Daily Papers

Bound states in the continuum (BIC) metasurfaces are typically described by geometry-specific parameters, hindering cross-geometry comparison, while ultranarrow qBIC features are easily diluted in full-spectrum learning. Here, 2015 samples from seven dielectric metasurface families are mapped to a shared reciprocal-lattice grid, where two frozen low-order Fourier channels capture resonance shifts with mean within-branch R^2 values of 0.871-0.999. Field-level analysis of two representative branches further confirms that these shifts are consistent with the Maxwell-Fourier perturbation picture. A five-channel K-space backbone models the broadband spectrum, while a local complex K-space expert parameterizes the qBIC resonance through a differentiable Fano layer. The expert reduces resonance-position mean absolute error (MAE) from 3.2 to 0.95 nm and the resonance-depth error by 14-fold on a geometry-blocked test set. The same coordinate supports spectrum-to-structure reconstruction.

Multispectral satellite images play a vital role in agriculture, fisheries, and environmental monitoring. However, their high dimensionality, large data volumes, and diverse spatial resolutions across multiple channels pose significant challenges for data compression and analysis. This paper presents ImpliSat, a unified framework specifically designed to address these challenges through efficient compression and reconstruction of multispectral satellite data. ImpliSat leverages Implicit Neural Representations (INR) to model satellite images as continuous functions over coordinate space, capturing fine spatial details across varying spatial resolutions. Furthermore, we introduce a Fourier modulation algorithm that dynamically adjusts to the spectral and spatial characteristics of each band, ensuring optimal compression while preserving critical image details.

Periodic signals play an important role in daily lives. Although conventional sequential models have shown remarkable success in various fields, they still come short in modeling periodicity; they either collapse, diverge or ignore details. In this paper, we introduce a novel framework inspired by Fourier series to generate periodic signals. We first decompose the given signals into multiple sines and cosines and then conditionally generate periodic signals with the output components. We have shown our model efficacy on three tasks: reconstruction, imputation and conditional generation. Our model outperforms baselines in all tasks and shows more stable and refined results.

A fundamental problem in quantum information is to understand the operational significance of quantum resources. Quantum resource theories (QRTs) provide a powerful theoretical framework that aids in analyzing and comprehending the operational meaning of these resources. Early resource theories primarily focused on analyzing static quantum resources. Recently, there has been growing interest in the study of dynamic quantum resources. In this paper, we utilize superchannel theory to describe the dynamic resource theory of quantum coherence. In this dynamic resource theory, we treat classical channels as free channels and consider two classes of free superchannels that preserve channel incoherence (maximally incoherent superchannels (MISC) and dephasing-covariant incoherent superchannels (DISC)) as free resources. We regard the quantum Fourier transform as the golden unit of dynamic coherence resources. We first establish the one-shot theory of dynamic coherence cost and dynamic coherence distillation, which involves converting the quantum Fourier transform into an arbitrary quantum channel using MISC and DISC. Next, we introduce a class of free superchannels known as δ-MISC, which asymptotically generate negligible dynamic coherence. Finally, we provide upper and lower bounds for the one-shot catalytic dynamic coherence cost of quantum channels under the action of these δ-MISC superchannels.

Spectral analysis provides one of the most effective paradigms for information-preserving dimensionality reduction, as simple descriptions of naturally occurring signals are often obtained via few terms of periodic basis functions. In this work, we study deep neural networks designed to harness the structure in frequency domain for efficient learning of long-range correlations in space or time: frequency-domain models (FDMs). Existing FDMs are based on complex-valued transforms i.e. Fourier Transforms (FT), and layers that perform computation on the spectrum and input data separately. This design introduces considerable computational overhead: for each layer, a forward and inverse FT. Instead, this work introduces a blueprint for frequency domain learning through a single transform: transform once (T1). To enable efficient, direct learning in the frequency domain we derive a variance-preserving weight initialization scheme and investigate methods for frequency selection in reduced-order FDMs. Our results noticeably streamline the design process of FDMs, pruning redundant transforms, and leading to speedups of 3x to 10x that increase with data resolution and model size. We perform extensive experiments on learning the solution operator of spatio-temporal dynamics, including incompressible Navier-Stokes, turbulent flows around airfoils and high-resolution video of smoke. T1 models improve on the test performance of FDMs while requiring significantly less computation (5 hours instead of 32 for our large-scale experiment), with over 20% reduction in average predictive error across tasks.

In this paper, we prove representation bottlenecks of a cascaded convolutional decoder network, considering the capacity of representing different frequency components of an input sample. We conduct the discrete Fourier transform on each channel of the feature map in an intermediate layer of the decoder network. Then, we introduce the rule of the forward propagation of such intermediate-layer spectrum maps, which is equivalent to the forward propagation of feature maps through a convolutional layer. Based on this, we find that each frequency component in the spectrum map is forward propagated independently with other frequency components. Furthermore, we prove two bottlenecks in representing feature spectrums. First, we prove that the convolution operation, the zero-padding operation, and a set of other settings all make a convolutional decoder network more likely to weaken high-frequency components. Second, we prove that the upsampling operation generates a feature spectrum, in which strong signals repetitively appears at certain frequencies.

We propose a neural network for both forward and inverse continuous nonlinear Fourier transforms, NFT and INFT respectively. We demonstrate the network's capability to perform NFT and INFT for a random mix of NFDM-QAM signals. The network transformations (NFT and INFT) exhibit true characteristics of these transformations; they are significantly different for low and high-power input pulses. The network shows adequate accuracy with an RMSE of 5e-3 for forward and 3e-2 for inverse transforms. We further show that the trained network can be used to perform general nonlinear Fourier transforms on arbitrary pulses beyond the training pulse types.

Many machine learning algorithms rely on the Fourier transform, which is also the inner core of many analog computing devices and, specifically, of optical computing. The square of the modulus of the Fourier spectrum naturally maps onto pairwise interactions, as in the renowned Ising model, making it applicable to combinatorial optimization. Here we show how the nonlinear Fourier transform, related to integrable nonlinear partial differential equations, significantly widens the computational capability and also enables a wide variety of computing gates. We study specifically the case of the nonlinear Schrödinger equation; we introduce an input composed by ``tokens'', i.e., batches of information encoded in a multi-step potential, and we show how the nonlinear spectral computing arising in the wave propagation not only generalizes combinatorial optimization to higher order problems, but also realizes boolean gates. In the weakly nonlinear regime, nonlinearity generates higher-order interactions as perturbative corrections to the pairwise terms. In the solitonic regime, the discrete nonlinear spectrum performs digital computation through soliton generation. By using four input tokens, we demonstrate the programmability of all 256 Boolean functions of three variables, including 3-SAT instances. Our results show that integrable nonlinear propagation provides an unprecedented single physical platform for higher-order and logical computation, opening the way to a rich variety of new algorithms for analog computing, machine learning, and classical and quantum information processing.

Quantum Fourier analysis is an important topic in mathematical physics. We introduce a systematic protocol for testing and measuring ``magic'' in quantum states and gates, using a quantum Fourier approach. Magic, as a quantum resource, is necessary to achieve a quantum advantage in computation. Our protocols are based on quantum convolutions and swap tests, implemented via quantum circuits. We describe this for both qubit and qudit systems. Our quantum Fourier approach offers a unified method to quantify magic, in stabilizer circuits, as well as in matchgate and bosonic Gaussian circuits.

Deep Learning-based Computer Vision field has recently been trying to explore larger kernels for convolution to effectively scale up Convolutional Neural Networks. Simultaneously, new paradigm of models such as Vision Transformers find it difficult to scale up to larger higher resolution images due to their quadratic complexity in terms of input sequence. In this report, Fast Fourier Transform is utilised in various ways to provide some solutions to these issues.

The "theoretical limit of time-frequency resolution in Fourier analysis" is thought to originate in certain mathematical and/or physical limitations. This, however, is not true. The actual origin arises from the numerical (technical) method deployed to reduce computation time. In addition, there is a gap between the theoretical equation for Fourier analysis and its numerical implementation. Knowing the facts brings us practical benefits. In this case, these related to boundary conditions, and complex integrals. For example, replacing a Fourier integral with a complex integral brings a hybrid method for the Laplace and Fourier transforms, and reveals another perspective on time-frequency analysis. We present such a perspective here with a simple demonstrative analysis.

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting G-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

This research addresses the challenge of developing a universal deepfake detector that can effectively identify unseen deepfake images despite limited training data. Existing frequency-based paradigms have relied on frequency-level artifacts introduced during the up-sampling in GAN pipelines to detect forgeries. However, the rapid advancements in synthesis technology have led to specific artifacts for each generation model. Consequently, these detectors have exhibited a lack of proficiency in learning the frequency domain and tend to overfit to the artifacts present in the training data, leading to suboptimal performance on unseen sources. To address this issue, we introduce a novel frequency-aware approach called FreqNet, centered around frequency domain learning, specifically designed to enhance the generalizability of deepfake detectors. Our method forces the detector to continuously focus on high-frequency information, exploiting high-frequency representation of features across spatial and channel dimensions. Additionally, we incorporate a straightforward frequency domain learning module to learn source-agnostic features. It involves convolutional layers applied to both the phase spectrum and amplitude spectrum between the Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (iFFT). Extensive experimentation involving 17 GANs demonstrates the effectiveness of our proposed method, showcasing state-of-the-art performance (+9.8\%) while requiring fewer parameters. The code is available at {\cred https://github.com/chuangchuangtan/FreqNet-DeepfakeDetection}.

We present an efficient frequency-based neural representation termed PREF: a shallow MLP augmented with a phasor volume that covers significant border spectra than previous Fourier feature mapping or Positional Encoding. At the core is our compact 3D phasor volume where frequencies distribute uniformly along a 2D plane and dilate along a 1D axis. To this end, we develop a tailored and efficient Fourier transform that combines both Fast Fourier transform and local interpolation to accelerate na\"ive Fourier mapping. We also introduce a Parsvel regularizer that stables frequency-based learning. In these ways, Our PREF reduces the costly MLP in the frequency-based representation, thereby significantly closing the efficiency gap between it and other hybrid representations, and improving its interpretability. Comprehensive experiments demonstrate that our PREF is able to capture high-frequency details while remaining compact and robust, including 2D image generalization, 3D signed distance function regression and 5D neural radiance field reconstruction.

Neural operators provide deep neural networks for learning mappings between function spaces. Among them, the Fourier Neural Operator (FNO) is particularly effective: its spectral convolution relies on low-dimensional Fourier-domain representations and can handle inputs at different resolutions. This design aligns well with settings where the Fourier basis diagonalizes the underlying operator, such as linear, constant-coefficient PDEs on periodic domains, in which Fourier modes evolve independently. However, nonlinear PDEs may benefit from an additional inductive bias, as they exhibit structured interactions between modes, governed by polynomial nonlinearities. To capture this inductive bias, we introduce the Higher-Order Spectral Convolution, a spectral mixer that extends FNO from diagonal modulation to explicit n-linear mode mixing, aligned with the dynamics of nonlinear PDEs. Our experiments on standard benchmarks show that the proposed Higher-Order FNO (HO-FNO) retains the efficiency of FNO-based architectures and consistently improves over other spectral neural operators. HO-FNO also performs on par with or better than state-of-the-art transformers and state-space models on several datasets, with stronger gains in highly nonlinear regimes, such as the Poisson equation with polynomial forcing, where a single HO-FNO layer outperforms FNO models with up to 16 layers. We open-source our code for reproducibility at: https://github.com/AlexColagrande/HO-FNO.

Compositional Pattern Producing Networks (CPPNs) are differentiable networks that independently map (x, y) pixel coordinates to (r, g, b) colour values. Recently, CPPNs have been used for creating interesting imagery for creative purposes, e.g., neural art. However their architecture biases generated images to be overly smooth, lacking high-frequency detail. In this work, we extend CPPNs to explicitly model the frequency information for each pixel output, capturing frequencies beyond the DC component. We show that our Fourier-CPPNs (F-CPPNs) provide improved visual detail for image synthesis.

Strategies for the generation of periodic discrete structures with identical two-point correlation are developed. Starting from a pair of root structures, which are not related by translation, phase inversion or axis reflections, child structures of arbitrary resolution (i.e., pixel or voxel numbers) and number of phases (i.e., material phases/species) can be generated by means of trivial embedding based phase extension, application of kernels and/or phase coalescence, such that the generated structures inherit the two-point-correlation equivalence. Proofs of the inheritance property are provided by means of the Discrete Fourier Transform theory. A Python 3 implementation of the results is offered by the authors through the Github repository https://github.com/DataAnalyticsEngineering/EQ2PC in order to make the provided results reproducible and useful for all interested readers. Examples for the generation of structures are demonstrated, together with applications in the homogenization theory of periodic media.

