Spectral Representation Method

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3 papers in the last four weeks, down 57% on the four weeks before. 0.0% of all new papers.

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Latest papers 46

Sep 28, 2026cs.LG

Neural networks for spectral optimization

Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional. PDE solvers might be used to tackle this optimization. It is however computationally expensive. We propose two neural network models which learn the spectrum directly from the geometry of the domain and can be used to optimize the domain from one or more eigenvalues. We investigate two representations. The first encodes the domain through Fourier coefficients and a light MLP, which is efficient on star-shaped geometries, achieving a precision of 0.2%. Through a rescaling of the coefficients the designed models satisfy the scaling law of the eigenvalues. Additionally, averaging the outputs of the trained surrogates over rotations and reflections induces invariance for these transformations. The second is a model that takes the landscape function, the indicator function and the gradient of the landscape function. A Gram-Schmidt process produces orthogonal eigenfunctions as output of the model along with the associated eigenvalues. The landscape model reaches 1% mean relative error on the first ten eigenvalues, compared with 4% for an FNO model. Replacing the landscape by an SDF worsened both prediction and optimization errors. The trained model also generalizes from synthetic shapes to domains given as classical image dataset. The resulting surrogates of both approaches recover classical spectral optima such as the disk for the first eigenvalue or the conjectured minima of higher eigenvalues. This confirms that our models produce accurate differentiable estimates of eigenvalues, which can be used in shape optimization problems involving spectral quantities.

Alexis de Villeroché, Beniamin Bogosel, Stéphane Breuils +2

Spectral Representation Method Neural Partial Differential Equation Solvers Spectral Neural Network Optimization Neural Network Landscapes

Sep 28, 2026stat.ML

Learning Conditional Expectation Operators via Functional Newton Updates

We introduce the Functional Spectral-Newton Method (FSNM) for learning the leading singular structure of a conditional expectation operator without fixing a basis or reproducing kernel Hilbert space. FSNM fits a low-rank representation of the centered joint-to-product density ratio kernel by alternating functional Newton updates. Each update reduces to a preconditioned regression, which we approximate with vector-valued regression trees in a stagewise boosting procedure. At the population level, we establish descent and an

O(1/T)

best-iterate block-stationarity rate under a relative weak-learner accuracy condition, and show that every nondegenerate local minimum over the full centered

L^2

spaces is a globally optimal rank-

d

approximation. Synthetic experiments show that FSNM recovers a low-rank density ratio and its leading spectral structure, and that the same learned kernel can answer multiple conditional queries without refitting.

Thiago Ramos, Alek Fröhlich, Daniel Perazzo +1

Federal University of São Carlos · CSML, Istituto Italiano di Tecnologia and University of Genoa · CSML, Istituto Italiano di Tecnologia and University College London

Density Ratio Estimation Spectral Representation Method Kernel Hilbert Spaces Low-Rank Structure Newton

Sep 20, 2026cs.LG

Predicting Out-of-Distribution Generalization of Neural Operators via Observable Spectral Error Decomposition

Neural operators have emerged as powerful surrogates for solving partial differential equations (PDEs), yet their reliability under distribution shift remains a critical barrier to deployment. Existing approaches to out-of-distribution (OOD) generalization in operator learning are largely empirical and black-box: they report aggregate error metrics without explaining why errors arise or when they will grow. We propose a structure-preserving framework that makes OOD generalization predictable and auditable. Our key idea is to parameterize the learned solution operator as a spectral filter

h_θ(λ)

acting on the eigenvalues of the underlying elliptic operator, implemented via Chebyshev polynomial expansions and trained with a weak-form objective. This parameterization admits an exact decomposition of the energy-norm error into two observable components: a model-dependent spectral approximation term and a distribution-dependent spectral weighting term induced by the input. From this decomposition we derive three diagnostics: a conservative in-band supremum

\vareps_{\mathrm{sup}}

, a global RMS proxy

\vareps_{\mathrm{rms}}

, and a sample-dependent effective metric

\vareps_{\mathrm{eff}}(f)

. These diagnostics can be computed without access to ground-truth solutions. Through four controlled experiments, we show that

\vareps_{\mathrm{eff}}(f)\|f\|

consistently predicts energy error under in-distribution, in-band spectral shift, out-of-band tail, and compound shifts, whereas global metrics can be systematically misleading. Our framework shifts OOD assessment of neural operators from black-box benchmarking to operator-structure diagnostics, providing a practical route to auditable scientific machine learning.

