of the architecture as possible to load the examples, e.g. by setting as many weights to zero as possible, and by removing as many nodes as possible (a node can be removed if all its incoming weights are zero.) [rumS7] [hin87]. The following shows that the VC dimension of such a "reduced" architecture is not much larger than what one would get if one knew a priori what nodes and edges could be deleted. Corollary 5: Let F be the class of all functions computed by linear threshold feedforward nets defined on a fixed underlying graph G with N' > 2 computation nodes and E' ~ N' edges, such that at most E > 2 edges have non-zero weights and at most N ~ 2 nodes have at least one incoming edge with a non-zero weight. Let W = E + N. Then the conclusion of Corollary 4 holds for sample size 32W l 32NE' m>-n--f f Prool sketch: We can bound dF( m) by considering the number of ways the N nodes and E edges that remain can be chosen from among those in the initial network. A crude upper bound is (N')N (E')E. Applying Corollary 2 to the remaining network gives dF(m) ~ (N')N(E')E(Nem/W)w. This is at most (N E'em/W)w. The rest of the analysis is similar to that in Corollary 4. This iridicates that minimizing non-zero weights may be a fruitful approach. Similar approaches in other learning contexts are discussed in [hauSS] and [litSS]. CONDITIONS NECESSARY FOR VALID GENERALIZATION The following general theorem gives a lower bound on the number of examples needed for distribution-free learning, regardless of the algorithm used. Theorem 6 [ehkvS7] (see also [behw87b]): Let F be a class of {-I, +1}-valued functions on ~n. with VCdim(F) > 2. Let A be any learning algorithm that takes as input a sequence of {-I, +1}-labeled examples over ~n. and produces as output a function from ~n. into {-I, +1}. Then for any 0 < f ~ l/S, 0 < 0 ~ l~ and [1- fl 1 VCdim(F) -1] m < maz n7' 3 ' e v 2e there exists (1) a function I E F and (2) a distribution D on ~n X {-I, +1} for which Prob((E, a) : a f. I(E)) = 0, such that given a random sample of size m chosen according to D, with probability at least 0, A produces a function with error greater than e. What Size Net Gives Valid Generalization? 87 This theorem can be used to obtain a lower bound on the number of examples needed to train a net, assuming that the examples are drawn from the worst-case distribution that is consistent with some function realizable on that net. We need only obtain lower bounds on the VC dimension of the associated architecture. In this section we will specialize by considering only fully-connected networks of linear threshold units that have only one hidden layer. Thus each hidden node will have an incoming edge from each input node and an outgoing edge to the output node, and no other edges will be present. In [b88] a slicing construction is given that shows that a one hidden layer net of threshold units with n inputs and 2j hidden units can shatter an arbitrary set of 2jn vectors in general position in ~". A corollary of this result is: Theorem 7: The class of one hidden layer linear threshold nets taking input from ~" with k hidden units has VC dimension at least 2L~Jn. Note that for large k and n, 2 L ~ J n is approximately equal to the total number W of weights in the network. A special case of considerable interest occurs when the domain is restricted to the hypercube: {+1,-1}". Lemma 6 of [lit88] shows that the class of Boolean· functions on {+1, _I}" represented by disjunctive normal form expressions with k terms, k < 0(2,,/2/Vn), where each term is the conjunction of n/2 literals, has VC dimension at least kn/4. Since these functions can be represented on a linear threshold net with one hidden layer of k units, this provides a lower bound on the VC dimension of this architecture. We also can use the slicing construction of [b88] to give a lower bound approaching kn/2. The actual result is somewhat stronger in that it shows that for large n a randomly chosen set of approximately kn/2 vectors is shattered with high probability. Theorem 8: With probability approaching 1 exponentially in n, a set S of m < 2,,/3 vectors chosen randomly and uniformly from {+1, _I}" can be shattered by the one hidden layer architecture with 2rm/l(n(1 1~0,,))J1linear threshold units in its hidden layer. Prool,ketch: With probability approaching 1 exponentially in n no pair of vectors in S are negations of each other. Assume n > eto. Let l' = In(lI~O,,)J. Divide S at random into r m/1' 1 disjoint subsets S1I ... , Srm/t'l each containing no more than l' vectors. We will describe a set T of ±1 vectors as Iliceable if the vectors in T are linearly independent and the subspace they span over the reals does not contain any ±l vector other than the vectors in T and their negations. In [od188] it is shown, for large n, that any random set of l' vectors has probability P = 4(;)(~)" +0(( 110)") of not being sliceable. Thus the probability that some S. is not sliceable is 0(mn2(~)"), which is exponentially small for m < 2,,/3. Hence with probability approaching 1 exponentially in n, each S, is sliceable, 1 ~ i $ r m/ 1'1. Consider any Boolean function I on S and let S: = {i E S, : f(i) = +1}, 88 Baum and Haussler 1 < i < r m/7' 1. If Si is sliceable and no pair of vectors in S are negations of each other then we may pass a plane through the points in st that doesn't contain any other points in S. Shifting this plane parallel to itself slightly we can construct two half spaces whose intersection forms a slice of~" containing st and no other points in S. Using threshold units at the hidden layer recognizing these two half spaces, with weights to the output unit +1 and -1 appropriately, the output unit receives input +2 for any point in the slice and 0 for any point not in the slice. Doing this for each S: and thresholding at 1 implements the function f. We can now apply Theorem 6 to show that any neural net learning algorithm using too few examples will be fooled by some reasonable distributions. Corollary 9: For any learning algorithm training Ii net with k linear threshold functions in its hidden layer, and 0 < l ~ 1/8, if the algorithm uses (a) fewer than 2llc/;'f,,-1 examples to learn a function from ~" to {-I, +1}, or (b) fewer than l"lll/2J(mQ,:I:(1/!