CNN-based generative modelling has evolved to produce synthetic images indistinguishable from real images in the RGB pixel space. Recent works have observed that CNN-generated images share a systematic shortcoming in replicating high frequency Fourier spectrum decay attributes. Furthermore, these works have successfully exploited this systematic shortcoming to detect CNN-generated images reporting up to 99% accuracy across multiple state-of-the-art GAN models. In this work, we investigate the validity of assertions claiming that CNN-generated images are unable to achieve high frequency spectral decay consistency. We meticulously construct a counterexample space of high frequency spectral decay consistent CNN-generated images emerging from our handcrafted experiments using DCGAN, LSGAN, WGAN-GP and StarGAN, where we empirically show that this frequency discrepancy can be avoided by a minor architecture change in the last upsampling operation. We subsequently use images from this counterexample space to successfully bypass the recently proposed forensics detector which leverages on high frequency Fourier spectrum decay attributes for CNN-generated image detection. Through this study, we show that high frequency Fourier spectrum decay discrepancies are not inherent characteristics for existing CNN-based generative models--contrary to the belief of some existing work--, and such features are not robust to perform synthetic image detection. Our results prompt re-thinking of using high frequency Fourier spectrum decay attributes for CNN-generated image detection. Code and models are available at https://keshik6.github.io/Fourier-Discrepancies-CNN-Detection/

Attention mechanism, especially channel attention, has gained great success in the computer vision field. Many works focus on how to design efficient channel attention mechanisms while ignoring a fundamental problem, i.e., channel attention mechanism uses scalar to represent channel, which is difficult due to massive information loss. In this work, we start from a different view and regard the channel representation problem as a compression process using frequency analysis. Based on the frequency analysis, we mathematically prove that the conventional global average pooling is a special case of the feature decomposition in the frequency domain. With the proof, we naturally generalize the compression of the channel attention mechanism in the frequency domain and propose our method with multi-spectral channel attention, termed as FcaNet. FcaNet is simple but effective. We can change a few lines of code in the calculation to implement our method within existing channel attention methods. Moreover, the proposed method achieves state-of-the-art results compared with other channel attention methods on image classification, object detection, and instance segmentation tasks. Our method could consistently outperform the baseline SENet, with the same number of parameters and the same computational cost. Our code and models will are publicly available at https://github.com/cfzd/FcaNet.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

Low-rank adaptation~(LoRA) has recently gained much interest in fine-tuning foundation models. It effectively reduces the number of trainable parameters by incorporating low-rank matrices A and B to represent the weight change, i.e., Delta W=BA. Despite LoRA's progress, it faces storage challenges when handling extensive customization adaptations or larger base models. In this work, we aim to further compress trainable parameters by enjoying the powerful expressiveness of the Fourier transform. Specifically, we introduce FourierFT, which treats Delta W as a matrix in the spatial domain and learns only a small fraction of its spectral coefficients. With the trained spectral coefficients, we implement the inverse discrete Fourier transform to recover Delta W. Empirically, our FourierFT method shows comparable or better performance with fewer parameters than LoRA on various tasks, including natural language understanding, natural language generation, instruction tuning, and image classification. For example, when performing instruction tuning on the LLaMA2-7B model, FourierFT surpasses LoRA with only 0.064M trainable parameters, compared to LoRA's 33.5M. Our code is released at https://github.com/Chaos96/fourierft.

We unify the discrete Fourier transform (DFT), discrete cosine transform (DCT), Walsh-Hadamard, Haar wavelet, Karhunen-Loève transform, and several others along with their continuous counterparts (Fourier transform, Fourier series, spherical harmonics, fractional Fourier transform) under one representation-theoretic principle: each is the eigenbasis of every covariance invariant under a specific finite or compact group, with columns constructed from the irreducible matrix elements of the group via the Peter-Weyl theorem. The unification rests on the Algebraic Diversity (AD) framework, which identifies the matched group of a covariance as the foundational object of second-order signal processing. The data-dependent KLT emerges as the trivial-matched-group limit; classical transforms emerge as the cyclic, dihedral, elementary abelian, iterated wreath, and hybrid wreath cases. Composition rules cover direct, wreath, and semidirect products. The Reed-Muller and arithmetic transforms appear as related change-of-basis transforms on the matched group of Walsh-Hadamard. A polynomial-time algorithm for matched-group discovery, the DAD-CAD relaxation cast as a generalized eigenvalue problem in double-commutator form, closes the operational loop: the matched group of any empirical covariance is discovered without expert judgment, with noise-aware variants via the commutativity residual δ and algebraic coloring index α for finite-SNR settings. The fractional Fourier transform is treated as the metaplectic SO(2) case with Hermite-Gauss matched basis, and a structural principle relates matched group size inversely to transform resolution. Modern applications (massive-MIMO, graph neural networks, transformer attention, point cloud and 3D vision, brain connectivity, single-cell genomics, quantum informatics) are sketched with their matched groups.

We consider the problem of computing the Walsh-Hadamard Transform (WHT) of some N-length input vector in the presence of noise, where the N-point Walsh spectrum is K-sparse with K = {O}(N^{delta}) scaling sub-linearly in the input dimension N for some 0<delta<1. Over the past decade, there has been a resurgence in research related to the computation of Discrete Fourier Transform (DFT) for some length-N input signal that has a K-sparse Fourier spectrum. In particular, through a sparse-graph code design, our earlier work on the Fast Fourier Aliasing-based Sparse Transform (FFAST) algorithm computes the K-sparse DFT in time {O}(Klog K) by taking {O}(K) noiseless samples. Inspired by the coding-theoretic design framework, Scheibler et al. proposed the Sparse Fast Hadamard Transform (SparseFHT) algorithm that elegantly computes the K-sparse WHT in the absence of noise using {O}(Klog N) samples in time {O}(Klog^2 N). However, the SparseFHT algorithm explicitly exploits the noiseless nature of the problem, and is not equipped to deal with scenarios where the observations are corrupted by noise. Therefore, a question of critical interest is whether this coding-theoretic framework can be made robust to noise. Further, if the answer is yes, what is the extra price that needs to be paid for being robust to noise? In this paper, we show, quite interestingly, that there is {\it no extra price} that needs to be paid for being robust to noise other than a constant factor. In other words, we can maintain the same sample complexity {O}(Klog N) and the computational complexity {O}(Klog^2 N) as those of the noiseless case, using our SParse Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm.

Ultra-High-Definition (UHD) photo has gradually become the standard configuration in advanced imaging devices. The new standard unveils many issues in existing approaches for low-light image enhancement (LLIE), especially in dealing with the intricate issue of joint luminance enhancement and noise removal while remaining efficient. Unlike existing methods that address the problem in the spatial domain, we propose a new solution, UHDFour, that embeds Fourier transform into a cascaded network. Our approach is motivated by a few unique characteristics in the Fourier domain: 1) most luminance information concentrates on amplitudes while noise is closely related to phases, and 2) a high-resolution image and its low-resolution version share similar amplitude patterns.Through embedding Fourier into our network, the amplitude and phase of a low-light image are separately processed to avoid amplifying noise when enhancing luminance. Besides, UHDFour is scalable to UHD images by implementing amplitude and phase enhancement under the low-resolution regime and then adjusting the high-resolution scale with few computations. We also contribute the first real UHD LLIE dataset, UHD-LL, that contains 2,150 low-noise/normal-clear 4K image pairs with diverse darkness and noise levels captured in different scenarios. With this dataset, we systematically analyze the performance of existing LLIE methods for processing UHD images and demonstrate the advantage of our solution. We believe our new framework, coupled with the dataset, would push the frontier of LLIE towards UHD. The code and dataset are available at https://li-chongyi.github.io/UHDFour.

We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the Random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs -- even in the case of 1-bit quantization -- allow a high accuracy approximation of the underlying kernels, and the approximation error decays at least polynomially fast as the dimension of the RFFs increases. We also show that the quantized RFFs can be further compressed, yielding an excellent trade-off between memory use and accuracy. Namely, the approximation error now decays exponentially as a function of the bits used. Moreover, we empirically show by testing the performance of our methods on several machine learning tasks that our method compares favorably to other state of the art quantization methods in this context.

It is well noted that coordinate based MLPs benefit -- in terms of preserving high-frequency information -- through the encoding of coordinate positions as an array of Fourier features. Hitherto, the rationale for the effectiveness of these positional encodings has been solely studied through a Fourier lens. In this paper, we strive to broaden this understanding by showing that alternative non-Fourier embedding functions can indeed be used for positional encoding. Moreover, we show that their performance is entirely determined by a trade-off between the stable rank of the embedded matrix and the distance preservation between embedded coordinates. We further establish that the now ubiquitous Fourier feature mapping of position is a special case that fulfills these conditions. Consequently, we present a more general theory to analyze positional encoding in terms of shifted basis functions. To this end, we develop the necessary theoretical formulae and empirically verify that our theoretical claims hold in practice. Codes available at https://github.com/osiriszjq/Rethinking-positional-encoding.

Implicit Neural Representations (INRs) have emerged as a powerful paradigm for representing signals such as images, audio, and 3D scenes. However, existing INR frameworks -- including MLPs with Fourier features, SIREN, and multiresolution hash grids -- implicitly assume a global and stationary spectral basis. This assumption is fundamentally misaligned with real-world signals whose frequency characteristics vary significantly across space, exhibiting local high-frequency textures, smooth regions, and frequency drift phenomena. We propose Neural Spectral Transport Representation (NSTR), the first INR framework that explicitly models a spatially varying local frequency field. NSTR introduces a learnable frequency transport equation, a PDE that governs how local spectral compositions evolve across space. Given a learnable local spectrum field S(x) and a frequency transport network F_θ enforcing nabla S(x) approx F_θ(x, S(x)), NSTR reconstructs signals by spatially modulating a compact set of global sinusoidal bases. This formulation enables strong local adaptivity and offers a new level of interpretability via visualizing frequency flows. Experiments on 2D image regression, audio reconstruction, and implicit 3D geometry show that NSTR achieves significantly better accuracy-parameter trade-offs than SIREN, Fourier-feature MLPs, and Instant-NGP. NSTR requires fewer global frequencies, converges faster, and naturally explains signal structure through spectral transport fields. We believe NSTR opens a new direction in INR research by introducing explicit modeling of space-varying spectrum.

Quantum computers can be used for supervised learning by treating parametrised quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate practical implications of this approach, many important theoretical properties of these models remain unknown. Here we investigate how the strategy with which data is encoded into the model influences the expressive power of parametrised quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data encoding gates in the circuit. By repeating simple data encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realise all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.

Despite the non-autoregressive potential of diffusion language models (dLLMs), existing decoding strategies demonstrate positional bias, failing to fully unlock the potential of arbitrary generation. In this work, we delve into the inherent spectral characteristics of dLLMs and present the first frequency-domain analysis showing that low-frequency components in hidden states primarily encode global structural information and long-range dependencies, while high-frequency components are responsible for characterizing local details. Based on this observation, we propose FourierSampler, which leverages a frequency-domain sliding window mechanism to dynamically guide the model to achieve a "structure-to-detail" generation. FourierSampler outperforms other inference enhancement strategies on LLADA and SDAR, achieving relative improvements of 20.4% on LLaDA1.5-8B and 16.0% on LLaDA-8B-Instruct. It notably surpasses similarly sized autoregressive models like Llama3.1-8B-Instruct.

Spectral bias is an important observation of neural network training, stating that the network will learn a low frequency representation of the target function before converging to higher frequency components. This property is interesting due to its link to good generalization in over-parameterized networks. However, in low dimensional settings, a severe spectral bias occurs that obstructs convergence to high frequency components entirely. In order to overcome this limitation, one can encode the inputs using a high frequency sinusoidal encoding. Previous works attempted to explain this phenomenon using Neural Tangent Kernel (NTK) and Fourier analysis. However, NTK does not capture real network dynamics, and Fourier analysis only offers a global perspective on the network properties that induce this bias. In this paper, we provide a novel approach towards understanding spectral bias by directly studying ReLU MLP training dynamics. Specifically, we focus on the connection between the computations of ReLU networks (activation regions), and the speed of gradient descent convergence. We study these dynamics in relation to the spatial information of the signal to understand how they influence spectral bias. We then use this formulation to study the severity of spectral bias in low dimensional settings, and how positional encoding overcomes this.

Single image-to-3D generation is pivotal for crafting controllable 3D assets. Given its underconstrained nature, we leverage geometric priors from a 3D novel view generation diffusion model and appearance priors from a 2D image generation method to guide the optimization process. We note that a disparity exists between the training datasets of 2D and 3D diffusion models, leading to their outputs showing marked differences in appearance. Specifically, 2D models tend to deliver more detailed visuals, whereas 3D models produce consistent yet over-smooth results across different views. Hence, we optimize a set of 3D Gaussians using 3D priors in spatial domain to ensure geometric consistency, while exploiting 2D priors in the frequency domain through Fourier transform for higher visual quality. This 2D-3D hybrid Fourier Score Distillation objective function (dubbed hy-FSD), can be integrated into existing 3D generation methods, yielding significant performance improvements. With this technique, we further develop an image-to-3D generation pipeline to create high-quality 3D objects within one minute, named Fourier123. Extensive experiments demonstrate that Fourier123 excels in efficient generation with rapid convergence speed and visual-friendly generation results.

Multi-Frequency Electrical Impedance Tomography (MF-EIT) is a non-invasive, low-cost modality that reconstructs electrical property distributions from boundary voltages. For stroke imaging, progress in 3D deep-learning reconstruction is limited by the lack of large-scale datasets with paired ground-truth (GT) volumes and by non-standardized pipelines for data generation, simulation, and evaluation. We introduce BREIT, a modular framework for 3D MF-EIT stroke reconstruction providing: (i) a neuroimaging-to-EIT pipeline that converts CT/MRI into frequency-dependent GT admittivity volumes; (ii) a self-contained Python 3D Complete Electrode Model (CEM) forward solver for simulating MF-EIT voltages; and (iii) a 3D D-bar implementation supporting non-uniform electrode layouts. Building on BREIT, we propose dFNO-bar, which integrates Fourier Neural Operators into D-bar by learning a mapping from scattering data t(ξ) to conductivity σ(x){=}Re{γ}. We evaluate dFNO-bar against D-bar, Deep D-bar, and Gauss--Newton reconstructions on UCLH-matched synthetic data, and observe higher brain SSIM with comparable CC across noise settings.

Harmonic retrieval techniques are the foundation of radio channel sounding, estimation and modeling. This paper introduces a Deep Learning approach for two-dimensional spectral estimation from frequency and time samples of a radio channel transfer function. Our work can estimate two-dimensional parameters from a signal containing an unknown number of paths. In contrast to existing deep learning-based methods, the signal parameters are not estimated via classification but instead in a quasi-grid-free manner. This alleviates the bias, spectral leakage, and ghost targets that grid-based approaches inherently produce. The proposed architecture also reliably estimates the number of spectral components in the measurement. Hence, the architecture jointly solves the model order selection problem and the parameter estimation task. Additionally, we propose a multi-channel windowing of the data during preprocessing, increasing the resulting estimator's robustness. We verify the performance compared to existing harmonic retrieval methods and also show how it can be integrated into an existing maximum likelihood estimator for efficient initialization of a gradient-based iteration.