Hang-Cheng Dong, Pengcheng Cheng

School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin 150001, China · School of Mathematics, Jilin University, Changchun 130012, China

Neural Operators Out-Of-Distribution Spectral Representation Method Decomposition

Sep 7, 2026cs.IR

EigenLI: Spectral Approximations to Late Interaction

Late-interaction models such as ColBERT achieve strong effectiveness by representing each document with many token-level vectors, but this expressivity leads to large indexing cost, storage footprints and expensive MaxSim scoring. We show that late-interaction representations exhibit an intrinsic low-rank structure: document token embeddings concentrate in a low-dimensional subspace that preserves most of the retrieval signal. Leveraging this observation, we introduce EigenLI, a spectral approximation framework that compresses late-interaction representations via document-specific low-dimensional subspaces. Unlike clustering or pooling methods, EigenLI identifies the dominant eigendirections of each document and uses them to construct reduced interaction representations. Empirically,

k

-EigenLI with

k \le 32

outperforms k-means and Ward clustering based pooling methods on ColBERTv2 and AnswerAI-ColBERT-small; GTE-ModernColBERT exhibits a different tradeoff at

k=32

, where clustering methods perform better. The same spectral construction also yields EigenLI-SV, an ANN-compatible single-vector representation derived from the second-order summary of the reduced structure. Across multiple datasets and all three text models, EigenLI-SV consistently outperforms comparable single-vector surrogates such as MUVERA.

Archish S, Sabyasachi Basu, Ankit Garg +2

Temple · Microsoft Research India

Subspace Spectral Decomposition Spectral Representation Method

Sep 3, 2026math.NA

Spectral Convergence of Random Feature Method in Multiple Dimensions

We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operator. On a single event determined only by the sampled features, one random space approximates every target in a prescribed source ball; moreover, for each target, a single coefficient vector defines an approximant that attains spectral accuracy simultaneously in all admissible error norms. For both regularity-adapted frequency distributions and uniform distributions on growing frequency windows, the resulting rates range from super-exponential to algebraic, depending on the regularity of the target. Second, we establish abstract error estimates for strong- and weak-form RFM discretizations, thereby converting the preceding approximation bounds into convergence estimates for multidimensional second-order elliptic boundary value and eigenvalue problems. Finally, for random feature matrices (RFMtxs), we prove super-exponential singular-value decay with Fourier features and exponential decay with

\tanh

features, together with corresponding condition-number lower bounds. The analysis identifies a common mechanism: the same spectral approximation that yields high accuracy also drives severe ill-conditioning.

Pingbing Ming, Hao Yu

Atom-Averaged Features Spectral Representation Method Regularity Singular Value Decomposition Convergence

Aug 13, 2026cs.LG

Sparse Orthogonal Regression Technique: A Spectral Framework for Equation Discovery, Approximation, and Integration

We develop the Sparse Orthogonal Regression Technique (SORT), a sparse spectral framework for learning orthonormal-basis expansions from noisy and irregularly sampled data. SORT estimates expansion coefficients directly from observations using L1-regularized regression, avoiding explicit quadrature or analytic inner-product evaluation. The central application is data-driven discovery of ordinary differential equations: vector fields are represented in chosen orthogonal bases and learned as sparse coefficient expansions. This provides a complementary route to symbolic regression, grammar-based discovery, and SINDy-style sparse identification by first recovering a compact spectral representation, which can later guide searches for simpler analytic forms. Across the dynamical-system experiments, SORT matches or improves upon library-based sparse-regression baselines when the basis is well adapted to the problem, and shows more stable degradation under sparse sampling, noisy derivative estimates, and representation mismatch. Specific examples illustrate why this representation is useful: if a finite library misses the problem-specific nonlinearity, the resulting model can fail. SORT is not immune to mismatch, but it shifts the problem away from brittle selection among generic terms to basis design adapted to the problem domain. The experiments also show that dominant low-order coefficients persist as model order increases, supporting order-consistent model growth. Beyond equation discovery, the same learned expansion supports nonlinear approximation and estimation of complex, high-dimensional integrals by coefficient readout. Overall, SORT provides a reusable intermediate representation for system identification, approximation, and integration, while making basis design an explicit part of the scientific modeling problem.