~~-10/(ln n»)J-1 examples to learn a function from {-I, +1}" to {-I, +1}, for k ~ O(2n / 3 ), then there exist distributions D for which (i) there exists a choice of weights such that the network exactly classifies its inputs according to D, but (ii) the learning algorithm will have probability at least .01 of finding a choice of weights which in fact has error greater than E. CONCLUSION We have given theoretical lower and upper bounds on the sample size vs. net size needed such that valid generalization can be expected. The exact constants we have given in these formulae are still quite crudej it may be expected that the actual values are closer to 1. The logarithmic factor in Corollary 4 may also not be needed, at least for the types of distributions and architectures seen in practice. Widrow's experience supports this conjecture [wid87]. However, closing the theoretical gap between the O( ': log ~) upper bound and the (2 ( 1f) lower bound on the worst case sample size for architectures with one hidden layer of threshold units remains an interesting open problem. Also, apart from our upper bound, the case of multiple hidden layers is largely open. Finally, our bounds are obtained under the assumption that the node functions are linear threshold functions (or at least Boolean valued). We conjecture that similar bounds also hold for classes of real valued functions such as sigmoid functions, and hope shortly to establish this. Acknowledgements We would like to thank Ron Rivest for suggestions on improving the bounds given in Corollaries 4 and 5 in an earlier draft of this paper, and Nick Littlestone for many helpful comments. The research of E. Baum was performed by the Jet Propulsion Laboratory, California Institute of Technology, as part of its Innovative Space What Size Net Gives Valid Generalization? 89 Technology Center, which is sponsored by the Strategic Defense Initiative Organization/Innovative Science and Technology through an agreement with the National Aeronautics and Space Administration (NASA). D. Haussler gratefully acknowledges the support of ONR grant NOOOI4-86-K-0454. Part of this work was done while E. Baum was visiting UC Santa Cruz. References [b88]BAUM, E. B., (1988) On the capabilities of multilayer perceptrons, J. of Complexity, 4, 1988, ppI93-215. [behw87a]BLUMER, A., EHRENFEUCHT, A. HAUSSLER, D., WARMUTH, M., (1987), Occam's Razor, Int: Proc. Let., 24, 1987, pp377-380. [behw87b]BLUMER, A., EHRENFEUCHT, A. HAUSSLER, D., WARMUTH, M., (1987), Learnability and the Vapnik-Chervonenkis dimension, UC Santa Cruz Tech. Rep. UCSC-CRL-87-20 (revised Oct., 1988) and J. ACM, to appear. [cov65]COVER, T., (1965), Geometrical and statistical properties of systems of linear inequalities with applications to pattern recognition, IEEE Trans. Elect. Comp., V14, pp326-334. [dev88]DEVROYE, L., (1988), Automatic pattern recognition, a study of the probability of error, IEEE Trans. P AMI, V10, N 4, pp530-543. [dswshhj87]DENKER J., SCHWARTZ D., WITTNER B., SOLLA S., HOP FIELD J., HOWARD R., JACKEL L., (1987), Automatic learning, rule extraction, and generalization, Complex Systems 1 pp877-922. [dh73]DUDA, R., HART, P., (1973), Pattern clallification and scene analysis, Wiley, New York. [ehkv87]EHRENFEUCHT, A., llAUSSLER, D., KEARNS, M., VALIANT, L., (1987), A general lower bound on the number of examples needed for learning, UC Santa Cruz Tech. Rep. UCSC-CRL-87-26 and Information and Computation, to appear. [hau88]HAUSSLER, D., (1988), Quantifying inductive bias: AI learning algorithms and Valiant's learning framework, Artificial Intelligence, 36, 1988, pp177-221. [hin87]HINTON, G., (1987), Connectionist learning procedures, Artificial Intelligence, to appear. [lit88]LITTLESTONE, N., (1988) Learning quickly when irrelevant attributes abound: a new linear threshold algorithm, Machine Learning, V2, pp285-318. [odI88]ODLYZKO, A., (1988), On subspaces spanned by random selections of ±1 vectors, J. Comb. Th. A, V47, Nt, pp124-133. [poI84]POLLARD, D., (1984), Convergence 0/ stochastic procelles, Springer-Verlag, New York. ' 90 Baum and Haussler [qs88]QUIAN, N., SEJNOWSKI, T. J., (1988), Predicting the secondary structure of globular protein using neural nets, Bull. Math. Biophys. 5, 115-137. [rum87]RUMELHART, D., (1987), personal communication. [sau72]SAUER, N., (1972), On the density of families of sets, J. Comb. Th. A, V13, 145-147. [sr87]SEJNOWSKI, T.J., ROSENBERG, C. R., (1987), NET Talk: a parallel network that learns to read aloud, Complex Systems, vi pp145-168. [ts88]TESAURO G., SEJNOWSKI, T. J.,(1988), A 'neural' network that learns to play backgammon, in Neural Information Procelling Sy,tem" ed. D.Z. Anderson, AlP, NY, pp794-803. [val84]VALIANT, L. G., (1984), A theory of the learnable, Comm. ACM V27, Nil pp1l34-1142. [vc71]VAPNIK, V.N., Chervonenkis, A. Ya., (1971), On the uniform convergence of relative frequencies of events to their probabilities, Th. Probe and its Applications, V17, N2, pp264-280. [vap82]VAPNIK, V.N., (1982), E,timation of Dependence, Ba,ed on Empirical Data, Springer Verlag, NY. [wd81]WENOCUR, R. S., DUDLEY, R. M., (1981) Some special Vapnik-Chervonenkis classes, Discrete Math., V33, pp313-318. [wid87]WIDROW, B, (1987) ADALINE and MADALINE - 1963, Plenary Speech, Vol I, Proc. IEEE 1st Int. Conf. on Neural Networks, San Diego, CA, pp143-158.