Attentional mechanisms are order-invariant. Positional encoding is a crucial component to allow attention-based deep model architectures such as Transformer to address sequences or images where the position of information matters. In this paper, we propose a novel positional encoding method based on learnable Fourier features. Instead of hard-coding each position as a token or a vector, we represent each position, which can be multi-dimensional, as a trainable encoding based on learnable Fourier feature mapping, modulated with a multi-layer perceptron. The representation is particularly advantageous for a spatial multi-dimensional position, e.g., pixel positions on an image, where L_2 distances or more complex positional relationships need to be captured. Our experiments based on several public benchmark tasks show that our learnable Fourier feature representation for multi-dimensional positional encoding outperforms existing methods by both improving the accuracy and allowing faster convergence.

We demonstrate a new computational illumination technique that achieves large space-bandwidth-time product, for quantitative phase imaging of unstained live samples in vitro. Microscope lenses can have either large field of view (FOV) or high resolution, not both. Fourier ptychographic microscopy (FPM) is a new computational imaging technique that circumvents this limit by fusing information from multiple images taken with different illumination angles. The result is a gigapixel-scale image having both wide FOV and high resolution, i.e. large space-bandwidth product (SBP). FPM has enormous potential for revolutionizing microscopy and has already found application in digital pathology. However, it suffers from long acquisition times (on the order of minutes), limiting throughput. Faster capture times would not only improve imaging speed, but also allow studies of live samples, where motion artifacts degrade results. In contrast to fixed (e.g. pathology) slides, live samples are continuously evolving at various spatial and temporal scales. Here, we present a new source coding scheme, along with real-time hardware control, to achieve 0.8 NA resolution across a 4x FOV with sub-second capture times. We propose an improved algorithm and new initialization scheme, which allow robust phase reconstruction over long time-lapse experiments. We present the first FPM results for both growing and confluent in vitro cell cultures, capturing videos of subcellular dynamical phenomena in popular cell lines undergoing division and migration. Our method opens up FPM to applications with live samples, for observing rare events in both space and time.

We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP fails to learn high frequencies both in theory and in practice. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.

Fourier Neural Operators (FNO) learn solution operators of partial differential equations by parameterizing global convolutions in the complex Fourier domain. For real-valued PDE solutions, the complex FFT carries representational redundancy through conjugate symmetry. We introduce the Hartley Neural Operator (HNO), the exact real-valued mirror of FNO: it replaces the FFT with the purely real Discrete Hartley Transform and learns a single real multiplier per retained spectral mode, with no complex arithmetic. Because the real Hartley spectrum is not halved by conjugate symmetry, HNO retains twice as many frequency corners as FNO but one real weight where FNO carries a complex pair, so the two operators are iso-parametric at equal width and differ only in spectral basis. Our central thesis is that the best basis is a property of the operator. Self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that the real Hartley multiplier diagonalizes exactly, and HNO is favored there. Time-dependent operators carry phase, from oscillation in the wave equation to transport in advection, Burgers, and Navier-Stokes, which a real diagonal multiplier cannot represent, so FNO is favored there, and increasingly so with the operator's phase content, leaving the phaseless heat equation as the borderline case. Training both operators identically and benchmarking across PDE classes, initial-condition families, and boundary conditions, we find an elliptic-versus-time-dependent split that is monotone in operator phase content and matches the Green's-function theory we develop. Rather than a universal winner, our findings give a predictive rule: match the spectral basis to the symmetry of the solution operator.

Anomaly detection in multivariate time series is challenging as heterogeneous subsequence anomalies may occur. Reconstruction-based methods, which focus on learning normal patterns in the frequency domain to detect diverse abnormal subsequences, achieve promising results, while still falling short on capturing fine-grained frequency characteristics and channel correlations. To contend with the limitations, we introduce CATCH, a framework based on frequency patching. We propose to patchify the frequency domain into frequency bands, which enhances its ability to capture fine-grained frequency characteristics. To perceive appropriate channel correlations, we propose a Channel Fusion Module (CFM), which features a patch-wise mask generator and a masked-attention mechanism. Driven by a bi-level multi-objective optimization algorithm, the CFM is encouraged to iteratively discover appropriate patch-wise channel correlations, and to cluster relevant channels while isolating adverse effects from irrelevant channels. Extensive experiments on 10 real-world datasets and 12 synthetic datasets demonstrate that CATCH achieves state-of-the-art performance. We make our code and datasets available at https://github.com/decisionintelligence/CATCH.

Vision transformers have delivered tremendous success in representation learning. This is primarily due to effective token mixing through self attention. However, this scales quadratically with the number of pixels, which becomes infeasible for high-resolution inputs. To cope with this challenge, we propose Adaptive Fourier Neural Operator (AFNO) as an efficient token mixer that learns to mix in the Fourier domain. AFNO is based on a principled foundation of operator learning which allows us to frame token mixing as a continuous global convolution without any dependence on the input resolution. This principle was previously used to design FNO, which solves global convolution efficiently in the Fourier domain and has shown promise in learning challenging PDEs. To handle challenges in visual representation learning such as discontinuities in images and high resolution inputs, we propose principled architectural modifications to FNO which results in memory and computational efficiency. This includes imposing a block-diagonal structure on the channel mixing weights, adaptively sharing weights across tokens, and sparsifying the frequency modes via soft-thresholding and shrinkage. The resulting model is highly parallel with a quasi-linear complexity and has linear memory in the sequence size. AFNO outperforms self-attention mechanisms for few-shot segmentation in terms of both efficiency and accuracy. For Cityscapes segmentation with the Segformer-B3 backbone, AFNO can handle a sequence size of 65k and outperforms other efficient self-attention mechanisms.

Deep learning has become an area of interest in most scientific areas, including physical sciences. Modern networks apply real-valued transformations on the data. Particularly, convolutions in convolutional neural networks discard phase information entirely. Many deterministic signals, such as seismic data or electrical signals, contain significant information in the phase of the signal. We explore complex-valued deep convolutional networks to leverage non-linear feature maps. Seismic data commonly has a lowcut filter applied, to attenuate noise from ocean waves and similar long wavelength contributions. Discarding the phase information leads to low-frequency aliasing analogous to the Nyquist-Shannon theorem for high frequencies. In non-stationary data, the phase content can stabilize training and improve the generalizability of neural networks. While it has been shown that phase content can be restored in deep neural networks, we show how including phase information in feature maps improves both training and inference from deterministic physical data. Furthermore, we show that the reduction of parameters in a complex network outperforms larger real-valued networks.

Conventional self-attention mechanisms incur quadratic complexity, limiting their scalability on long sequences. We introduce FFTNet, an adaptive spectral filtering framework that leverages the Fast Fourier Transform (FFT) to achieve global token mixing in O(nlog n) time. By transforming inputs into the frequency domain, FFTNet exploits the orthogonality and energy preservation guaranteed by Parseval's theorem to capture long-range dependencies efficiently. A learnable spectral filter and modReLU activation dynamically emphasize salient frequency components, providing a rigorous and adaptive alternative to traditional self-attention. Experiments on the Long Range Arena and ImageNet benchmarks validate our theoretical insights and demonstrate superior performance over fixed Fourier and standard attention models.

The transformer model is known to be computationally demanding, and prohibitively costly for long sequences, as the self-attention module uses a quadratic time and space complexity with respect to sequence length. Many researchers have focused on designing new forms of self-attention or introducing new parameters to overcome this limitation, however a large portion of them prohibits the model to inherit weights from large pretrained models. In this work, the transformer's inefficiency has been taken care of from another perspective. We propose Fourier Transformer, a simple yet effective approach by progressively removing redundancies in hidden sequence using the ready-made Fast Fourier Transform (FFT) operator to perform Discrete Cosine Transformation (DCT). Fourier Transformer is able to significantly reduce computational costs while retain the ability to inherit from various large pretrained models. Experiments show that our model achieves state-of-the-art performances among all transformer-based models on the long-range modeling benchmark LRA with significant improvement in both speed and space. For generative seq-to-seq tasks including CNN/DailyMail and ELI5, by inheriting the BART weights our model outperforms the standard BART and other efficient models. Our code is publicly available at \url{https://github.com/LUMIA-Group/FourierTransformer}

Fourier Neural Operators (FNOs) have proven to be an efficient and effective method for resolution-independent operator learning in a broad variety of application areas across scientific machine learning. A key reason for their success is their ability to accurately model long-range dependencies in spatio-temporal data by learning global convolutions in a computationally efficient manner. To this end, FNOs rely on the discrete Fourier transform (DFT), however, DFTs cause visual and spectral artifacts as well as pronounced dissipation when learning operators in spherical coordinates since they incorrectly assume a flat geometry. To overcome this limitation, we generalize FNOs on the sphere, introducing Spherical FNOs (SFNOs) for learning operators on spherical geometries. We apply SFNOs to forecasting atmospheric dynamics, and demonstrate stable auto\-regressive rollouts for a year of simulated time (1,460 steps), while retaining physically plausible dynamics. The SFNO has important implications for machine learning-based simulation of climate dynamics that could eventually help accelerate our response to climate change.

The short-time Fourier transform (STFT) is widely used for analyzing non-stationary signals. However, its performance is highly sensitive to its parameters, and manual or heuristic tuning often yields suboptimal results. To overcome this limitation, we propose a unified differentiable formulation of the STFT that enables gradient-based optimization of its parameters. This approach addresses the limitations of traditional STFT parameter tuning methods, which often rely on computationally intensive discrete searches. It enables fine-tuning of the time-frequency representation (TFR) based on any desired criterion. Moreover, our approach integrates seamlessly with neural networks, allowing joint optimization of the STFT parameters and network weights. The efficacy of the proposed differentiable STFT in enhancing TFRs and improving performance in downstream tasks is demonstrated through experiments on both simulated and real-world data.

Phenomenological modeling of variable stars allows determination of a set of the parameters, which are needed for classification in the "General Catalogue of Variable Stars" and similar catalogs. We apply a recent method NAV ("New Algol Variable") to eclipsing binary stars of different types. Although all periodic functions may be represented as Fourier series with an infinite number of coefficients, this is impossible for a finite number of the observations. Thus one may use a restricted Fourier series, i.e. a trigonometric polynomial (TP) of order s either for fitting the light curve, or to make a periodogram analysis. However, the number of parameters needed drastically increases with decreasing width of minimum. In the NAV algorithm, the special shape of minimum is used, so the number of parameters is limited to 10 (if the period and initial epoch are fixed) or 12 (not fixed). We illustrate the NAV method by application to a recently discovered Algol-type eclipsing variable 2MASS J11080308-6145589 (in the field of previously known variable star RS Car) and compare results to that obtained using the TP fits. For this system, the statistically optimal number of parameters is 44, but the fit is still worse than that of the NAV fit. Application to the system GSC 3692-00624 argues that the NAV fit is better than the TP one even for the case of EW-type stars with much wider eclipses. Model parameters are listed.

Neural operators learn mappings between function spaces, which is practical for learning solution operators of PDEs and other scientific modeling applications. Among them, the Fourier neural operator (FNO) is a popular architecture that performs global convolutions in the Fourier space. However, such global operations are often prone to over-smoothing and may fail to capture local details. In contrast, convolutional neural networks (CNN) can capture local features but are limited to training and inference at a single resolution. In this work, we present a principled approach to operator learning that can capture local features under two frameworks by learning differential operators and integral operators with locally supported kernels. Specifically, inspired by stencil methods, we prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions. Both these approaches preserve the properties of operator learning and, hence, the ability to predict at any resolution. Adding our layers to FNOs significantly improves their performance, reducing the relative L2-error by 34-72% in our experiments, which include a turbulent 2D Navier-Stokes and the spherical shallow water equations.

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from O(L^6) to O(L^3), where L is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size n is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in R^K, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a ell_2 distance of at most varepsilon from the true simplex (for any varepsilon>0). Also, we theoretically show that in order to achieve this bound, it is sufficient to have ngeleft(K^2/varepsilon^2right)e^{Omegaleft(K/SNR^2right)} samples, where SNR stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as SNRgeOmegaleft(K^{1/2}right), the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in ashtiani2018nearly, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

Implicit neural representations (INRs) have arisen as useful methods for representing signals on Euclidean domains. By parameterizing an image as a multilayer perceptron (MLP) on Euclidean space, INRs effectively represent signals in a way that couples spatial and spectral features of the signal that is not obvious in the usual discrete representation, paving the way for continuous signal processing and machine learning approaches that were not previously possible. Although INRs using sinusoidal activation functions have been studied in terms of Fourier theory, recent works have shown the advantage of using wavelets instead of sinusoids as activation functions, due to their ability to simultaneously localize in both frequency and space. In this work, we approach such INRs and demonstrate how they resolve high-frequency features of signals from coarse approximations done in the first layer of the MLP. This leads to multiple prescriptions for the design of INR architectures, including the use of complex wavelets, decoupling of low and band-pass approximations, and initialization schemes based on the singularities of the desired signal.

Recent advancements in neural vocoding are predominantly driven by Generative Adversarial Networks (GANs) operating in the time-domain. While effective, this approach neglects the inductive bias offered by time-frequency representations, resulting in reduntant and computionally-intensive upsampling operations. Fourier-based time-frequency representation is an appealing alternative, aligning more accurately with human auditory perception, and benefitting from well-established fast algorithms for its computation. Nevertheless, direct reconstruction of complex-valued spectrograms has been historically problematic, primarily due to phase recovery issues. This study seeks to close this gap by presenting Vocos, a new model that directly generates Fourier spectral coefficients. Vocos not only matches the state-of-the-art in audio quality, as demonstrated in our evaluations, but it also substantially improves computational efficiency, achieving an order of magnitude increase in speed compared to prevailing time-domain neural vocoding approaches. The source code and model weights have been open-sourced at https://github.com/charactr-platform/vocos.