Sabin Roman, Ljupco Todorovski, Saso Dzeroski

Department of Knowledge Technologies, Jožef Stefan Institute · Faculty of Mathematics and Physics, University of Ljubljana

System Identification Symbolic Regression Spectral Representation Method Ordinary Differential Equations Least Squares Orthogonality+1

Aug 9, 2026stat.ML

Multi-kernel spectral clustering: Entrywise eigenvector perturbation bounds and exact recovery

Kernel spectral clustering with a single bandwidth can be inadequate for data exhibiting multiple characteristic pairwise-distance scales, a problem particularly prevalent in the high-dimensional regime. We address this issue through a multi-kernel formulation that aggregates kernels with different bandwidths. The bandwidths are selected as prescribed empirical quantiles of the pairwise squared distances, thereby capturing the relevant distance scales without requiring prior population-scale information. We develop a rigorous theoretical analysis of the resulting method under a general high-dimensional, multi-scale mixture model with heterogeneous cluster centers and covariance geometries. We construct a blockwise constant, low-rank informative approximation to the empirical multi-kernel matrix and establish row-wise

\ell_{2,\infty}

perturbation bounds for its leading spectral components, as well as for the associated normalized Laplacian matrix. These bounds yield observation-level control of the spectral embedding, which is more informative than conventional global eigenspace perturbation estimates. Under suitable eigen-gap and cluster-separation conditions, we show that approximate

K

-means applied to the multi-kernel spectral embedding achieves exact recovery with high probability.

Zeqin Lin, Guangming Pan, Zhixiang Zhang +1

1Nanyang Technological University · University of Macau · 3Dalian University of Technology

Kernel Method Spectral Representation Method Mixture Models Distance

Aug 7, 2026cs.AI

From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning

Neural operators have become a central tool for solving partial differential equations (PDEs), with spectral operators offering efficient global mixing across spatial locations. However, many PDEs contain physics-sensitive local structures that are critical to the underlying physical behavior. For example, in Darcy flow, local material interfaces are often reflected by sharp changes in the permeability field and can strongly influence the solution. Existing spectral operators primarily adapt modal mixing based on center-point representations, making them insufficiently responsive to such localized structural variations. We propose the Edge-Conditioned Spectral Operator (ESO), a novel spectral operator framework that modulates global spectral mixing using local edge-wise variations. By incorporating the Pairwise-Variation Modal Mixer (PVMM) to inject local edge information into spectral mode selection, ESO preserves the global approximation capability of spectral neural operators while enabling the learned kernel to adapt to physics-sensitive local structures. Furthermore, we introduce a task-adaptive Physics-Aware Reweighting (PAR) that emphasizes physically important regions, identified by taskspecific physical quantities. Across nine PDE benchmarks, ESO consistently achieves state-of-the-art performance. Visual and region-wise analyses further demonstrate that ESO reduces solution errors near coefficient jumps, high-gradient flow structures, and other physically sensitive regions. The code is available at https://github.com/Tanpig-X/ESO.

Zhentao Tan, Ruijie Quan, Yi Yang

Collaborative Innovation Center of Artificial Intelligence (CCAI), Zhejiang University, China

Partial Differential Equations Spectral Representation Method Edge

Aug 7, 2026math.NA

Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples

We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations

\mathfrak L_βu=f

using linearized ReLU

^k

neural networks on the sphere, where

\mathfrak L_β

is a positive elliptic spectral multiplier of order

β

. Given a parameter set

Θ_n=\{θ_{j}^*\}_{j=1}^n\subset\mathbb S^d

, we approximate

u

in the linearized network space

L_n^k(Θ_n)

by the discrete residual on the collocation points

\{η_i^*\}_{i=1}^m

\begin{equation*} u_{n,m}\in\arg\min_{v_n\in L_n^k(Θ_n)}\frac1m\sum_{i=1}^m\left(f(η_i^)-\mathfrak L_βv_n(η_i^)\right)^2. \end{equation*} With

k>\frac{d-1}{2}+β

, for antipodally quasi-uniform network parameter sets and any quasi-uniform collocation points with

m\gtrsim n

, we prove that \begin{equation*} |u-u_{n,m}|{\mathcal H^β(\mathbb S^d)}\eqsim|f-\mathfrak L_βu{n,m}|{\mathcal L^2(\mathbb S^d)}\lesssim n^{-\frac{r}{d}} \begin{cases} |f|{\mathcal W^{r,p}(\mathbb S^d)},&\frac{d}{p}<r\leq \frac{d}{2},~p>2,\ |f|{\mathcal H^r(\mathbb S^d)},&r>\frac{d}{2}. \end{cases} \end{equation*} We also establish a high-probability residual estimate, up to a logarithmic factor and an arbitrarily small smoothness loss, for i.i.d.\ uniformly distributed collocation points. The key analytical ingredient is a Bernstein inequality for linearized ReLU $^k$ network spaces. If $\underline h$ denotes the antipodal separation distance of the network parameters, then \begin{equation*} |v_n|{\mathcal H^r(\mathbb S^d)}\lesssim\underline h^{-(r-s)}|v_n|_{\mathcal H^s(\mathbb S^d)},\qquad 0\leq s<r<k+\tfrac12. \end{equation*}