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678 A LOW-POWER CMOS CIRCUIT WHICH EMULATES TEMWORALELECTIDCALPROPERTIES OF NEURONS Jack L. Meador and Clint S. Cole Electrical and Computer Engineering Dept. Washington State University Pullman WA. 99164-2752 ABSTRACf This paper describes a CMOS artificial neuron. The circuit is directly derived from the voltage-gated channel model of neural membrane, has low power dissipation, and small layout geometry. The principal motivations behind this work include a desire for high performance, more accurate neuron emulation, and the need for higher density in practical neural network implementations. INTRODUCTION Popular neuron models are based upon some statistical measure of known natural behavior. Whether that measure is expressed in terms of average firing rate or a firing probability, the instantaneous neuron activation is only represented in an abstract sense. Artificial electronic neurons derived from these models represent this excitation level as a binary code or a continuous voltage at the output of a summing amplifier. While such models have been shown to perform well for many applications, and form an integral part of much current work, they only partially emulate the manner in which natural neural networks operate. They ignore, for example, differences in relative arrival times of neighboring action potentials -- an important characteristic known to exist in natural auditory and visual networks {Sejnowski, 1986}. They are also less adaptable to fme-grained, neuron-centered learning, like the post-tetanic facilitation observed in natural neurons. We are investigating the implementation and application of neuron circuits which better approximate natural neuron function. BACKGROUND The major temporal artifacts associated with natural neuron function include the spacio-temporal integration of synaptic activity, the generation of an action potential (AP), and the post-AP hyperpolarization (refractory) period (Figure 1). Integration, manifested as a gradual membrane depolarization, occurs when the neuron accumulates sodium ions which migrate through pores in its cellular membrane. The rate of ion migration is related to the level of presynaptic AP bombardment, and is also known to be a non-linear function of transmembrane potential. Efferent AP generation occurs when the voltage-sensitive membrane of the axosomal hillock reaches some threshold potential whereupon a rapid increase in sodium permeability leads to A Low-Power CMOS Circuit Which Emulates Neurons 679 complete depolarization. Immediately thereafter, sodium pores "close" simultaneously with increased potassium permeability, thereby repolarizing the membrane toward its resting potential. The high potassium permeability during AP generation leads to the transient post-AP hyperpolarization state known as the refractory period. Actl vat Ion Threshold v ( 1) ( 3) Figure 1. Temporal artifacts associated with neuron function. (1) gradual depolarization, (2) AP generation, (3) refractory period. Several analytic and electronic neural models have been proposed which embody these characteristics at varying levels of detail. These neuromimes have been used to good advantage in studying neuron behavior. However, with the advent of artificial neural networks (ANN) for computing, emphasis has switched from modeling neurons for physiologic studies to developing practical neural network implementations. As the desire for high performance ANNs grows, models amenable to hardware implementation become more attractive. The general idea behind electronic neuromimes is not new. Beginning in 1937 with work by Harmon {Harmon, 1937},lIectronic circuits have been used to model and study neuronal behavior. In the late 196(Ys, Lewis {Lewis, 1968} developed a circuit which simulated the Hodgkin-Huxley model for a single neuron, followed by MacQregor's circuit {MacGregor, 1973} in the early 1970's which modelled a group of 50 neurons. With the availability of VLSI in the 1980's, electronic neural implementations have largely moved to the realm of integrated circuits. Two different strategies have been documented: analog implementations employing operational amplifiers {Graf, et at, 1987,1988; Sivilotti, et at, 1986; Raffel, 1988; Schwartz, et al, 1988}; and digital implementations such as systolic arrays {Kung, 1988}. More recently, impulse neural implementations are receiving increased attention. like other models, these neuromimes generate outputs based on some non-linear function of the weighted net inputs. However, interneuron communication is realized through impulse streams rather than continuous voltages or binary numbers {Murray, 1988; N. El-Leithy, 1987}. Impulse networks communicate neuron activation as variable pulse repetition rates. The impulse neuron circuits which shall be discussed offer both small geometry and low power dissipation as well as a closer approximation to natural neuron function. 