CNet is a C++/CUDA framework for building deep complex-valued neural networks (CVNNs) and optimizing complex functions by gradient descent with Wirtinger derivatives. Complex models are underexplored yet natural where data is intrinsically complex -- RF/IQ communications, MRI k-space, radar/SAR, audio spectra -- and phase carries information real networks discard. CNet is physics-native: a network is a cascade of complex (often unitary) operations on an amplitude vector, and classification is a Born-rule measurement p_k = |z_k|^2/||z||^2 rather than a softmax over real logits. Its library of complex layers makes conv(x,k) = IFFT(FFT(x).FFT(k)) learnable via signal-processing primitives -- spectral-padding local kernels (Pad), the inverse DFT, a magnitude nonlinearity |z|^2 (CModulus2), holomorphic powers z^M (CPower), and mean pooling (MeanPool) -- with GPU kernels for every layer, a GPU true-Adam optimizer, and a reduced-memory inference mode. Three studies. (1) A fully complex FNet-style causal language model, built on a new O(N log N) causal Fourier mixer, matches a param-matched real-valued causal FNet on tiny-shakespeare in under half the steps. We introduce Born-rule attention: a content-based causal mixer whose weights are quantum-measurement probabilities |<Q_k,K_j>|^2 of one token against another -- the first attention mechanism built on the Born rule. With rotary position embeddings and per-token complex normalization, a fully complex Born-rule model reaches 1.52 nats/char on tiny-shakespeare, matching softmax attention and beating the Fourier mixer by 0.23. (2,3) Two bottleneck analyses -- radio-modulation classification (RML2016.10a) and the Fourier phase problem of coherent-diffraction imaging -- isolate where CVNNs need new operators. The complex machinery learns the correct structure; open problems are concrete and operator-level. Code: https://github.com/crasmarum/CNet

Despite the remarkable success achieved by neural networks, particularly those represented by MLP and Transformer, we reveal that they exhibit potential flaws in the modeling and reasoning of periodicity, i.e., they tend to memorize the periodic data rather than genuinely understanding the underlying principles of periodicity. However, periodicity is a crucial trait in various forms of reasoning and generalization, underpinning predictability across natural and engineered systems through recurring patterns in observations. In this paper, we propose FAN, a novel network architecture based on Fourier Analysis, which empowers the ability to efficiently model and reason about periodic phenomena. By introducing Fourier Series, the periodicity is naturally integrated into the structure and computational processes of the neural network, thus achieving a more accurate expression and prediction of periodic patterns. As a promising substitute to multi-layer perceptron (MLP), FAN can seamlessly replace MLP in various models with fewer parameters and FLOPs. Through extensive experiments, we demonstrate the effectiveness of FAN in modeling and reasoning about periodic functions, and the superiority and generalizability of FAN across a range of real-world tasks, including symbolic formula representation, time series forecasting, and language modeling.

Within the graph learning community, conventional wisdom dictates that spectral convolutional networks may only be deployed on undirected graphs: Only there could the existence of a well-defined graph Fourier transform be guaranteed, so that information may be translated between spatial- and spectral domains. Here we show this traditional reliance on the graph Fourier transform to be superfluous and -- making use of certain advanced tools from complex analysis and spectral theory -- extend spectral convolutions to directed graphs. We provide a frequency-response interpretation of newly developed filters, investigate the influence of the basis used to express filters and discuss the interplay with characteristic operators on which networks are based. In order to thoroughly test the developed theory, we conduct experiments in real world settings, showcasing that directed spectral convolutional networks provide new state of the art results for heterophilic node classification on many datasets and -- as opposed to baselines -- may be rendered stable to resolution-scale varying topological perturbations.

Parameter-efficient fine-tuning (PEFT) has emerged as a crucial paradigm for adapting large language models (LLMs) under constrained computational budgets. However, standard PEFT methods often struggle in multi-task fine-tuning settings, where diverse optimization objectives induce task interference and limited parameter budgets lead to representational deficiency. While recent approaches incorporate mixture-of-experts (MoE) to alleviate these issues, they predominantly operate in the spatial domain, which may introduce structural redundancy and parameter overhead. To overcome these limitations, we reformulate adaptation in the spectral domain. Our spectral analysis reveals that different tasks exhibit distinct frequency energy distributions, and that LLM layers display heterogeneous frequency sensitivities. Motivated by these insights, we propose FourierMoE, which integrates the MoE architecture with the inverse discrete Fourier transform (IDFT) for frequency-aware adaptation. Specifically, FourierMoE employs a frequency-adaptive router to dispatch tokens to experts specialized in distinct frequency bands. Each expert learns a set of conjugate-symmetric complex coefficients, preserving complete phase and amplitude information while theoretically guaranteeing lossless IDFT reconstruction into real-valued spatial weights. Extensive evaluations across 28 benchmarks, multiple model architectures, and scales demonstrate that FourierMoE consistently outperforms competitive baselines in both single-task and multi-task settings while using significantly fewer trainable parameters. These results highlight the promise of spectral-domain expert adaptation as an effective and parameter-efficient paradigm for LLM fine-tuning.

This paper introduces the idea of applying signal processing inside a Large Language Model (LLM). With the recent explosion of generative AI, our work can help bridge two fields together, namely the field of signal processing and large language models. We draw parallels between classical Fourier-Transforms and Fourier Transform-like learnable time-frequency representations for every intermediate activation signal of an LLM. Once we decompose every activation signal across tokens into a time-frequency representation, we learn how to filter and reconstruct them, with all components learned from scratch, to predict the next token given the previous context. We show that for GPT-like architectures, our work achieves faster convergence and significantly increases performance by adding a minuscule number of extra parameters when trained for the same epochs. We hope this work paves the way for algorithms exploring signal processing inside the signals found in neural architectures like LLMs and beyond.

We present a novel type of neural fields that uses general radial bases for signal representation. State-of-the-art neural fields typically rely on grid-based representations for storing local neural features and N-dimensional linear kernels for interpolating features at continuous query points. The spatial positions of their neural features are fixed on grid nodes and cannot well adapt to target signals. Our method instead builds upon general radial bases with flexible kernel position and shape, which have higher spatial adaptivity and can more closely fit target signals. To further improve the channel-wise capacity of radial basis functions, we propose to compose them with multi-frequency sinusoid functions. This technique extends a radial basis to multiple Fourier radial bases of different frequency bands without requiring extra parameters, facilitating the representation of details. Moreover, by marrying adaptive radial bases with grid-based ones, our hybrid combination inherits both adaptivity and interpolation smoothness. We carefully designed weighting schemes to let radial bases adapt to different types of signals effectively. Our experiments on 2D image and 3D signed distance field representation demonstrate the higher accuracy and compactness of our method than prior arts. When applied to neural radiance field reconstruction, our method achieves state-of-the-art rendering quality, with small model size and comparable training speed.

Initialization of parameters in deep neural networks has been shown to have a big impact on the performance of the networks (Mishkin & Matas, 2015). The initialization scheme devised by He et al, allowed convolution activations to carry a constrained mean which allowed deep networks to be trained effectively (He et al., 2015a). Orthogonal initializations and more generally orthogonal matrices in standard recurrent networks have been proved to eradicate the vanishing and exploding gradient problem (Pascanu et al., 2012). Majority of current initialization schemes do not take fully into account the intrinsic structure of the convolution operator. Using the duality of the Fourier transform and the convolution operator, Convolution Aware Initialization builds orthogonal filters in the Fourier space, and using the inverse Fourier transform represents them in the standard space. With Convolution Aware Initialization we noticed not only higher accuracy and lower loss, but faster convergence. We achieve new state of the art on the CIFAR10 dataset, and achieve close to state of the art on various other tasks.

The use of mmWave frequencies is one of the key strategies to achieve the fascinating 1000x increase in the capacity of future 5G wireless systems. While for traditional sub-6 GHz cellular frequencies several well-developed statistical channel models are available for system simulation, similar tools are not available for mmWave frequencies, thus preventing a fair comparison of independently developed transmission and reception schemes. In this paper we provide a simple albeit accurate statistical procedure for the generation of a clustered MIMO channel model operating at mmWaves, for both the cases of slowly and rapidly time-varying channels. Matlab scripts for channel generation are also provided, along with an example of their use.

We prove a weighted Wasserstein spectral gap for the three-dimensional primitive equations on mathscr E={uin H^1cap L^8: partial_zuin L^4} under finite-rank additive Brownian forcing that is phase-complete finite Fourier and satisfies a quantitative quadratic saturation condition. In particular, four real forcing directions suffice and this number is optimal within the phase-complete class. Consequently, the system is uniquely ergodic and exponentially mixing in the H^1 topology. The main difficulties stem from the lack of sufficiently strong moment bounds and the highly degenerate nature of the noise, placing the problem outside the direct reach of standard methods. We introduce a new quantitative stable--compact asymptotic coupling criterion that captures the delicate balance between the available control of the growth of initial perturbations and the accuracy--cost of compensating them by perturbing the noise. For the primitive equations, this leads to a quantitative infinite-dimensional Hörmander analysis requiring tracking the degree and coefficient cost of Lie-bracket polynomials with the Fourier frequency.

Diffusion models have achieved remarkable success in image synthesis, but the generated high-quality images raise concerns about potential malicious use. Existing detectors often struggle to capture discriminative clues across different models and settings, limiting their generalization to unseen diffusion models and robustness to various perturbations. To address this issue, we observe that diffusion-generated images exhibit progressively larger differences from natural real images across low- to high-frequency bands. Based on this insight, we propose a simple yet effective representation by enhancing the Frequency Forgery Clue (F^2C) across all frequency bands. Specifically, we introduce a frequency-selective function which serves as a weighted filter to the Fourier spectrum, suppressing less discriminative bands while enhancing more informative ones. This approach, grounded in a comprehensive analysis of frequency-based differences between natural real and diffusion-generated images, enables general detection of images from unseen diffusion models and provides robust resilience to various perturbations. Extensive experiments on various diffusion-generated image datasets demonstrate that our method outperforms state-of-the-art detectors with superior generalization and robustness.

Holographic Metasurface Transceivers (HMTs) are emerging as cost-effective substitutes to large antenna arrays for beamforming in Millimeter and TeraHertz wave communication. However, to achieve desired channel gains through beamforming in HMT, phase-shifts of a large number of elements need to be appropriately set, which is challenging. Also, these optimal phase-shifts depend on the location of the receivers, which could be unknown. In this work, we develop a learning algorithm using a {\it fixed-budget multi-armed bandit framework} to beamform and maximize received signal strength at the receiver for far-field regions. Our algorithm, named \Algo exploits the parametric form of channel gains of the beams, which can be expressed in terms of two {\it phase-shifting parameters}. Even after parameterization, the problem is still challenging as phase-shifting parameters take continuous values. To overcome this, {\it\HB} works with the discrete values of phase-shifting parameters and exploits their unimodal relations with channel gains to learn the optimal values faster. We upper bound the probability of {\it\HB} incorrectly identifying the (discrete) optimal phase-shift parameters in terms of the number of pilots used in learning. We show that this probability decays exponentially with the number of pilot signals. We demonstrate that {\it\HB} outperforms state-of-the-art algorithms through extensive simulations.

This article focuses on estimating relative transfer functions (RTFs) for beamforming applications. Traditional methods often assume that spectra are uncorrelated, an assumption that is often violated in practical scenarios due to factors such as time-domain windowing or the non-stationary nature of signals, as observed in speech. To overcome these limitations, we propose an RTF estimation technique that leverages spectral and spatial correlations through subspace analysis. Additionally, we derive Cram\'er--Rao bounds (CRBs) for the RTF estimation task, providing theoretical insights into the achievable estimation accuracy. These bounds reveal that channel estimation can be performed more accurately if the noise or the target signal exhibits spectral correlations. Experiments with both real and synthetic data show that our technique outperforms the narrowband maximum-likelihood estimator, known as covariance whitening (CW), when the target exhibits spectral correlations. Although the proposed algorithm generally achieves accuracy close to the theoretical bound, there is potential for further improvement, especially in scenarios with highly spectrally correlated noise. While channel estimation has various applications, we demonstrate the method using a minimum variance distortionless (MVDR) beamformer for multichannel speech enhancement. A free Python implementation is also provided.

Convolutional Neural Networks (CNN) have been successful in processing data signals that are uniformly sampled in the spatial domain (e.g., images). However, most data signals do not natively exist on a grid, and in the process of being sampled onto a uniform physical grid suffer significant aliasing error and information loss. Moreover, signals can exist in different topological structures as, for example, points, lines, surfaces and volumes. It has been challenging to analyze signals with mixed topologies (for example, point cloud with surface mesh). To this end, we develop mathematical formulations for Non-Uniform Fourier Transforms (NUFT) to directly, and optimally, sample nonuniform data signals of different topologies defined on a simplex mesh into the spectral domain with no spatial sampling error. The spectral transform is performed in the Euclidean space, which removes the translation ambiguity from works on the graph spectrum. Our representation has four distinct advantages: (1) the process causes no spatial sampling error during the initial sampling, (2) the generality of this approach provides a unified framework for using CNNs to analyze signals of mixed topologies, (3) it allows us to leverage state-of-the-art backbone CNN architectures for effective learning without having to design a particular architecture for a particular data structure in an ad-hoc fashion, and (4) the representation allows weighted meshes where each element has a different weight (i.e., texture) indicating local properties. We achieve results on par with the state-of-the-art for the 3D shape retrieval task, and a new state-of-the-art for the point cloud to surface reconstruction task.

In this work we introduce Steerable Transformers, an extension of the Vision Transformer mechanism that maintains equivariance to the special Euclidean group SE(d). We propose an equivariant attention mechanism that operates on features extracted by steerable convolutions. Operating in Fourier space, our network utilizes Fourier space non-linearities. Our experiments in both two and three dimensions show that adding a steerable transformer encoder layer to a steerable convolution network enhances performance.