Xinliang Liu, Tong Mao, Jinchao Xu

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China · Shenzhen Loop Area Institute, Shenzhen 518000, P. R. China

Neural Approximations Spectral Representation Method Bayesian Quadrature Least Squares Rectified Linear Unit Networks Empirical Study

Aug 6, 2026cs.SD

Explicit and Stable Pseudospectral Time-Domain Method for the Föppl-von Kármán Equations

Modal synthesis is a widely-used technique for simulation of musical instrument dynamics. In the linear case, a modal decomposition leads to an uncoupled system of damped and forced harmonic oscillators which can be efficiently solved by standard time-stepping methods. However, extensions to nonlinear problems are challenging due to the presence of products of modal expansions in the governing equations. In the case of the Föppl-von Kármán plate, the nonlinear coupling between the modes is described by a fourth-order tensor and is prohibitively expensive to evaluate in the modal domain. In this work, we propose a pseudospectral method in which the products are evaluated on a grid in the spatial domain while spatial derivatives are computed exactly in the modal domain. Discrete sine and cosine transforms between the modal and spatial domains are used to impose simply supported boundary conditions for the plate. Finally, we prove non-negativity of the nonlinear potential energy of the system and employ a scalar auxiliary variable technique for explicit and stable time integration in the modal domain. As a result, we reduce the computational cost of modal synthesis while preserving its advantages like a precise control over the simulated frequency range. Sound examples are presented.

Victor Zheleznov, Stefan Bilbao

Acoustics and Audio Group, University of Edinburgh, Edinburgh, UK · STMS (UMR9912), IRCAM, CNRS, Sorbonne Universit´e, Paris, France

Finite Element Method Spectral Representation Method Fokker--Planck Equation Implicit

Aug 5, 2026cs.LG

Spectral Distillation: From Nonlinear Dynamics to Linear State-Space Models

Can nonlinear dynamical systems be learned through a compact linear state-space representation, without directly solving a non-convex system-identification problem? We give a provable pipeline for doing so. Starting from observations of an unknown nonlinear dynamical system, we first learn an implicit spectral predictor using Observation Spectral Filtering (OSF), a convex method that competes with the best linear observer for the system. We then apply spectral-to-LDS distillation to convert this predictor into an explicit recurrent linear dynamical system. Our main theorem shows that the average prediction error of the distilled LDS decomposes into an exponentially-small distillation term and the OSF learning term governed by the Luenberger complexity of the best observer. The guarantee is dimension-free: it depends on observer complexity rather than on the latent dimension needed to represent the nonlinear system. To our knowledge, this yields the first end-to-end provable method for extracting a best-in-hindsight LDS representation of nonlinear dynamics through convex learning followed by provable distillation. Experiments on linear LDS benchmarks and MuJoCo behavior cloning show that the train-then-distill pipeline produces compact LDS predictors that match or outperform directly trained baselines.

Liane Galanti, Devan Shah, Shlomo Fortgang +1

Princeton University

System Identification Nonlinear Dynamics State Space Models Spectral Representation Method Least Squares Dataset Distillation

Aug 4, 2026cs.LG

Physics-informed reduced-order modelling with equivariant spectral submanifolds

Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.

Georg Maierhofer

Department of Applied Mathematics and Theoretical Physics · University of Cambridge, UK

Reduced-Order Models Spectral Representation Method

Jul 30, 2026math.AP

Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation

We study fractional parabolic initial-value problems with lower-order drift and potential terms in anisotropic spectral Barron spaces, defined by weighted space--time Fourier

L^1

norms adapted to parabolic scaling. We prove existence, uniqueness, and maximal regularity with a gain of one derivative in time and

γ

derivatives in space, where

γ>0

is the order of the fractional Laplacian. The evolution is defined only for

t\geq0

, whereas the finite-time norm requires a global extension with sufficient temporal Fourier decay. We construct a finite reflected semigroup extension using a Vandermonde system to match derivatives at

t=0

, obtaining temporal Fourier estimates uniform in the semigroup parameter. Combined with Fourier multiplier estimates for the damped principal operator, it yields maximal regularity. Dimension-independent multiplication estimates support a finite regularity bootstrap, while interpolation and sufficient damping absorb the lower-order terms in the base estimate. The a priori estimate and the method of continuity yield maximal regularity without smallness assumptions on the lower-order coefficients. A frequency-localized counterexample shows that a uniform-in-time spatial Barron bound on the forcing does not imply the corresponding two-derivative solution bound, even for the one-dimensional heat equation. Using this regularity, Fourier sampling yields

n^{-1/2}

approximation rates for the solution in mixed space--time Sobolev norms using shallow networks with suitable activations. Sampling in a product Hilbert space yields a population-level PINN consistency estimate for shallow cosine networks on a bounded cylinder. There exists a single width-

n

network for which the sum of the squared mixed-Sobolev solution error, the squared

L^2

-norm of the residual for the whole-space fractional equation, and the squared initial-data error is

O(n^{-1})

.