680 Meador and Cole A CMOS IMPULSE NEURON An impulse neuron circuit developed for use in CMOS neural networks is shown in Figure 2. In this circuit, membrane ion current is modeled by charge flowing to and from Ca. Potassium and sodium influx is represented by current flow from V dd to the capacitor, and ion efflux by flow from the capacitor to ground. The Field EffectTransistors (FETs) connected between V dd, Vsr, and the capacitor emulate voltageand chemically-gated ion channels found in natural neural membrane. In the Figure, PET 1 corresponds to the post-synaptic chemicaIly-gated ion channels associated with one synapse. PETs 2, 3, and 4 emulate the voltage-gated channels distributed throughout a neuron membrane. The following equations summarize circuit operation: Ca dVa/dt=/31E (Vr,Va)+/3:uF(Va)-/34G (Va) E(Vr,Va)= (Vr-Va-V",)(V dd-Va)-(V dd-Va)2 /2 F(Y. ) ={(V dd-Vtp) (Va-V dd)-(Va-V dd)2 /2 if g(t) ~O a 0 otherwlSe G(V)={(Vdd-V",)Va-Va2/2 if h(t~)=O a 0 otherwlSe g(t) =h (t)(l-h (t-C)) 1 0 if Va(t) > Vth; h()Vtl<Va(t)<Vth and h (Va(t-e))=O t 1 if Va(t)<Vtl; Vtl<Va(t)<Vth and h (~(t-e))=l Vdd {52 10 Exci tator y Synapse {53 {51 + Vs + Va Ca 1 /5. Figure 2. A CMOS impulse neuron with one excitatory synapse-PET. (1) (2) (3) (4) (5) (6) Axon A Low-Power CMOS Circuit Which Emulates Neurons 681 Equation (1) expresses how changes in Va (which emulates instantaneous neuron excitation) depend upon the sum of three current components controlled by these PETs. E, F, and G in equations (2) through (4) express PET drain-source currents as functions terminal voltages. Equations (3) and (5) rely upon the assumption that PET 2 and PET 3 are implemented as a single dual-gate device where the transconductance f3n=fJ2f33/f.P2+/33). Non-saturated PET operation is assumed for these equations even though the PETs will momentarily pass through saturation at the onset of conduction in the actual circuit. The Schmitt trigger circuit establishes a nonlinear positive feedback path responsible for action potential initiation. The upper threshold of the trigger (VIII) emulates the natural neuron activation threshold while the lower threshold (Va) emulates the maximum hyperpolarization voltage. Equation (6) expresses the hysterisis present in the Schmitt trigger transfer characteristic. When Vs reaches the upper Schmitt threshold, PET 3 turns on, creating a current path from V tid to Cs, and emulating the upswing of a natural action potential spike. A moment later, PET 2 turns off, starting the action potential downswing. Simultaneously, PET 4 turns on, begining the absolute refractory period where Cs is discharged toward the maximum hyperpolarization potential. When that potential is reached, the Schmitt trigger turns off PET 4 and the impulse firing cycle is complete. The capacitor terminal voltage Va emulates all gross temporal artifacts associated with membrane potential, including spacio-temporal integration, the action potential spike, and a refractory period. The instantaneous net excitation to the neuron is represented by the total current flowing into the summing node on the floating plate of the capacitor. Charge packets are transferred from V tid to the capacitor by the excitatory synapse PET. Excitatory packet magnitude is dependent upon the transconductance PI. Inhibitory synapses (not shown) operate similarly, but instead reduce capacitor voltage by drawing charge to Va. A buffered action potential signal useful for driving many synapse PETs is available at the axon output. The membrane potential components (E,F,and G) of the circuit equations describe nonlinear relationships between post-synaptic excitation (E), membrane potential (F and G), and membrane ion currents. The functional forms of these components are equivalent to those found between terminal voltages and currents in non-saturated PETs. It is notable that natural voltage-gated channels do not necessarily follow the same current-voltage relationship of a PET. Even though more accurate models and emulations of natural membrane conductance exist, it seems unlikely at this time that they would help further improve neural network implementation. There is little doubt that more complex circuitry would be required to better approximate the true nonlinear relationship found in the biochemistry of natural neural membrane. That need conflicts directly with the goal of high-density integration. IMPULSE NEURAL NE1WORKS ~ Organizing a collection of neuron circuits into a useful network confIguration requires some weight specification method. Weight values can be either directly specified by the designer or learned by the network. A method particularly suited for use with the fIXed PET -synapses of the foregoing circuit is to fust learn weights using an "off-line" 682 Meador and Cole simulation, then translate the numerical results to physical FET transconductances. To do this, the activation function of an impulse neuron is derived and used in a modified back-propagation learning procedure. IMPULSE NEURON ACfIVATION FUNCTION Learning algorithms typically require some expression of the neuron activation function. Neuron activation can be expressed as a numerical value, a binary pattern, or a circuit voltage. In an impulse neuron, activation is expressed in terms of firing rate. The more frequently an impulse neuron circuit fues, the greater its activation. Impulse neuron activation is a nonlinear