Many recent multivariate time series anomaly detection (MTSAD) models incorporate cross-channel modeling, under the implicit assumption that the structure of anomalies may be spread across multiple channels. We evaluate this assumption on eight widely used public benchmarks by introducing a per-segment diagnostic framework that flags, for each labeled anomaly, whether at least one channel deviates individually from its normal history, whether the cross-channel correlation structure changes, or both. The framework shows that no cross-channel rupture occurs without an accompanying univariate deviation across a range of reasonable thresholds. A complementary metric also reveals that on six of the eight benchmarks, at least half of the labeled anomaly segments deviate univariately on 89% to 100% of their timesteps, reaching 100% on three of these datasets. To verify that our framework captures cross-channel structure when present, we construct synthetic data of phase-shifted sinusoidal channels with shared noise. Each anomalous segment is altered through one of two channel-wise corruptions that preserve the per-channel marginal distribution while breaking cross-channel structure, and our framework correctly characterizes these segments as cross-channel-only. On these data, channel-dependent (CD) models successfully exploit the cross-channel signal whereas channel-independent (CI) ones fail. The CI/CD comparison of a recent SOTA detector on real benchmarks further confirms that CD modeling brings no measurable gain. We conclude that current MTSAD benchmarks are unsuitable for validating cross-channel modeling capabilities, and we call for the development of more structurally diverse evaluation sets. The code for this study is publicly available.

What spatial frequency information do humans and neural networks use to recognize objects? In neuroscience, critical band masking is an established tool that can reveal the frequency-selective filters used for object recognition. Critical band masking measures the sensitivity of recognition performance to noise added at each spatial frequency. Existing critical band masking studies show that humans recognize periodic patterns (gratings) and letters by means of a spatial-frequency filter (or "channel'') that has a frequency bandwidth of one octave (doubling of frequency). Here, we introduce critical band masking as a task for network-human comparison and test 14 humans and 76 neural networks on 16-way ImageNet categorization in the presence of narrowband noise. We find that humans recognize objects in natural images using the same one-octave-wide channel that they use for letters and gratings, making it a canonical feature of human object recognition. On the other hand, the neural network channel, across various architectures and training strategies, is 2-4 times as wide as the human channel. In other words, networks are vulnerable to high and low frequency noise that does not affect human performance. Adversarial and augmented-image training are commonly used to increase network robustness and shape bias. Does this training align network and human object recognition channels? Three network channel properties (bandwidth, center frequency, peak noise sensitivity) correlate strongly with shape bias (53% variance explained) and with robustness of adversarially-trained networks (74% variance explained). Adversarial training increases robustness but expands the channel bandwidth even further away from the human bandwidth. Thus, critical band masking reveals that the network channel is more than twice as wide as the human channel, and that adversarial training only increases this difference.

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new Spatiotemporal Fourier Neural Operator (SFNO) that learns maps between Bochner spaces, and a new learning framework to address these issues. This new paradigm leverages wisdom from traditional numerical PDE theory and techniques to refine the pipeline of commonly adopted end-to-end neural operator training and evaluations. Specifically, in the learning problems for the turbulent flow modeling by the Navier-Stokes Equations (NSE), the proposed architecture initiates the training with a few epochs for SFNO, concluding with the freezing of most model parameters. Then, the last linear spectral convolution layer is fine-tuned without the frequency truncation. The optimization uses a negative Sobolev norm for the first time as the loss in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is almost exact thanks to the Parseval identity. This design allows the neural operators to effectively tackle low-frequency errors while the relief of the de-aliasing filter addresses high-frequency errors. Numerical experiments on commonly used benchmarks for the 2D NSE demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers.

Physics-informed surrogate models can accelerate computational fluid dynamics simulations. However, many existing methods reproduce global flow patterns more reliably than localized multiscale structures. This study presents a physics-informed Fourier-wavelet transformer for next-step velocity-field reconstruction in real-world flow benchmarks. The proposed formulation combines hybrid Fourier-wavelet spectral encoding with physics-biased self-attention based on partial differential equation residual diagnostics. It also uses self-supervised pretraining through Masked Physics Prediction and Equation Consistency Prediction. The experiments are conducted on two real benchmark cases: cylinder-wake flow and fluid-structure interaction. All approaches are evaluated under a shared local protocol and compared with spectral, transformer-based, operator-learning, and physics-informed neural-network baselines. On the cylinder-wake benchmark, the proposed model achieves the best aggregate accuracy, with an all-channel normalized mean-squared error of 0.05875 and an all-channel Pearson correlation coefficient of 0.97019. On the fluid-structure-interaction benchmark, it gives the lowest all-channel normalized mean-squared error of 2.70 times 10^{-4}, compared with 4.02 times 10^{-4} for the strongest baseline. Component-wise field comparisons and scale-separated diagnostics further show stronger recovery of localized wake structures, including near-body, wake-core, and far-wake features. The results demonstrate improved real-world flow reconstruction while maintaining a practical accuracy-cost tradeoff.

Large Language Models struggle with memory demands from the growing Key-Value (KV) cache as context lengths increase. Existing compression methods homogenize head dimensions or rely on attention-guided token pruning, often sacrificing accuracy or introducing computational overhead. We propose FourierAttention, a training-free framework that exploits the heterogeneous roles of transformer head dimensions: lower dimensions prioritize local context, while upper ones capture long-range dependencies. By projecting the long-context-insensitive dimensions onto orthogonal Fourier bases, FourierAttention approximates their temporal evolution with fixed-length spectral coefficients. Evaluations on LLaMA models show that FourierAttention achieves the best long-context accuracy on LongBench and Needle-In-A-Haystack (NIAH). Besides, a custom Triton kernel, FlashFourierAttention, is designed to optimize memory via streamlined read-write operations, enabling efficient deployment without performance compromise.

Accurate air quality forecasts are vital for public health alerts, exposure assessment, and emissions control. In practice, observational data are often missing in varying proportions and patterns due to collection and transmission issues. These incomplete spatiotemporal records impede reliable inference and risk assessment and can lead to overconfident extrapolation. To address these challenges, we propose an end to end framework, the channel gated learning unit based spatiotemporal bayesian neural field (CGLUBNF). It uses Fourier features with a graph attention encoder to capture multiscale spatial dependencies and seasonal temporal dynamics. A channel gated learning unit, equipped with learnable activations and gated residual connections, adaptively filters and amplifies informative features. Bayesian inference jointly optimizes predictive distributions and parameter uncertainty, producing point estimates and calibrated prediction intervals. We conduct a systematic evaluation on two real world datasets, covering four typical missing data patterns and comparing against five state of the art baselines. CGLUBNF achieves superior prediction accuracy and sharper confidence intervals. In addition, we further validate robustness across multiple prediction horizons and analysis the contribution of extraneous variables. This research lays a foundation for reliable deep learning based spatio-temporal forecasting with incomplete observations in emerging sensing paradigms, such as real world vehicle borne mobile monitoring.

Multivariate time series (MTS) forecasting has shown great importance in numerous industries. Current state-of-the-art graph neural network (GNN)-based forecasting methods usually require both graph networks (e.g., GCN) and temporal networks (e.g., LSTM) to capture inter-series (spatial) dynamics and intra-series (temporal) dependencies, respectively. However, the uncertain compatibility of the two networks puts an extra burden on handcrafted model designs. Moreover, the separate spatial and temporal modeling naturally violates the unified spatiotemporal inter-dependencies in real world, which largely hinders the forecasting performance. To overcome these problems, we explore an interesting direction of directly applying graph networks and rethink MTS forecasting from a pure graph perspective. We first define a novel data structure, hypervariate graph, which regards each series value (regardless of variates or timestamps) as a graph node, and represents sliding windows as space-time fully-connected graphs. This perspective considers spatiotemporal dynamics unitedly and reformulates classic MTS forecasting into the predictions on hypervariate graphs. Then, we propose a novel architecture Fourier Graph Neural Network (FourierGNN) by stacking our proposed Fourier Graph Operator (FGO) to perform matrix multiplications in Fourier space. FourierGNN accommodates adequate expressiveness and achieves much lower complexity, which can effectively and efficiently accomplish the forecasting. Besides, our theoretical analysis reveals FGO's equivalence to graph convolutions in the time domain, which further verifies the validity of FourierGNN. Extensive experiments on seven datasets have demonstrated our superior performance with higher efficiency and fewer parameters compared with state-of-the-art methods.

Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We present the theoretical foundations of NFT and show that the existence of a linear equivariant feature, which has been assumed ubiquitously in equivariance learning, is equivalent to the existence of a group invariant kernel on the dataspace. We also provide experimental results to demonstrate the application of NFT in typical scenarios with varying levels of knowledge about the acting group.

Generative diffusion models have emerged as a powerful tool for high-quality image synthesis, yet their iterative nature demands significant computational resources. This paper proposes an efficient time step sampling method based on an image spectral analysis of the diffusion process, aimed at optimizing the denoising process. Instead of the traditional uniform distribution-based time step sampling, we introduce a Beta distribution-like sampling technique that prioritizes critical steps in the early and late stages of the process. Our hypothesis is that certain steps exhibit significant changes in image content, while others contribute minimally. We validated our approach using Fourier transforms to measure frequency response changes at each step, revealing substantial low-frequency changes early on and high-frequency adjustments later. Experiments with ADM and Stable Diffusion demonstrated that our Beta Sampling method consistently outperforms uniform sampling, achieving better FID and IS scores, and offers competitive efficiency relative to state-of-the-art methods like AutoDiffusion. This work provides a practical framework for enhancing diffusion model efficiency by focusing computational resources on the most impactful steps, with potential for further optimization and broader application.

Gaussian quantum channels constitute a cornerstone of continuous-variable quantum information science, underpinning a wide array of protocols in quantum optics and quantum metrology. While the action of such channels on arbitrary states is well-characterized under full channel knowledge, we address the inverse problem, namely, the precise estimation of fundamental channel parameters, including the beam splitter transmissivity and the two-mode squeezing amplitude. Employing the quantum Fisher information (QFI) as a benchmark for metrological sensitivity, we demonstrate that the symmetry inherent in mode mixing critically governs the amplification of QFI, thereby enabling high-precision parameter estimation. In addition, we investigate quantum thermometry by estimating the average photon number of thermal states, revealing that the transmissivity parameter significantly modulates estimation precision. Our results underscore the metrological utility of two-mode Gaussian states and establish a robust framework for parameter inference in noisy and dynamically evolving quantum systems.

We prove a single-logarithm bound for the eigenvalues in the plunge region of Fourier concentration operators, removing the extra logarithmic factor in the Kulikov--Dam Larsen bound for two general admissible sets throughout each fixed exponential window. For bounded spatial and frequency sets A,BsubsetR^d of positive measure with finite upper codimension-one Minkowski boundary content, the number Λ_varepsilon(cA,B) of eigenvalues in (varepsilon,1-varepsilon) satisfies Λ_varepsilon(cA,B)le C(d,A,B,β)c^{d-1}D(1+log_+(c/D)), where D=log(1/(varepsilon(1-varepsilon))), for every fixed β>0, cge2, 0<varepsilon<1/2, and Dleβc, with log_+t=max(0,log t). The argument uses positive band-limited smoothing and requires no symmetry, convexity, or connectedness. In dimension two, the corresponding exponential-window estimate has a constant uniform in the window parameter. We also determine the optimal coordinate-Lipschitz constant d-1 for recursive decompositions of balls and standard simplices in every dimension, and obtain box-frequency estimates with explicit geometric constants. For C^1 convex bodies in three dimensions, the optimal infimum of recursive coordinate slopes is 2. Uniform real-order Bessel estimates give independent disk and ball bounds at every spectral threshold, including an explicit deep-tail term. For smooth strictly convex planar frequency bodies with positive curvature, we prove correlation estimates for aligned Gaussian packets at arbitrary relative scales and quantitative lower bounds on the cost of estimating spatial blocks separately. Together these results connect the global spectral bound with sharp geometric constructions and the cancellation retained by grouping boundary interactions.

Diffusion models offer impressive controllability for image tasks, primarily through noise predictions that encode task-specific information and classifier-free guidance enabling adjustable scaling. This scaling mechanism implicitly defines a ``scaling space'' whose potential for fine-grained semantic manipulation remains underexplored. We investigate this space, starting with inversion-based editing where the difference between conditional/unconditional noise predictions carries key semantic information. Our core contribution stems from a Fourier analysis of noise predictions, revealing that its low- and high-frequency components evolve differently throughout diffusion. Based on this insight, we introduce FreSca, a straightforward method that applies guidance scaling independently to different frequency bands in the Fourier domain. FreSca demonstrably enhances existing image editing methods without retraining. Excitingly, its effectiveness extends to image understanding tasks such as depth estimation, yielding quantitative gains across multiple datasets.

Orthogonal time frequency space (OTFS) modulation was shown to provide significant error performance advantages over orthogonal frequency division multiplexing (OFDM) in delay--Doppler channels. In order to detect OTFS modulated data, the channel impulse response needs to be known at the receiver. In this paper, we propose embedded pilot-aided channel estimation schemes for OTFS. In each OTFS frame, we arrange pilot, guard, and data symbols in the delay--Doppler plane to suitably avoid interference between pilot and data symbols at the receiver. We develop such symbol arrangements for OTFS over multipath channels with integer and fractional Doppler shifts, respectively. At the receiver, channel estimation is performed based on a threshold method and the estimated channel information is used for data detection via a message passing (MP) algorithm. Thanks to our specific embedded symbol arrangements, both channel estimation and data detection are performed within the same OTFS frame with a minimum overhead. We compare by simulations the error performance of OTFS using the proposed channel estimation and OTFS with ideally known channel information and observe only a marginal performance loss. We also demonstrate that the proposed channel estimation in OTFS significantly outperforms OFDM with known channel information. Finally, we present extensions of the proposed schemes to MIMO and multi-user uplink/downlink.