Jae-Hwan Choi, Hyojae Lim, Jinsol Seo +2

Neural Approximations Partial Differential Equations Spectral Representation Method Regularity Fourier

Jul 23, 2026cs.LG

Regularized Optimization on Grassmann Manifold: Theory, Algorithm and Applications

Spectral methods are among the most widely used techniques for community detection, clustering, and graph learning. Their performance, however, critically depends on the accurate estimation of the underlying spectral subspace and can deteriorate substantially in the presence of noise, outliers, or model perturbations. To address this limitation, we propose a Regularized Projection Matrix Approximation (RPMA) framework for robust estimation of rank-

K

projection matrices. RPMA extends classical spectral projection by incorporating a regularization term, producing projection estimates that are more robust, sparse, and interpretable. We formulate the proposed model as an optimization problem on the manifold of rank-

K

projection matrices and exploit its geometric equivalence to the Grassmann manifold. Based on this manifold characterization, we derive the first- and second-order optimality conditions, establish the local stability of the regularized leading eigenspace, and characterize the stability of the critical-point landscape under sufficiently small regularization. To efficiently solve the resulting nonconvex optimization problem, we develop a Riemannian gradient projection algorithm with backtracking line search, together with a more efficient Cayley--Sherman--Morrison--Woodbury (Cayley--SMW) gradient method that avoids repeated eigendecompositions. Extensive experiments on both synthetic and real-world datasets demonstrate that RPMA substantially improves the recovery accuracy of projection matrices and consistently outperforms conventional spectral projection methods for community detection and clustering under noisy environments.

Zhuan Liang, Zheng Zhai

Department of Statistics, Faculty of Arts and Sciences at Beijing Normal University, Zhuhai

Spectral Representation Method Community Detection Low-Rank Structure Data Manifold Regularization Manifolds

Jul 20, 2026cs.LG

fSRD: Fuzzy Spectral Region Decomposition -- Automated Multi Operator Koopman Representations via an Adaptive Spectral Learning Architecture

Highly nonlinear chaotic dynamical systems remain difficult to model due to fundamental trade-offs between complexity, expressivity, and data efficiency. Modern machine learning methods achieve strong predictive performance but often rely on a-priori system knowledge or curated data with limited interpretability. Koopman operator theory offers a promising direction via linear representation in an infinite-dimensional observable space. However, many data-driven Koopman methods seek globally valid operators for which useful finite-dimensional spectral embeddings remain difficult to identify under these constraints. To overcome associated limitations, we introduce Fuzzy Spectral Region Decomposition (fSRD), a fully automated learning framework for estimating finite Koopman representation via multiple operators. The proposed method realizes a data-adaptive framework for assembling locally invariant embeddings, termed Invariant Decomposition. fSRD achieves highly accurate linear reconstructions of nonlinear systems while learning finite-dimensional representations of their induced evolution operators, bridging interpretable operator-theoretic models with expressive data-driven sequence learning. These embeddings are adaptively constructed via a global fuzzy tree model, drawing inspiration from fuzzy neural architectures to learn the induced dynamics while prioritizing parsimonious solutions. Empirical results across canonical chaotic systems (e.g., Lorenz and Duffing) and high-dimensional real-world data demonstrate strong predictive accuracy, interpretability, and robust expressivity across data-rich and data-limited regimes, highlighting the method's generality.