function of the excitation imparted through its synapse connections. An analytical expression of this nonlinear function can be derived using a rectangular approximation of neuron impulse waveforms. It is fust necessary to defme a unit-impulse as one impulse conducted by a synapse FET having some pre-determined reference transconductance (f3~). In Figure 3, To represents an invariant activation impulse width which is assumed to be identical for all neurons. T 1 represents the variable time period required for the neuron to accumulate the equivalent of K unit-impulses input excitation prior to firing. It can be assumed that net input comes from a single excitatory synapse with no other excitation. It shall also be assumed that impulses arrive at a constant rate, so T 1 =K /W;jR; (7) where R/ is the firing rate of the source neuron and W;j is the weight of the synapse connecting neuron; and neuron j. The firing rate of the receiving neuron will be Rj = 1/ (To + T 1). Substituting for T 1 this becomes: (8) F"tgure 3 compares this function with the logistic activation function. The impulse activation function approaches zero at the rate of 1/ K when R; approaches zero. The function also approaches an asymptote of Rj = l/T 0 as R; increases without bound. Any non-synaptic source which causes current flow from V tit to Co will shift the curve to the left, and reflect a spontaneous firing rate at zero input excitation. A similar current source to VoU will shift the function to the right, reflecting a positive firingonset threshold. Circuit-level simulations show a clear correspondence to these analytical results. This functional form is also evident in activation curves experimentally observed with natural neurons {Guyton, 1986}. Various natural neurons are known to exhibit both spontaneous firing and fuing-onset thresholds as well. The impulse activation function constant, K, is determined by several factors including fJ~, Co, and To. Assuming that To« T h no leakage current exists, and that a FET conducting in its non-saturated region can be approximated by a resistor, the following expression for K is obtained: (9) A Low-Power CMOS Circuit Which Emulates Neurons 683 where Rchon =l/pteJ<Vdd -V".), Ca is the summing capacitance, Va Vth are the low and high threshold voltages of the Schmitt trigger, and V". is the gate threshold voltage for an excitatory PET-synapse. A more accurate K value can be obtained by using the non-saturated PET current equation and solving a nonlinear differential equation. 1.0 0 .5 o rj~ ~ 1, tTo~ 1 ri Logistic Activation j J rectangUlar Impulse Train Impulse Activation ~--~----------------------------------~---------------------------------------------ri o Figure 3. Rectangular impulse train approximation for impulse activation function derivation. Unlike the logistic function which asymptotically approaches zero, impulse activation is equal to zero over a range of net excitation. BACK·PROPAGATION IN IMPULSE NE1WORKS A back-propagation algorithm has been used to learn connection weights for impulse neural networks. At this time, weight values are non-adaptive (they are fixed at circuit fabrication) because they are implemented as invariant PET transconductances. Adaptive synapses compatible with impulse neuron circuits are in the early stages of development, but are not available at this time. Much can be learned about these networks using non-adaptive prototypes, however. As a result, weight learning is performed offline as part of the network design process. The back-propagation procedure used to learn weights for impulse networks differs from the generalized delta rule {Rumelhart, 1986} in two ways. The fust difference is the use of the impulse activation function instead of the logistic function. Any activation nonlinearity is a viable candidate for use with the generalized delta rule as long as it is differentiable. This is where difficulties mount with the impulse activation function. First of all, it is not differentiable at zero. What seems to be more important, however, is that its first derivative equals zero over a range of 684 Meador and Cole inputs. Examination of the generalized delta rule (which performs gradient-descent) reveals that when the fust derivative of neuron activation becomes zero, connections associated with that-neuron will cease to adapt. Once this happens, the procedure will most probably never arrive at a problem solution. To work around this problem, a second deviation from the generalized delta rule was implemented. This involves a departure from using the true first derivative when the impulse activation becomes zero. A small constant can be used to guarantee that learning continues even though the associated neuron activation is zero: Act = l/(To + K /Net) A ,_{(l/(To+K/Net)' if Net >0 ct e otherwise (10) (11) The use of these equations yields a back-propagation algorithm for impulse networks which does not perform true gradient descent, yet which so far has been observed to learn solutions to logic problems such as XOR and the 4-2-4 encoder. Investigation of other offline learning algorithms for impulse networks continues. Currently, this algorithm fulfills the immediate need for an offline procedure which can be used in the design of multi-layer impulse neural networks. IMPLEMENTATION Two requirements for high density integration are low power dissipation and small circuit geometry. CMOS impulse neurons use switching circuits having no continuous power