Low-discrepancy (LD) sequences have been extensively used as efficient experimental designs across many scientific disciplines. QMCPy (https://qmcsoftware.github.io/QMCSoftware/) is an accessible Python library which provides a unified implementation of randomized LD sequences, automatic variable transformations, adaptive Quasi-Monte Carlo error estimation algorithms, and fast kernel methods. This article focuses on recent updates to QMCPy which broaden support for randomized LD sequences and add new tools to enable fast kernel methods using LD sequences. Specifically, we give a unified description of the supported LD lattices, digital nets, and Halton point sets, along with randomization options including random permutations / shifts, linear matrix scrambling (LMS), and nested uniform scrambling (NUS). We also support higher-order digital nets, higher-order scrambling with LMS or NUS, and Halton scrambling with LMS or NUS. For fast kernel methods, we provide shift-invariant (SI) and digitally-shift-invariant (DSI) kernels, including a new set of higher-order smoothness DSI kernels. When SI and DSI kernels are respectively paired with n LD lattice and digital net points, the resulting Gram matrices permit multiplication and inversion at only O(n log n) cost. These fast operations utilize QMCPy's implementation of the fast Fourier transform in bit-reversed order (FFTBR), inverse FFTBR (IFFTBR), and fast Walsh--Hadamard transform (FWHT).

Fourier PlenOctrees have shown to be an efficient representation for real-time rendering of dynamic Neural Radiance Fields (NeRF). Despite its many advantages, this method suffers from artifacts introduced by the involved compression when combining it with recent state-of-the-art techniques for training the static per-frame NeRF models. In this paper, we perform an in-depth analysis of these artifacts and leverage the resulting insights to propose an improved representation. In particular, we present a novel density encoding that adapts the Fourier-based compression to the characteristics of the transfer function used by the underlying volume rendering procedure and leads to a substantial reduction of artifacts in the dynamic model. Furthermore, we show an augmentation of the training data that relaxes the periodicity assumption of the compression. We demonstrate the effectiveness of our enhanced Fourier PlenOctrees in the scope of quantitative and qualitative evaluations on synthetic and real-world scenes.

Spiking Neural Networks (SNNs) have emerged as an energy-efficient alternative to conventional neural networks, demonstrating strong performance in computer vision and robotics. More recently, SNNs have been applied to time series forecasting (TSF), with methods exploring spiking temporal backbones, spike-compatible positional encodings, Fourier-domain processing, and redesigned neuron dynamics. However, existing SNN forecasting approaches process variables independently, lacking explicit mechanisms for modeling inter-variable dependencies. This is a critical limitation in multivariate settings, where cross-variable correlations carry substantial predictive information. We propose Spiking Fourier Graph Operators (SpikF-GO), which addresses this gap by combining a hypervariate graph formulation in which every scalar observation becomes a graph node with spike-driven spectral processing. SpikF-GO introduces a Hard Concrete frequency gate for learnable sparse frequency selection and a Complex LIF gate that applies independent spiking neurons to real and imaginary Fourier components, preserving binary, event-driven computation throughout the spectral domain. We further present a variant incorporating Central Pattern Generator-based positional encodings for stronger long-range temporal modeling. Evaluated on eight benchmarks under a unified experimental protocol, SpikF-GO achieves the best average rank among all SNN methods and outperforms its ANN counterpart, FourierGNN, at reduced energy cost. SpikF-GO maintains competitive accuracy even at substantially smaller embedding dimensions, thereby achieving significant energy reductions. To our knowledge, this is among the first works to bring graph-based multivariate modeling into the spiking domain for TSF and the first to provide a unified comparison across SNN forecasting architectures under a common experimental protocol.

Using the Riesz-Feller fractional derivative, we apply the factorization algorithm to the fractional quantum harmonic oscillator along the lines previously proposed by Olivar-Romero and Rosas-Ortiz, extending their results. We solve the non-Hermitian fractional eigenvalue problem in the k space by introducing in that space a new class of Hermite `polynomials' that we call Riesz-Feller Hermite `polynomials'. Using the inverse Fourier transform in Mathematica, interesting analytic results for the same eigenvalue problem in the x space are also obtained. Additionally, a more general factorization with two different Levy indices is briefly introduced

TFDWT is an open-source Python library that allows the construction of TensorFlow Layers for Fast Discrete Wavelet Transform (DWT) and Inverse Discrete Wavelet Transform (IDWT) in end-to-end backpropagation learning networks. By definition, a multiresolution signal representation using a multi-rate discrete wavelet system creates enriched joint natural-frequency representations. These layers facilitate the construction of multilevel DWT filter banks and Wavelet Packet Transform (WPT) filter banks for a spatial-frequency representation of the inputs and features in shallow or deep networks. The discrete wavelet system partitions the frequency plane into subbands using orthogonal dilated and translated lowpass scaling and highpass wavelet function. The realization of a fast discrete wavelet system is a two-band perfect reconstruction multi-rate filter bank with FIR filters corresponding to the impulse responses of the scaling and wavelet function with downsampling and upsampling operations. A filter bank for a higher dimensional input is a seamless extension by successive separable circular convolutions across each independent axis. The command `pip install TFDWT' installs the latest version of the package.

Reliable clinical deployment of deep medical image models is hindered by distribution shifts across scanners, sites, and acquisition protocols. Existing domain generalization (DG) methods often focus on style or intensity diversification, but they can still leave networks dependent on domain-specific texture correlations. Inspired by evidence that Fourier phase encodes semantic structure, we introduce PhaseAT, a phase-aware adversarial training framework for medical DG. PhaseAT forms phase-perturbed training views in the Fourier domain by iteratively updating a bounded phase perturbation while keeping the amplitude spectrum unchanged, thereby stressing spatial organization under matched appearance statistics. Perturbations are applied only to the luminance channel in YCbCr color space to avoid chromatic artifacts. Additionally, a simple phase-saliency mask concentrates updates on the most influential frequencies. The model is trained with a weighted combination of losses on clean and phase-perturbed samples, supporting both single-source and multi-source DG. We validate our method on two challenging medical datasets and demonstrate that PhaseAT achieves over 20% improvement in single-source domain generalization, outperforming several state-of-the-art DG methods. The code implementation is available at: https://github.com/ahmed-sharshar/PhaseAT.

Learned image compression (LIC) has gained traction as an effective solution for image storage and transmission in recent years. However, existing LIC methods are redundant in latent representation due to limitations in capturing anisotropic frequency components and preserving directional details. To overcome these challenges, we propose a novel frequency-aware transformer (FAT) block that for the first time achieves multiscale directional ananlysis for LIC. The FAT block comprises frequency-decomposition window attention (FDWA) modules to capture multiscale and directional frequency components of natural images. Additionally, we introduce frequency-modulation feed-forward network (FMFFN) to adaptively modulate different frequency components, improving rate-distortion performance. Furthermore, we present a transformer-based channel-wise autoregressive (T-CA) model that effectively exploits channel dependencies. Experiments show that our method achieves state-of-the-art rate-distortion performance compared to existing LIC methods, and evidently outperforms latest standardized codec VTM-12.1 by 14.5%, 15.1%, 13.0% in BD-rate on the Kodak, Tecnick, and CLIC datasets.

Modeling long-range dependencies in time series generation poses a fundamental trade-off between representational capacity and computational efficiency. Traditional temporal diffusion models suffer from local entanglement and the O(L^2) cost of attention mechanisms. We address these limitations by introducing PaCoDi (Parallel Complex Diffusion), a spectral-native architecture that decouples generative modeling in the frequency domain. PaCoDi fundamentally alters the problem topology: the Fourier Transform acts as a diagonalizing operator, converting locally coupled temporal signals into globally decorrelated spectral components. Theoretically, we prove the Quadrature Forward Diffusion and Conditional Reverse Factorization theorem, demonstrating that the complex diffusion process can be split into independent real and imaginary branches. We bridge the gap between this decoupled theory and data reality using a Mean Field Theory (MFT) approximation reinforced by an interactive correction mechanism. Furthermore, we generalize this discrete DDPM to continuous-time Frequency SDEs, rigorously deriving the Spectral Wiener Process describe the differential spectral Brownian motion limit. Crucially, PaCoDi exploits the Hermitian Symmetry of real-valued signals to compress the sequence length by half, achieving a 50% reduction in attention FLOPs without information loss. We further derive a rigorous Heteroscedastic Loss to handle the non-isotropic noise distribution on the compressed manifold. Extensive experiments show that PaCoDi outperforms existing baselines in both generation quality and inference speed, offering a theoretically grounded and computationally efficient solution for time series modeling.

We consider distributed image transmission over a noisy multiple access channel (MAC) using deep joint source-channel coding (DeepJSCC). It is known that Shannon's separation theorem holds when transmitting independent sources over a MAC in the asymptotic infinite block length regime. However, we are interested in the practical finite block length regime, in which case separate source and channel coding is known to be suboptimal. We introduce a novel joint image compression and transmission scheme, where the devices send their compressed image representations in a non-orthogonal manner. While non-orthogonal multiple access (NOMA) is known to achieve the capacity region, to the best of our knowledge, non-orthogonal joint source channel coding (JSCC) scheme for practical systems has not been studied before. Through extensive experiments, we show significant improvements in terms of the quality of the reconstructed images compared to orthogonal transmission employing current DeepJSCC approaches particularly for low bandwidth ratios. We publicly share source code to facilitate further research and reproducibility.

Steganography is the science of hiding a secret message within an ordinary public message, which is referred to as Carrier. Traditionally, digital signal processing techniques, such as least significant bit encoding, were used for hiding messages. In this paper, we explore the use of deep neural networks as steganographic functions for speech data. We showed that steganography models proposed for vision are less suitable for speech, and propose a new model that includes the short-time Fourier transform and inverse-short-time Fourier transform as differentiable layers within the network, thus imposing a vital constraint on the network outputs. We empirically demonstrated the effectiveness of the proposed method comparing to deep learning based on several speech datasets and analyzed the results quantitatively and qualitatively. Moreover, we showed that the proposed approach could be applied to conceal multiple messages in a single carrier using multiple decoders or a single conditional decoder. Lastly, we evaluated our model under different channel distortions. Qualitative experiments suggest that modifications to the carrier are unnoticeable by human listeners and that the decoded messages are highly intelligible.

Joint source-channel coding (JSCC) offers a promising avenue for enhancing transmission efficiency by jointly incorporating source and channel statistics into the system design. A key advancement in this area is the deep joint source and channel coding (DeepJSCC) technique that designs a direct mapping of input signals to channel symbols parameterized by a neural network, which can be trained for arbitrary channel models and semantic quality metrics. This paper advances the DeepJSCC framework toward a semantics-aligned, high-fidelity transmission approach, called semantics-guided diffusion DeepJSCC (SGD-JSCC). Existing schemes that integrate diffusion models (DMs) with JSCC face challenges in transforming random generation into accurate reconstruction and adapting to varying channel conditions. SGD-JSCC incorporates two key innovations: (1) utilizing some inherent information that contributes to the semantics of an image, such as text description or edge map, to guide the diffusion denoising process; and (2) enabling seamless adaptability to varying channel conditions with the help of a semantics-guided DM for channel denoising. The DM is guided by diverse semantic information and integrates seamlessly with DeepJSCC. In a slow fading channel, SGD-JSCC dynamically adapts to the instantaneous signal-to-noise ratio (SNR) directly estimated from the channel output, thereby eliminating the need for additional pilot transmissions for channel estimation. In a fast fading channel, we introduce a training-free denoising strategy, allowing SGD-JSCC to effectively adjust to fluctuations in channel gains. Numerical results demonstrate that, guided by semantic information and leveraging the powerful DM, our method outperforms existing DeepJSCC schemes, delivering satisfactory reconstruction performance even at extremely poor channel conditions.

Surrogate modeling of wave propagation and scattering (i.e. the wave speed and source to wave field map) in heterogeneous media has significant potential in applications such as seismic imaging and inversion. High-contrast settings, such as subsurface models with salt bodies, exhibit strong scattering and phase sensitivity that challenge existing neural operators. We propose a hybrid architecture that decomposes the scattering operator into two separate contributions: a smooth background propagation and a high-contrast scattering correction. The smooth component is learned with a Fourier Neural Operator (FNO), which produces globally coupled feature tokens encoding background wave propagation; these tokens are then passed to a vision transformer, where attention is used to model the high-contrast scattering correction dominated by strong, spatial interactions. Evaluated on high-frequency Helmholtz problems with strong contrasts, the hybrid model achieves substantially improved phase and amplitude accuracy compared to standalone FNOs or transformers, with favorable accuracy-parameter scaling.

In solving partial differential equations (PDEs), Fourier Neural Operators (FNOs) have exhibited notable effectiveness compared to Convolutional Neural Networks (CNNs). This paper presents clear empirical evidence through spectral analysis to elucidate the superiority of FNO over CNNs: FNO is significantly more capable of learning low-frequencies. This empirical evidence also unveils FNO's distinct low-frequency bias, which limits FNO's effectiveness in learning high-frequency information from PDE data. To tackle this challenge, we introduce SpecBoost, an ensemble learning framework that employs multiple FNOs to better capture high-frequency information. Specifically, a secondary FNO is utilized to learn the overlooked high-frequency information from the prediction residual of the initial FNO. Experiments demonstrate that SpecBoost noticeably enhances FNO's prediction accuracy on diverse PDE applications, achieving an up to 71% improvement.

Brain-computer interfaces (BCIs) offer transformative potential, but decoding neural signals presents significant challenges. The core premise of this paper is built around demonstrating methods to elucidate the underlying low-dimensional geometric structure present in high-dimensional brainwave data in order to assist in downstream BCI-related neural classification tasks. We demonstrate two pipelines related to electroencephalography (EEG) signal processing: (1) a preliminary pipeline removing noise from individual EEG channels, and (2) a downstream manifold learning pipeline uncovering geometric structure across networks of EEG channels. We conduct preliminary validation using two EEG datasets and situate our demonstration in the context of the BCI-relevant imagined digit decoding problem. Our preliminary pipeline uses an attention-based EEG filtration network to extract clean signal from individual EEG channels. Our primary pipeline uses a fast Fourier transform, a Laplacian eigenmap, a discrete analog of Ricci flow via Ollivier's notion of Ricci curvature, and a graph convolutional network to perform dimensionality reduction on high-dimensional multi-channel EEG data in order to enable regularizable downstream classification. Our system achieves competitive performance with existing signal processing and classification benchmarks; we demonstrate a mean test correlation coefficient of >0.95 at 2 dB on semi-synthetic neural denoising and a downstream EEG-based classification accuracy of 0.97 on distinguishing digit- versus non-digit- thoughts. Results are preliminary and our geometric machine learning pipeline should be validated by more extensive follow-up studies; generalizing these results to larger inter-subject sample sizes, different hardware systems, and broader use cases will be crucial.