Charles Bokor, Mark Cary, Denise Morrey +1

School of Engineering, Computing and Mathematics Oxford Brookes University Oxford OX33 1HX, UK · Department of Aeronautical and Automotive Engineering Loughborough University Loughborough LE11 3TU, UK

Koopman Operator Spectral Representation Method Chaos Representation Space Interpretability

Jul 15, 2026math.NA

Spectral-Informed Neural Networks Outperform Spectral Methods in High-dimensional PDEs

For low-dimensional problems (

d\leq3

), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems (

4 \leq d \lesssim 10

), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-dimensional problems (

d\gg 10

), spectral methods suffer frome the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising approach to overcome this challenge, offering scalability to high dimensions, but often suffer from limited accuracy and efficiency. Recently proposed spectral-informed neural networks (SINNs) combine spectral methods with PINNs, operating directly in the spectral domain to avoid spatial derivative computations and to reduce memory consumption. In this work, we introduce Modified SINNs, which integrate coefficient decay scaling and basis embeddings motivated by harmonic analysis to enhance accuracy in high-dimensional problems and enable accurate approximation of unknown spectral coefficients. Numerical experiments on steady and time-dependent partial differential equations demonstrate that Modified SINNs outperform sparse grid spectral methods on middle-dimensional problems with incomplete spectral information and achieve superior accuracy compared to PINNs on high-dimensional problems.

Tianchi Yu, Ivan Oseledets

1AXXX · 2Applied AI Institute · 3INM

Spectral Representation Method Partial Differential Equations High-Dimensional Neural Network

Jul 15, 2026cs.LG

Factorized Spectral Representations for Reinforcement Learning

Learning a compact model of the world from interaction data is central to sample-efficient deep reinforcement learning. Spectral representation methods have become the leading paradigm for representation learning in continuous control by taking a matrix view of the transition kernel, with state-action pairs on one side and next states on the other, and learning a low-rank factorization through self-supervised contrastive objectives. We take this view one step further. The transition kernel is naturally a three-mode tensor over states, actions, and next states, and a CP decomposition gives one feature map per mode. We propose FaStR, which fits this decomposition with a noise contrastive objective, producing separate state, action, and next-state encoders that together form a single spectral representation. The factored form yields a smaller hypothesis class, and the sample size needed for representation learning shrinks by a factor that scales with the smaller of the state and action dimensions. Empirically, FaStR delivers its largest gains on high-dimensional locomotion tasks whose dynamics align with the factored structure, and the learned state encoder transfers intact across actuator shift while only the action encoder is retrained.

Junyi Wu, Dan Li

Department of Industrial and Systems Engineering University of Washington Seattle, WA, USA

Spectral Representation Method Transitions

Jul 8, 2026math.NA

Near-Optimal Learning of Gaussian Sobolev Operators

A key question in operator learning is how to design surrogate operators with provable approximation guarantees in reasonable computational time. Whereas smooth operators can be approximated efficiently, i.e., with at least algebraic convergence in the amount of training data, learning finitely regular operators is known to be less efficient. The reason is an intrinsic curse of sample complexity, which allows only subalgebraic sample complexity rates. This fact makes it all the more important to develop algorithms which provably achieve these rates. In this work, we present a fully data-driven algorithm, termed Hermite-PCA approximation, for learning Gaussian Sobolev operators with near-optimal sample complexity. It employs principal component analysis and weighted least-squares methods and is therefore computationally efficient. Moreover, it is spectral, in the sense that it achieves faster (and near-optimal) convergence the higher the Sobolev regularity. We provide a full error analysis of this algorithm, taking into account all sources of error, along with numerical experiments that verify our theoretical results and empirically confirm the efficacy of Hermite-PCA approximation for learning Sobolev operators.

Ben Adcock, Michael Griebel, Gregor Maier

Department of Mathematics, Simon Fraser University, Burnaby BC, Canada · Institute for Numerical Simulation, University of Bonn, Bonn, Germany · Fraunhofer Institute for Algorithms and Scientific Computing (SCAI), Sankt Augustin, Germany

Probabilistic Operator Learning Logarithmic Sobolev Inequalities Spectral Representation Method Proximal Operators Optimal Sample Complexity Convergence

Jul 4, 2026cs.LG

PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling

Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric

A(x)=Λ(x)U(x)

. The diagonal factor

Λ(x)

controls anisotropic scaling, while the orthogonal factor

U(x)

is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.

Varvara Nazarenko, Timur Lidzhiev, Alexander Tarakanov

Faculty of Computer Science, HSE University, Moscow, Russia · AI VK, Moscow, Russia

Spectral Representation Method Physics-Informed Learning Spectral Decomposition Inductive Bias

Jul 1, 2026stat.ML

From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-

T

solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan Waniorek

University of Chicago

Fourier Neural Operators Nonlinear Operators Spectral Representation Method Neural Operators Neural Approximations Fourier+2