dissipation. A conventional op-amp circuit must draw constant current to achieve linear bias. An op-amp also requires larger circuit geometries for gain accuracy over typical fabrication process variations. Such is not the case for nonlinear switching circuits. As a result, these neurons and others like them are expected to help improve analog neural network integration density. An impulse neuron circuit has been designed which eliminates FETs 2 and 3 of Figure 2 in exchange for reduced layout area. In this circuit, Va no longer exhibits an activation potential spike. This spike seems irrelevant given the buffered impulse available at the axon output. The modified neuron circuit occupies 200 X 25 lambda chip area. A fIXed PET -synapse occupies a 16 by 18 lambda rectangle. With these dimensions a full-interconnect layout containing 40 neurons and 1600 fIXed connections will fit on a MOSIS 2-micron tiny chip. XOR and 4-2-4 networks of these circuits are being developed for 2-micron CMOS. CONCLUSION The motivation of this work is to improve neural network implementation technology by designing CMOS circuits derived from the temporal characteristics of natural neurons. The results obtained thus far include: Two CMOS circuits which closely correspond to the voltage-gat ed-channel model of natural neural membrane. A Low-Power CMOS Circuit Which Emulates Neurons 685 Simulations which show that these impulse neurons emulate gross artifacts of natural neuron function. Initial work on a back-propagation algorithm which learns logic solutions using the impulse neuron activation function. The development of prototype impulse network I.Cs. Future goals involve extending this investigation to plastic synapse and neuron circuits, alternate algorithms for both offline and online learning, and practical implementations. Rererences H. P. Graf W. Hubbard L. D. Jackel P. G. N. deVegvar. A CMOS associative Memory Chip. IEEE ICNN Con. Proc., pp. 461-468, (1987). H. P. Graf L. D. Jackel W. E. Hubbard. VLSI Implementation of a Neural Network Model. IEEE Computer, pp. 41-49, (1988). A.C. Guyton. Chapt. 10. Organization of the Nervous System: Basic Functions of Synapses. Textbook o[ Physiology, p.l36. (1986) N. EI-Liethy, R.W. Newcomb, M. Zaghlou. A Basic MOS Neural-Type Junction A Perspective on Neural-Type Microsystems. IEEE ICNN Con. Proc., pp. 469-477, (1987). E. R. Lewis. Using Electronic Circuits to Model Simple Neuroelectric Interactions. Proc. IEEE 56, pp. 931-949, (1968). R. J. MacGregor R. M. Oliver. A General-Purpose Electronic Model for Arbitrary Configurations of Neurons. 1. Theor. Bioi. 38, pp. 527-538 (1973). S. Y. Kung. and J. N. Hwang. Parallel Achitectures for Artificial Neural Nets. IEEE ICNN Con. Proc., pp. 11-165 to 11-172, (1988). J. Mann R. Lippman B. Berger J. Raffel. A Self-Organizing Neural Net Chip. IEEE Cust.Integr. Ckts. Conf" pp. 10.3.1-10.3.5 (1988). A. F. Murray A. V. W. Smith. Asynchronous VLSI Neural Networks Using PulseStream Arithmetic. IEEE lnl. of Sol. St. Phys. 23, pp. 688-697, (1988). J. I. Raffel. Electronic Implementation of Neuromorphic Systems. IEEE Cust. Integr. Ckts. Conf" pp. 10.1.1-10.1.7, (1988). D. Rumelhart, G.E. Hinton, and RJ. Williams. Learning Internal Representations by Error Propagation. Parallel Distributed Processing, Vol 1: Foundations, pp. 318-364, (1986). O. H. Schmitt. Mechanical Solution of the Equations of Nerve Impulse Propagation. Am. 1. Physiol. 119, pp. 399-400, (1937). D. B. Schwartz R. E. Howard. A Programmable Analog Neural Network Chip. IEEE Cust. Integr. Ckts. Conf., pp. 10.2.1-1.2.4, (1988). 686 Meador and Cole T J. Sejnowski. Open Questions About Computation in Cerebral Cortex. Parallel Distributed Processing Vol. 2:Psychological and Biological Models, pp. 378-385, (1986). M. A. Sivilotti M. R. Emerling C. A. Mead. VISI Architectures for Implementation of Neural Networks. Am. Ins. of Phys., 408-413, (1986).
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LINEAR LEARNING: LANDSCAPES AND ALGORITHMS Pierre Baldi Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91109 What follows extends some of our results of [1] on learning from examples in layered feed-forward networks of linear units. In particular we examine what happens when the ntunber of layers is large or when the connectivity between layers is local and investigate some of the properties of an autoassociative algorithm. Notation will be as in [1] where additional motivations and references can be found. It is usual to criticize linear networks because "linear functions do not compute" and because several layers can always be reduced to one by the proper multiplication of matrices. However this is not the point of view adopted here. It is assumed that the architecture of the network is given (and could perhaps depend on external constraints) and the purpose is to understand what happens during the learning phase, what strategies are adopted by a synaptic weights modifying algorithm, ... [see also Cottrell et al. (1988) for an example of an application and the work of Linsker (1988) on the emergence of feature detecting units in linear networks}. Consider first a two layer network with n input units, n output units and p hidden units (p < n). Let (Xl, YI), ... , (XT, YT) be the set of centered input-output training patterns. The problem is then to find two matrices of weights A and B minimizing the error function E: E(A, B) = L IIYt - ABXtIl2. (1) l<t<T 65 66 Baldi Let ~x x, ~XY, ~yy, ~y x denote the usual