Modeling and synthesizing low-light raw noise is a fundamental problem for computational photography and image processing applications. Although most recent works have adopted physics-based models to synthesize noise, the signal-independent noise in low-light conditions is far more complicated and varies dramatically across camera sensors, which is beyond the description of these models. To address this issue, we introduce a new perspective to synthesize the signal-independent noise by a generative model. Specifically, we synthesize the signal-dependent and signal-independent noise in a physics- and learning-based manner, respectively. In this way, our method can be considered as a general model, that is, it can simultaneously learn different noise characteristics for different ISO levels and generalize to various sensors. Subsequently, we present an effective multi-scale discriminator termed Fourier transformer discriminator (FTD) to distinguish the noise distribution accurately. Additionally, we collect a new low-light raw denoising (LRD) dataset for training and benchmarking. Qualitative validation shows that the noise generated by our proposed noise model can be highly similar to the real noise in terms of distribution. Furthermore, extensive denoising experiments demonstrate that our method performs favorably against state-of-the-art methods on different sensors.

The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number N and allowed average decoding error ε equals the vanishing-error capacity at mean photon number N/(1-ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.

Unlike color photography images, which are consistently encoded into RGB channels, biological images encompass various modalities, where the type of microscopy and the meaning of each channel varies with each experiment. Importantly, the number of channels can range from one to a dozen and their correlation is often comparatively much lower than RGB, as each of them brings specific information content. This aspect is largely overlooked by methods designed out of the bioimage field, and current solutions mostly focus on intra-channel spatial attention, often ignoring the relationship between channels, yet crucial in most biological applications. Importantly, the variable channel type and count prevent the projection of several experiments to a unified representation for large scale pre-training. In this study, we propose ChAda-ViT, a novel Channel Adaptive Vision Transformer architecture employing an Inter-Channel Attention mechanism on images with an arbitrary number, order and type of channels. We also introduce IDRCell100k, a bioimage dataset with a rich set of 79 experiments covering 7 microscope modalities, with a multitude of channel types, and channel counts varying from 1 to 10 per experiment. Our proposed architecture, trained in a self-supervised manner, outperforms existing approaches in several biologically relevant downstream tasks. Additionally, it can be used to bridge the gap for the first time between assays with different microscopes, channel numbers or types by embedding various image and experimental modalities into a unified biological image representation. The latter should facilitate interdisciplinary studies and pave the way for better adoption of deep learning in biological image-based analyses. Code and Data to be released soon.

The received in-phase and quadrature (I/Q) baseband signals inherently encode physical-layer and channel characteristics of wireless links. Learning robust and transferable representations directly from such raw signals, however, remains challenging due to heterogeneous communication systems, diverse propagation environments, and limited labeled data. To address this, we present LWM-Spectro, a transformer-based foundation model pretrained on large-scale I/Q data represented as time-frequency spectrograms. The model leverages self-supervised masked modeling, contrastive learning, and a mixture-of-experts (MoE) architecture to learn general-purpose wireless representations. These representations transfer effectively to downstream tasks such as modulation classification and joint SNR/mobility recognition, even with minimal supervision. Across tasks, LWM-Spectro consistently outperforms state-of-the-art deep learning baselines in both few-shot and data-rich regimes, providing a unified foundation for wireless learning.

Quantum kernels are among the most promising methods in quantum machine learning, providing a natural way to embed data into high-dimensional Hilbert spaces via quantum circuits and capturing non-linear relationships in data. Their learning behaviour is governed by the function class they represent, which is formally characterized by the structure of the associated Reproducing Kernel Hilbert Space (RKHS). However, a theoretical characterization of this function class, and of how it depends on the choice of feature map and its hyperparameters, is still lacking. In this work, we focus on the Instantaneous Quantum Polynomial (IQP) feature map and derive a Fourier representation of the associated RKHS generators as finite sums of trigonometric functions evaluated on polynomially transformed input variables, with a finite set of accessible frequencies determined by the circuit construction and scaled by the bandwidth hyperparameter. This representation identifies three bandwidth regimes with qualitatively distinct expressive properties from localised, through multimodal, to periodic. Numerical experiments on synthetic datasets with controlled periodic and multimodal structure, across data dimensions and feature-map configurations, reproduce the three regimes in both supervised classification and unsupervised anomaly detection, benchmarked against a classical Gaussian kernel. These results make the relationship between expressivity and data structure explicit for IQP kernels: expressive power translates into a practical benefit only when the accessible frequencies match with the structure of the data, which provides a criterion for selecting the bandwidth when the periodic or multimodal structure of data is known.

The spectral smoothness properties of the low-frequency array of the Square Kilometer Array (SKA), namely SKA-Low, are an important issue for its scientific objectives to be attainable. A large array of 256 log-periodic dipole antennas, installed on top of a 42~m circular ground plane, will work as an SKA-Low station in the frequency range 50-350 MHz. In this article, the ground plane induced effects are examined in terms of antenna beam spectral characteristics, while different antenna placements are considered. Results are produced both at isolated antenna and at array level in the band 50-100 MHz, by employing an approximate method for the speeding-up of array simulations. We attempt to distinguish the ground plane effect from that of mutual coupling among antennas, which appears to be more severe at specific frequencies, using 2 figures of merit. The Discrete Fourier Transform (DFT) components of gain pattern ratios identify the fundamental spatial components of the ripple, while the Envelope Correlation Coefficient quantifies the penalty to considering an infinite ground plane.

TorchOptics is an open-source Python library for differentiable Fourier optics simulations, developed using PyTorch to enable GPU-accelerated tensor computations and automatic differentiation. It provides a comprehensive framework for modeling, analyzing, and designing optical systems using Fourier optics, with applications in imaging, diffraction, holography, and signal processing. The library leverages PyTorch's automatic differentiation engine for gradient-based optimization, enabling the inverse design of complex optical systems. TorchOptics supports end-to-end optimization of hybrid models that integrate optical systems with machine learning architectures for digital post-processing. The library includes a wide range of optical elements and spatial profiles, and supports simulations with polarized light and fields with arbitrary spatial coherence.

Learning physical simulations has been an essential and central aspect of many recent research efforts in machine learning, particularly for Navier-Stokes-based fluid mechanics. Classic numerical solvers have traditionally been computationally expensive and challenging to use in inverse problems, whereas Neural solvers aim to address both concerns through machine learning. We propose a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluate a large set of basis functions in the context of (a) a compressible 1D SPH simulation, (b) a weakly compressible 2D SPH simulation, and (c) an incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries included in the basis functions are key aspects of stability and accuracy. Our broad evaluation shows that Fourier-based continuous convolutions outperform all other architectures regarding accuracy and generalization. Finally, using these Fourier-based networks, we show that prior inductive biases, such as window functions, are no longer necessary. An implementation of our approach, as well as complete datasets and solver implementations, is available at https://github.com/tum-pbs/SFBC.

We present a comprehensive analysis of how two-layer neural networks learn features to solve the modular addition task. Our work provides a full mechanistic interpretation of the learned model and a theoretical explanation of its training dynamics. While prior work has identified that individual neurons learn single-frequency Fourier features and phase alignment, it does not fully explain how these features combine into a global solution. We bridge this gap by formalizing a diversification condition that emerges during training when overparametrized, consisting of two parts: phase symmetry and frequency diversification. We prove that these properties allow the network to collectively approximate a flawed indicator function on the correct logic for the modular addition task. While individual neurons produce noisy signals, the phase symmetry enables a majority-voting scheme that cancels out noise, allowing the network to robustly identify the correct sum. Furthermore, we explain the emergence of these features under random initialization via a lottery ticket mechanism. Our gradient flow analysis proves that frequencies compete within each neuron, with the "winner" determined by its initial spectral magnitude and phase alignment. From a technical standpoint, we provide a rigorous characterization of the layer-wise phase coupling dynamics and formalize the competitive landscape using the ODE comparison lemma. Finally, we use these insights to demystify grokking, characterizing it as a three-stage process involving memorization followed by two generalization phases, driven by the competition between loss minimization and weight decay.

It is well-known that if a network aims to learn how to deblur, it should understand the blur process. Blurring is naturally caused by the convolution of the sharp image with the blur kernel. Thus, allowing the network to learn the blur process in the kernel-level can significantly improve the image deblurring performance. But, current deep networks are still at the pixel-level learning stage, either performing end-to-end pixel-level restoration or stage-wise pseudo kernel-level restoration, failing to enable the deblur model to understand the essence of the blur. To this end, we propose Fourier Kernel Estimator (FKE), which considers the activation operation in Fourier space and converts the convolution problem in the spatial domain to a multiplication problem in Fourier space. Our FKE, jointly optimized with the deblur model, enables the network to learn the kernel-level blur process with low complexity and without any additional supervision. Furthermore, we change the convolution object of the kernel from ``image" to network extracted ``feature", whose rich semantic and structural information is more suitable to blur process learning. With the convolution of the feature and the estimated kernel, our model can learn the essence of blur in kernel-level. To further improve the efficiency of feature extraction, we design a decoupled multi-scale architecture with multiple hierarchical sub-unets with a reversible strategy, which allows better multi-scale encoding and decoding in low training memory. Extensive experiments indicate that our method achieves state-of-the-art motion deblurring results and show potential for handling other kernel-related problems. Analysis also shows our kernel estimator is able to learn physically meaningful kernels. The code will be available at https://github.com/DeepMed-Lab-ECNU/Single-Image-Deblur.

Watermarking the outputs of generative models is a crucial technique for tracing copyright and preventing potential harm from AI-generated content. In this paper, we introduce a novel technique called Tree-Ring Watermarking that robustly fingerprints diffusion model outputs. Unlike existing methods that perform post-hoc modifications to images after sampling, Tree-Ring Watermarking subtly influences the entire sampling process, resulting in a model fingerprint that is invisible to humans. The watermark embeds a pattern into the initial noise vector used for sampling. These patterns are structured in Fourier space so that they are invariant to convolutions, crops, dilations, flips, and rotations. After image generation, the watermark signal is detected by inverting the diffusion process to retrieve the noise vector, which is then checked for the embedded signal. We demonstrate that this technique can be easily applied to arbitrary diffusion models, including text-conditioned Stable Diffusion, as a plug-in with negligible loss in FID. Our watermark is semantically hidden in the image space and is far more robust than watermarking alternatives that are currently deployed. Code is available at github.com/YuxinWenRick/tree-ring-watermark.

Multivariate time series forecasting plays a crucial role in various fields such as finance, traffic management, energy, and healthcare. Recent studies have highlighted the advantages of channel independence to resist distribution drift but neglect channel correlations, limiting further enhancements. Several methods utilize mechanisms like attention or mixer to address this by capturing channel correlations, but they either introduce excessive complexity or rely too heavily on the correlation to achieve satisfactory results under distribution drifts, particularly with a large number of channels. Addressing this gap, this paper presents an efficient MLP-based model, the Series-cOre Fused Time Series forecaster (SOFTS), which incorporates a novel STar Aggregate-Redistribute (STAR) module. Unlike traditional approaches that manage channel interactions through distributed structures, e.g., attention, STAR employs a centralized strategy to improve efficiency and reduce reliance on the quality of each channel. It aggregates all series to form a global core representation, which is then dispatched and fused with individual series representations to facilitate channel interactions effectively.SOFTS achieves superior performance over existing state-of-the-art methods with only linear complexity. The broad applicability of the STAR module across different forecasting models is also demonstrated empirically. For further research and development, we have made our code publicly available at https://github.com/Secilia-Cxy/SOFTS.

In this note we examine the autoregressive generalization of the FNet algorithm, in which self-attention layers from the standard Transformer architecture are substituted with a trivial sparse-uniformsampling procedure based on Fourier transforms. Using the Wikitext-103 benchmark, we demonstratethat FNetAR retains state-of-the-art performance (25.8 ppl) on the task of causal language modelingcompared to a Transformer-XL baseline (24.2 ppl) with only half the number self-attention layers,thus providing further evidence for the superfluity of deep neural networks with heavily compoundedattention mechanisms. The autoregressive Fourier transform could likely be used for parameterreduction on most Transformer-based time-series prediction models.

Parameterized quantum circuits are widely investigated in quantum machine learning, yet how their architecture can provide task-aligned inductive biases remains poorly understood. We study data reuploading quantum circuits with two distinct inputs: a signal that varies between samples and a fixed, task-defined spectral query that specifies how the signal should be probed. We show that encoding the query can change the circuit's hypothesis class over the signal by modulating how its existing signal frequencies are combined, without introducing new sample-dependent information. This mechanism matches tasks built on nonuniform Fourier transforms, whose evaluation frequencies specify the query and whose sampling positions guide its encoding. We examine this connection in a classification task inspired by synthetic aperture radar, where acquisition geometry determines a query for classifying unfocused spectral patches before image formation. Across small-scale spectral and radar-like tasks, aligned queries improve learning from limited data over parameter-matched circuits with mismatched or absent queries. These results provide a route from known spectral task structure to the design and validation of inductive biases in parameterized quantum circuits.

Implicit neural representations (INRs) use neural networks to provide continuous and resolution-independent representations of complex signals with a small number of parameters. However, existing INR models often fail to capture important frequency components specific to each task. To address this issue, in this paper, we propose a Fourier Kolmogorov Arnold network (FKAN) for INRs. The proposed FKAN utilizes learnable activation functions modeled as Fourier series in the first layer to effectively control and learn the task-specific frequency components. In addition, the activation functions with learnable Fourier coefficients improve the ability of the network to capture complex patterns and details, which is beneficial for high-resolution and high-dimensional data. Experimental results show that our proposed FKAN model outperforms three state-of-the-art baseline schemes, and improves the peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) for the image representation task and intersection over union (IoU) for the 3D occupancy volume representation task, respectively.