Jun 27, 2026math.NA

Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation

Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; the problem is that many of them may have nothing to do with the infinite-dimensional operator. In this paper we make spectral reliability the objective of dictionary learning. We train neural-network dictionaries not merely to predict the next snapshot, but to minimize Residual Dynamic Mode Decomposition residuals: operator-level a posteriori errors that test whether computed eigenvalues and modes are genuine Koopman spectral objects. To keep the learned observables from collapsing into an unstable coordinate system, the loss also penalizes the condition number of the lifted data matrix. Thus the method couples two requirements that should not be separated: small Koopman residuals and a well-conditioned representation. The result is a learned dictionary that is expressive, numerically stable, and spectrally disciplined. Across conservative and dissipative benchmark systems, the method sharply reduces spectral pollution, improves residual pseudospectral inclusion, and lowers forecast error relative to standard fixed dictionaries. On sea-surface temperature data, it gives cleaner Koopman diagnostics and substantially better one-step forecasts from noisy observations with no governing equations. The message is simple: neural Koopman learning should be judged not by prediction alone, but by whether its spectral claims can be certified. Residuals provide the certificate; conditioning makes it computable.

George Coote, Matthew J. Colbrook

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, United Kingdom

Koopman Operator Unsupervised Dictionary Learning Dynamic Mode Decomposition Spectral Representation Method Residual Learning Sea-Surface Temperature Data

Jun 22, 2026cs.LG

Solve for the Hyperparameter, Skip the Search: Kolmogorov-Optimal Scaling Laws for Spline Regression

Hyperparameter tuning almost always means search: fit the model at every value on a grid, score each by cross-validation, and keep the winner. For spline regression that search is unnecessary. The optimal resolution can be solved for in closed form, to the accuracy an exhaustive search reaches, at a fraction of the compute. Three ingredients make this possible: classical approximation theory pins the squared bias to a known power of the resolution G, exactly the Kolmogorov n-width of the smoothness class; the basis dimension is an explicit polynomial in G; and leave-one-out error follows from a single fit via the PRESS identity. Balancing the two known curves gives the minimizer analytically. We extend this calculus to many coordinates by replacing ambient input dimension with interaction order, the number of active low-order components in an ANOVA decomposition, yielding a scaling law in which the optimal resolution and error are power functions of the effective density (sample size per active component), with input dimension absent from the exponent. The law becomes an algorithm. KORE (Kolmogorov-optimal Order-aware Resolution Estimation) fits two pilot resolutions, solves a leverage-calibrated 2x2 system for the bias and noise scales, and evaluates the closed-form plug-in resolution with a tiny leave-one-out certificate: about a dozen fits instead of a full grid sweep, with a consistency guarantee as the sample grows. Across additive and sparse pairwise targets up to 80 input dimensions, KORE matches exhaustive 3-fold cross-validation and the full classical ladder (GCV, Mallows' Cp, AIC, BIC) while fitting roughly 8x fewer models; on 36 real tabular datasets it ranks first among 21 methods in accuracy per unit of compute, ahead of tuned boosters and kernel machines. When complexity lives in low interaction order, solving for the resolution beats searching for it.

Yong Yi Bay, Kathleen A. Yearick

PhD, University of Illinois at Urbana-Champaign

Kernel Ridge Regression General Grid Search Framework Splines Hyperparameter Spectral Representation Method Exponential Family+3

Jun 3, 2026cs.LG

Measuring Model Robustness via Fisher Information: Spectral Bounds, Theoretical Guarantees, and Practical Algorithms

The robustness of deep neural networks is crucial for safety-critical deployments, yet existing evaluation methods are often attack-dependent and lack interpretability. We propose a principled, attack-agnostic robustness metric based on the spectral norm of the Fisher Information Matrix (FIM), which quantifies the worst-case sensitivity of the model's output distribution to input perturbations. Theoretically, we establish that the FIM equals the variance of the input Jacobian and derive closed-form spectral bounds for common architectures, including VGG, ResNet, DenseNet, and Transformer, providing the first theoretical robustness ranking. To enable scalable evaluation, we develop efficient algorithms, including power iteration and Hutchinson-based estimation, that support both white-box and black-box settings. Extensive experiments across multiple datasets, including CIFAR, ImageNet, and medical images, and across multiple architectures show a strong correlation between our metric and adversarial vulnerability. Our framework serves as an interpretable diagnostic tool that complements attack-based evaluations, offering insights into architectural sensitivity and guiding the design of more robust models. Code is available at: https://github.com/franz-chang/SRP/.