covariance matrices. The main result of [1] is a description of the landscape of E, characerised by a multiplicity of saddle points and an absence of local minima. More precisely, the following four facts are true. Fact 1: For any fixed n x p matrix A the function E(A, B) is convex in the coefficients of B and attains its minimum for any B satisfying the equation A'AB~xx = A/~yX. (2) If in addition ~x X is invertible and A is of full rank p, then E is strictly convex and has a unique minimum reached when (3) Fact 2: For any fixed p x n matrix B the function E(A, B) is convex in the coefficients of A and attains its minimum for any A satisfying the equation AB~xxB' = ~YXB'. (4) If in addition ~xx is invertible and B is of full rank p, then E is strictly convex and has a unique minimum reached when (5) Fact 3: Assume that ~x X is invertible. If two matrices A and B define a critical point of E (i.e. a point where 8E / 8aij = 8E /8bij = 0) then the global map W = AB is of the form (6) where P A denotes the matrix of the orthogonal projection onto the subspace spanned by the columns of A and A satisfies (7) Linear Learning: Landscapes and Algorithms 67 with ~ = ~y X ~x~~XY. If A is of full rank p, then A and B define a critical point of E if and only if A satisties (7) and B = R(A), or equivalently if and only if A and W satisfy (6) and (7). Notice that in (6), the matrix ~y X ~x~ is the slope matrix for the ordinary least square regression of Y on X. Fact 4: Assume that ~ is full rank with n distinct eigenvalues At > ... > An. If I = {i t , ... ,ip}(l < it < ... < ip < n) is any ordered p-index set, let Uz = [Uit , ••• , Uip ] denote the matrix formed by the orthononnal eigenvectors of ~ associated with the eigenvalues Ail' ... , Aip • Then two full rank matrices A and B define a critical point of E if and only if there exist an ordered p-index set I and an invertible p x p matrix C such that A=UzC For such a critical point we have E(A,B) = tr(~yy) - L Ai. iEZ (8) (9) (10) (11 ) Therefore a critical point of W of rank p is always the product of the ordinary least squares regression matrix followed by an orthogonal projection onto the subspace spanned by p eigenvectors of~. The map W associated with the index set {I, 2, ... ,p} is the unique local and global minimum of E. The remaining (;) -1 p-index sets correspond to saddle points. All additional critical points defined by matrices A and B which are not of full rank are also saddle points and can be characerized in terms of orthogonal projections onto subspaces spanned by q eigenvectors with q < p. 68 Baldi Deep Networks Consider now the case of a deep network with a first layer of n input units, an (m + 1 )-th layer of n output units and m - 1 hidden layers with an error function given by E(AI, ... ,An)= L IIYt-AIAl ... AmXtll2. (12) l<t<T It is worth noticing that, as in fact 1 and 2 above, if we fix any m-1 of the m matrices AI, ... , Am then E is convex in the remaining matrix of connection weights. Let p (p < n) denote the ntunber of units in the smallest layer of the network (several hidden layers may have p units). In other words the network has a bottleneck of size p. Let i be the index of the corresponding layer and set A = AI A2 ... Am-i+1 B = Am-i+2 ... Am (13) When we let AI, ... , Am vary, the only restriction they impose on A and B is that they be of rank at most p. Conversely, any two matrices A and B of rank at most p can always be decomposed (and in many ways) in products of the form of (13). It results that any local minima of the error function of the deep network should yield a local minima for the corresponding "collapsed" .three layers network induced by (13) and vice versa. Therefore E(AI , ... , Am) does not have any local minima and the global minimal map W· = AIA2 ... Am is unique and given by (10) with index set {I, 2, ... , p}. Notice that of course there is a large number of ways of decomposing W· into a product of the form A I A2 ... Am . Also a saddle point of the error function E(A, B) does not necessarily generate a saddle point of the corresponding E (A I , ... , Am) for the expressions corresponding to the two gradients are very different. Linear Learning: Landscapes and Algorithms 69 Forced Connections. Local Connectivity Assume now an error function of the form E(A) = L IIYt - AXt[[2 l~t~T (14) for a two layers network but where the value of some of the entries of A may be externally prescribed. In particular this includes the case of local connectivity described by relations of the form aij = 0 for any output unit i and any input unit j which are not connected. Clearly the error function E(A) is convex in A. Every constraint of the form aij =cst defines an hyperplane in the space of all possible A. The intersection of all these constraints is therefore a convex set. Thus minimizing E under the given constraints is still a convex optimization problem and so there are no local minima. It should be noticed that, in the case of a network with even only three constrained layers with two matrices A and B and a set of constraints of the form aij =cst on A and bk1 =cst for B, the set of admissible matrices of the form W = AB is, in general, not convex anymore. It is not unreasonable to conjecture that local minima may then arise, though this question needs to be investigated in greater detail. Algorithmic Aspects One of the nice features of the error landscapes described so far is the absence of local minima and the existence, up to