In remote sensing rotated object detection, mainstream methods suffer from two bottlenecks, directional incoherence at detector neck and task conflict at detecting head. Ulitising fourier rotation equivariance, we introduce Fourier Angle Alignment, which analyses angle information through frequency spectrum and aligns the main direction to a certain orientation. Then we propose two plug and play modules : FAAFusion and FAA Head. FAAFusion works at the detector neck, aligning the main direction of higher-level features to the lower-level features and then fusing them. FAA Head serves as a new detection head, which pre-aligns RoI features to a canonical angle and adds them to the original features before classification and regression. Experiments on DOTA-v1.0, DOTA-v1.5 and HRSC2016 show that our method can greatly improve previous work. Particularly, our method achieves new state-of-the-art results of 78.72% mAP on DOTA-v1.0 and 72.28% mAP on DOTA-v1.5 datasets with single scale training and testing, validating the efficacy of our approach in remote sensing object detection. The code is made publicly available at https://github.com/gcy0423/Fourier-Angle-Alignment .

Using the supersymmetric method of random matrix theory within the Heidelberg approach framework we provide statistical description of stationary intensity sampled in locations inside an open wave-chaotic cavity, assuming that the time-reversal invariance inside the cavity is fully broken. In particular, we show that when incoming waves are fed via a finite number M of open channels the probability density {cal P}(I) for the single-point intensity I decays as a power law for large intensities: {cal P}(I)sim I^{-(M+2)}, provided there is no internal losses. This behaviour is in marked difference with the Rayleigh law {cal P}(I)sim exp(-I/I) which turns out to be valid only in the limit Mto infty. We also find the joint probability density of intensities I_1, ldots, I_L in L>1 observation points, and then extract the corresponding statistics for the maximal intensity in the observation pattern. For Lto infty the resulting limiting extreme value statistics (EVS) turns out to be different from the classical EVS distributions.

We solve the Schrödinger equation for a particle in the circle with a quantum mechanical bootstrap method. The main idea is that in physical situations one needs solutions of the recursion relations for the Fourier coefficients of the measure that vanish exponentially fast. We show that this results in a (generalized) continued fraction representation of the solution. The solution is an analytical function of the couplings and the energy. We also show how one this method directly computes the density of states and checking that the answer is physical is reduced to checking an inequality for a single variable. We also show how to reconstruct the wave function and how to extract the quasimomentum of the particle as a function of the energy. We illustrate our solution in the simplest case of the Mathieu equation, where we show that the method can easily resolve exponentially small bands in the band spectrum.

In Image-to-Video (I2V) generation, a video is created using an input image as the first-frame condition. Existing I2V methods concatenate the full information of the conditional image with noisy latents to achieve high fidelity. However, the denoisers in these methods tend to shortcut the conditional image, which is known as conditional image leakage, leading to performance degradation issues such as slow motion and color inconsistency. In this work, we further clarify that conditional image leakage leads to overfitting to in-domain data and decreases the performance in out-of-domain scenarios. Moreover, we introduce Fourier-Guided Latent Shifting I2V, named FlashI2V, to prevent conditional image leakage. Concretely, FlashI2V consists of: (1) Latent Shifting. We modify the source and target distributions of flow matching by subtracting the conditional image information from the noisy latents, thereby incorporating the condition implicitly. (2) Fourier Guidance. We use high-frequency magnitude features obtained by the Fourier Transform to accelerate convergence and enable the adjustment of detail levels in the generated video. Experimental results show that our method effectively overcomes conditional image leakage and achieves the best generalization and performance on out-of-domain data among various I2V paradigms. With only 1.3B parameters, FlashI2V achieves a dynamic degree score of 53.01 on Vbench-I2V, surpassing CogVideoX1.5-5B-I2V and Wan2.1-I2V-14B-480P. Github page: https://pku-yuangroup.github.io/FlashI2V/

We present an algorithm, Fourier Activity Recognition (FAR), for UAV video activity recognition. Our formulation uses a novel Fourier object disentanglement method to innately separate out the human agent (which is typically small) from the background. Our disentanglement technique operates in the frequency domain to characterize the extent of temporal change of spatial pixels, and exploits convolution-multiplication properties of Fourier transform to map this representation to the corresponding object-background entangled features obtained from the network. To encapsulate contextual information and long-range space-time dependencies, we present a novel Fourier Attention algorithm, which emulates the benefits of self-attention by modeling the weighted outer product in the frequency domain. Our Fourier attention formulation uses much fewer computations than self-attention. We have evaluated our approach on multiple UAV datasets including UAV Human RGB, UAV Human Night, Drone Action, and NEC Drone. We demonstrate a relative improvement of 8.02% - 38.69% in top-1 accuracy and up to 3 times faster over prior works.

Despite their ubiquity throughout science and engineering, only a handful of partial differential equations (PDEs) have analytical, or closed-form solutions. This motivates a vast amount of classical work on numerical simulation of PDEs and more recently, a whirlwind of research into data-driven techniques leveraging machine learning (ML). A recent line of work indicates that a hybrid of classical numerical techniques and machine learning can offer significant improvements over either approach alone. In this work, we show that the choice of the numerical scheme is crucial when incorporating physics-based priors. We build upon Fourier-based spectral methods, which are known to be more efficient than other numerical schemes for simulating PDEs with smooth and periodic solutions. Specifically, we develop ML-augmented spectral solvers for three common PDEs of fluid dynamics. Our models are more accurate (2-4x) than standard spectral solvers at the same resolution but have longer overall runtimes (~2x), due to the additional runtime cost of the neural network component. We also demonstrate a handful of key design principles for combining machine learning and numerical methods for solving PDEs.

Recent vision transformers, large-kernel CNNs and MLPs have attained remarkable successes in broad vision tasks thanks to their effective information fusion in the global scope. However, their efficient deployments, especially on mobile devices, still suffer from noteworthy challenges due to the heavy computational costs of self-attention mechanisms, large kernels, or fully connected layers. In this work, we apply conventional convolution theorem to deep learning for addressing this and reveal that adaptive frequency filters can serve as efficient global token mixers. With this insight, we propose Adaptive Frequency Filtering (AFF) token mixer. This neural operator transfers a latent representation to the frequency domain via a Fourier transform and performs semantic-adaptive frequency filtering via an elementwise multiplication, which mathematically equals to a token mixing operation in the original latent space with a dynamic convolution kernel as large as the spatial resolution of this latent representation. We take AFF token mixers as primary neural operators to build a lightweight neural network, dubbed AFFNet. Extensive experiments demonstrate the effectiveness of our proposed AFF token mixer and show that AFFNet achieve superior accuracy and efficiency trade-offs compared to other lightweight network designs on broad visual tasks, including visual recognition and dense prediction tasks.

Despite the recent visually-pleasing results achieved, the massive computational cost has been a long-standing flaw for diffusion probabilistic models (DPMs), which, in turn, greatly limits their applications on resource-limited platforms. Prior methods towards efficient DPM, however, have largely focused on accelerating the testing yet overlooked their huge complexity and sizes. In this paper, we make a dedicated attempt to lighten DPM while striving to preserve its favourable performance. We start by training a small-sized latent diffusion model (LDM) from scratch, but observe a significant fidelity drop in the synthetic images. Through a thorough assessment, we find that DPM is intrinsically biased against high-frequency generation, and learns to recover different frequency components at different time-steps. These properties make compact networks unable to represent frequency dynamics with accurate high-frequency estimation. Towards this end, we introduce a customized design for slim DPM, which we term as Spectral Diffusion (SD), for light-weight image synthesis. SD incorporates wavelet gating in its architecture to enable frequency dynamic feature extraction at every reverse steps, and conducts spectrum-aware distillation to promote high-frequency recovery by inverse weighting the objective based on spectrum magni tudes. Experimental results demonstrate that, SD achieves 8-18x computational complexity reduction as compared to the latent diffusion models on a series of conditional and unconditional image generation tasks while retaining competitive image fidelity.

Inverse problems in image reconstruction are fundamentally complicated by unknown noise properties. Classical iterative deconvolution approaches amplify noise and require careful parameter selection for an optimal trade-off between sharpness and grain. Deep learning methods allow for flexible parametrization of the noise and learning its properties directly from the data. Recently, self-supervised blind-spot neural networks were successfully adopted for image deconvolution by including a known point-spread function in the end-to-end training. However, their practical application has been limited to 2D images in the biomedical domain because it implies large kernels that are poorly optimized. We tackle this problem with Fast Fourier Transform convolutions that provide training speed-up in 3D microscopy deconvolution tasks. Further, we propose to adopt a Siamese invariance loss for deconvolution and empirically identify its optimal position in the neural network between blind-spot and full image branches. The experimental results show that our improved framework outperforms the previous state-of-the-art deconvolution methods with a known point spread function.

Which functions can be used as activations in deep neural networks? This article explores families of functions based on orthonormal bases, including the Hermite polynomial basis and the Fourier trigonometric basis, as well as a basis resulting from the tropicalization of a polynomial basis. Our study shows that, through simple variance-preserving initialization and without additional clamping mechanisms, these activations can successfully be used to train deep models, such as GPT-2 for next-token prediction on OpenWebText and ConvNeXt for image classification on ImageNet. Our work addresses the issue of exploding and vanishing activations and gradients, particularly prevalent with polynomial activations, and opens the door for improving the efficiency of large-scale learning tasks. Furthermore, our approach provides insight into the structure of neural networks, revealing that networks with polynomial activations can be interpreted as multivariate polynomial mappings. Finally, using Hermite interpolation, we show that our activations can closely approximate classical ones in pre-trained models by matching both the function and its derivative, making them especially useful for fine-tuning tasks. These activations are available in the torchortho library, which can be accessed via: https://github.com/K-H-Ismail/torchortho.

Acoustic beamforming models typically assume wide-sense stationarity of speech signals within short time frames. However, voiced speech is better modeled as a cyclostationary (CS) process, a random process whose mean and autocorrelation are T_1-periodic, where alpha_1=1/T_1 corresponds to the fundamental frequency of vowels. Higher harmonic frequencies are found at integer multiples of the fundamental. This work introduces a cyclic multichannel Wiener filter (cMWF) for speech enhancement derived from a cyclostationary model. This beamformer exploits spectral correlation across the harmonic frequencies of the signal to further reduce the mean-squared error (MSE) between the target and the processed input. The proposed cMWF is optimal in the MSE sense and reduces to the MWF when the target is wide-sense stationary. Experiments on simulated data demonstrate considerable improvements in scale-invariant signal-to-distortion ratio (SI-SDR) on synthetic data but also indicate high sensitivity to the accuracy of the estimated fundamental frequency alpha_1, which limits effectiveness on real data.

Conventional communication systems, including both separation-based coding and AI-driven joint source-channel coding (JSCC), are largely guided by Shannon's rate-distortion theory. However, relying on generic distortion metrics fails to capture complex human visual perception, often resulting in blurred or unrealistic reconstructions. In this paper, we propose Joint Source-Channel-Generation Coding (JSCGC), a novel paradigm that shifts the focus from deterministic reconstruction to probabilistic generation. JSCGC leverages a generative model at the receiver as a generator rather than a conventional decoder to parameterize the data distribution, enabling direct maximization of mutual information under channel constraints while controlling stochastic sampling to produce outputs residing on the authentic data manifold with high fidelity. We further derive a theoretical lower bound on the maximum semantic inconsistency with given transmitted mutual information, elucidating the fundamental limits of communication in controlling the generative process. Extensive experiments on image transmission demonstrate that JSCGC substantially improves perceptual quality and semantic fidelity, significantly outperforming conventional distortion-oriented JSCC methods.

Diffusion models are proficient at generating high-quality images. They are however effective only when operating at the resolution used during training. Inference at a scaled resolution leads to repetitive patterns and structural distortions. Retraining at higher resolutions quickly becomes prohibitive. Thus, methods enabling pre-existing diffusion models to operate at flexible test-time resolutions are highly desirable. Previous works suffer from frequent artifacts and often introduce large latency overheads. We propose two simple modules that combine to solve these issues. We introduce a Frequency Modulation (FM) module that leverages the Fourier domain to improve the global structure consistency, and an Attention Modulation (AM) module which improves the consistency of local texture patterns, a problem largely ignored in prior works. Our method, coined Fam diffusion, can seamlessly integrate into any latent diffusion model and requires no additional training. Extensive qualitative results highlight the effectiveness of our method in addressing structural and local artifacts, while quantitative results show state-of-the-art performance. Also, our method avoids redundant inference tricks for improved consistency such as patch-based or progressive generation, leading to negligible latency overheads.

The coexistence of heterogeneous cellular standards (2G-5G) in shared spectrum demands sophisticated RF source separation techniques, yet no public dataset exists for data-driven research on this problem. We present RFSS (RF Signal Source Separation), an open-source dataset of 100,000 multi-source RF signal samples generated with full 3GPP standards compliance. The dataset covers GSM (TS 45.004), UMTS (TS 25.211), LTE (TS 36.211), and 5G NR (TS 38.211), with 2-4 simultaneous sources per sample plus 4,000 single-source reference samples, at 30.72 MHz sample rate. Each sample passes through independent 3GPP TDL multipath fading channels and realistic hardware impairments: carrier frequency offset, I/Q imbalance, phase noise, DC offset, and PA nonlinearity (Rapp model). Two mixing modes are provided: co-channel (all sources at baseband) and adjacent-channel (each source frequency-shifted to its standard-specific carrier). The dataset totals 103 GB in HDF5 format with a 70/15/15 train/validation/test split. We benchmark five methods: FastICA, Frobenius-norm NMF, Conv-TasNet, DPRNN, and a CNN-LSTM baseline, evaluated using permutation-invariant SI-SINR (PI-SI-SINR). Conv-TasNet achieves -21.18 dB PI-SI-SINR on 2-source mixtures versus -34.91 dB for ICA, a 13.7 dB improvement. On co-channel mixtures, Conv-TasNet reaches -12.34 dB versus -28.04 dB for ICA and -16.19 dB for NMF. The dataset and evaluation code are publicly released at submission time.