equivalence, of a unique global minimum which can be understood in terms of principal component analysis and least square regression. However the landscapes are also characterized by a large number of saddle points which could constitute a problem for a simple gradient descent algorithm during the learning phase. The proof in [1] shows that the lower is the E value corresponding to a saddle point, the more difficult it is to escape from it because of a reduction in the possible number of directions of escape (see also [Chauvin, 1989] in a context of Hebbian learning). To assert how relevant these issues are for practical implementations requires further simulation experiments. On a more 70 Baldi speculative side, it remains also to be seen whether, in a problem of large size, the number and spacing of saddle points encountered during the first stages of a descent process could not be used to "get a feeling" for the type of terrain being descented and as a result to adjust the pace (i. e. the learning rate). We now turn to a simple algorithm for the auto-associative case in a three layers network, i. e. the case where the presence of a teacher can be avoided by setting Yt = Xt and thereby trying to achieve a compression of the input data in the hidden layer. This technique is related to principal component analysis, as described in [1]. If Yt = Xt, it is easy to see from equations (8) and (9) that, if we take the matrix C to be the identity, then at the optimum the matrices A and B are transpose of each other. This heuristically suggests a possible fast algorithm for auto-association, where at each iteration a gradient descent step is applied only to one of the connection matrices while the other is updated in a symmetric fashion using transposition and avoiding to back-propagate the error in one of the layers (see [Williams, 1985] for a similar idea). More formally, the algorithm could be concisely described by A(O) = random B(O) = A'(O) 8E A(k+l)=A(k)-11 8A B(k+l)=A'(k+l) (15) Obviously a similar algorithm can be obtained by setting B(k + 1) = B(k) -118E/8B and A(k + 1) = B'(k + 1). It may actually even be bet ter to alternate the gradient step, one iteration with respect to A and one iteration with respect to B. A simple calculation shows that (15) can be rewritten as A(k + 1) = A(k) + 11(1 W(k))~xxA(k) B(k + 1) = B(k) + 11B(k)~xx(I - W(k)) (16) Linear Learning: Landscapes and Algorithms 71 where W(k) = A(k)B(k). It is natural from what we have already seen to examine the behavior of this algorithm on the eigenvectors of ~xx. Assume that u is an eigenvector of both ~xx and W(k) with eigenvalues ,\ and /-l( k). Then it is easy to see that u is an eigenvector of W(k + 1) with eigenvalue (17) For the algorithm to converge to the optimal W, /-l( k + 1) must converge to 0 or 1. Thus one has to look at the iterates of the function f( x) = x[l + 7],\(1 - x)]2. This can be done in detail and we shall only describe the main points. First of all, f' (x) = 0 iff x = 0 or x = Xa = 1 + (1/7],\) or x = Xb = 1/3 + (1/37],\) and f"(x) = 0 iff x = Xc = 2/3 + (2/37],\) = 2Xb. For the fixed points, f(x) = x iff x = 0, x = 1 or x = Xd = 1 + (2/7],\). Notice also that f(xa) = a and f(1 + (1/7],\)) = (1 + (1/7],\)(1 - 1? Points corresponding to the values 0,1, X a , Xd of the x variable can readily be positioned on the curve of f but the relative position of Xb (and xc) depends on the value assumed by 7],\ with respect to 1/2. Obviously if J1(0) = 0,1 or Xd then J1( k) = 0,1 or Xd, if J1(0) < 0 /-l( k) ~ -00 and if /-l( k) > Xd J1( k) ~ +00. Therefore the algorithm can converge only for a < /-leO) < Xd. When the learning rate is too large, i. e. when 7],\ > 1/2 then even if /-leO) is in the interval (0, Xd) one can see that the algorithm does not converge and may even exhibit complex oscillatory behavior. However when 7],\ < 1/2, if 0 < J1(0) < Xa then J1( k) ~ 1, if /-leO) = Xa then /-l( k) = a and if Xa < J1(0) < Xd then /-l(k) ~ 1. In conclusion, we see that if the algorithm is to be tested, the learning rate should be chosen so that it does not exceed 1/2,\, where ,\ is the largest eigenvalue of ~xx. Even more so than back propagation, it can encounter problems in the proximity of saddle points. Once a non-principal eigenvector of ~xx is learnt, the algorithm rapidly incorporates a projection along that direction which cannot be escaped at later stages. Simulations are required to examine the effects of "noisy gradients" (computed after the presentation of only a few training examples), multiple starting points, variable learning rates, momentum terms, and so forth. 72 Baldi Aknowledgement Work supported by NSF grant DMS-8800323 and in part by ONR contract 411P006-01. References (1) Baldi, P. and Hornik, K. (1988) Neural Networks and Principal Component Analysis: Learning from Examples without Local Minima. Neural Networks, Vol. 2, No. 1. (2) Chauvin, Y. (1989) Another Neural Model as a Principal Component Analyzer. Submitted for publication. (3) Cottrell, G. W., Munro, P. W. and Zipser, D. (1988) Image Compression by Back Propagation: a Demonstration of Extensional Programming. In: Advances in Cognitive Science, Vol. 2, Sharkey, N. E. ed., Norwood, NJ Abbex. (4) Linsker, R. (1988) Self-Organization in a Perceptual Network. Computer 21 (3), 105-117. ( 5) Willi ams, R. J. (1985) Feature Discovery Through Error-Correction Learning. ICS Report 8501, University of California., San Diego.
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