Chong Zhang, Xiang Li, Jia Wang +2

Xi’an Jiaotong-Liverpool University · The Chinese University of Hong Kong

Robustness Verification Fisher Information Matrix Fisher Information Spectral Norm Spectral Representation Method Worst-Case

Jun 2, 2026cs.LG

Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations

Neural operator surrogates (NO) approximate PDE solutions orders of magnitude faster than numerical solvers, but suffer from spectral bias: high-frequency content is systematically attenuated, limiting reliability where fine-scale structure matters. Sparse sensor measurements of the field are often available too, offering pointwise accuracy without spectral distortion but covering only a small fraction of the domain. We address this by treating NO predictions as auxiliary observations in a diffusion posterior sampling framework. Our method, FreqNO-DPS (https://github.com/niccoloperrone/FreqNO-DPS), combines an unconditional score-based diffusion prior, trained on high-fidelity simulations, with diffusion posterior sampling (DPS) conditioned on sparse observations and guided by a frozen neural operator. Naive integration reintroduces the surrogate's spectral bias; we resolve this with a closed-form, spectrally shaped guidance score that weights the surrogate by its frequency-dependent accuracy and needs no denoiser backpropagation. A distribution-free analysis bounds the approximation error across the frequency-diffusion-time plane and shows the guidance's frequency dependence is preserved regardless of distributional assumptions. On 3D elastic wavefield prediction at 5% and 2% sensor coverage, the method reaches near-zero spectral bias across all bands, where both the surrogate and sensor-only DPS show systematic high-frequency attenuation. Isotropic guidance, the natural baseline, improves pointwise accuracy but carries the bias into the posterior nearly intact, confirming that frequency-dependent calibration is essential, not merely beneficial. The framework needs only paired surrogate/reference data and exploits no problem-specific structure beyond the residual's approximate spectral diagonality, verifiable for new surrogates via the coherence diagnostic we provide.

Niccolò Perrone, Fanny Lehmann, Stefania Fresca +1

Université Paris-Saclay, CentraleSupélec, CNRS, ENS Paris-Saclay · Laboratoire de Mécanique Paris-Saclay UMR 9026 · 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France +5

Neural Operators Neural Partial Differential Equation Solvers Spectral Representation Method Diffusion Sampling Surrogates

Jun 1, 2026cs.LG

Pseudospectral Bounds for Transient Amplification in Coupled Gradient Descent

Coupled gradient descent - where the update of one parameter depends on another - arises naturally in bilevel optimization, two-time-scale stochastic approximation, and generative adversarial networks. When the coupled Jacobian is block-triangular, asymptotic stability is determined by the spectral radii of the diagonal blocks, yet transient amplification before convergence can be arbitrarily large due to non-normality. We develop a sharp pseudospectral theory for block-triangular Jacobians J = [[A, 0], [C, D]], proving Kreiss-constant bounds of the form K(J) <= 2/(1-γ) + ||C||/(4(1-γ)) when ρ(A), ρ(D) <= γ< 1 and A, D are symmetric, and establishing matching minimax lower bounds. We characterize the critical coupling threshold for spectral instability and extend the theory to nearly self-referential systems via a Neumann-series perturbation framework. As a consequence, we obtain a finite-horizon O(K(J)^2 log(1/δ)) iteration complexity bound. Framed as scaling laws for stochastic two-time-scale optimization, our results expose a non-asymptotic, instance-dependent regime of high-dimensional learning dynamics that is invisible to spectral-radius analysis. Experiments on linear-quadratic problems, IQC-based comparisons, and neural-network training confirm the theory.

Ahanaf Hasan Ariq

Ideal School and College

Spectral Representation Method Gradient Descent Jacobian

May 29, 2026physics.comp-ph

Data-Driven Spectral Prediction for Accelerating Large-Scale Electronic Structure Calculations

Simulating large molecular systems comprising thousands of atoms requires highly scalable methodologies. While modern Density Functional Theory (DFT) codes exhibit linear scaling, solving the associated large, sparse generalized eigenproblems remains a critical computational bottleneck on exascale architectures. In the context of the LimitX project, we propose a data-driven framework to accelerate these calculations. By shifting the machine learning target from discrete eigenvalues to the coefficients of an interpolating Chebyshev polynomial, and by comparing both all-atom and fragment-based structural representations, we successfully overcome the dimensionality constraints of large-scale spectral prediction. We investigate three machine learning models (Kernel Ridge Regression, Graph Neural Networks, and Random Forests) trained on a novel 2 TB dataset of protein dimers. The predicted spectra provide initial guesses that effectively bypass early Self-Consistent Field (SCF) iterations in BigDFT. Ultimately, these spectral predictors will be deployed to dynamically optimize upcoming rational filter-based eigensolvers, such as FrASE, which is currently in initial development.

Abhiram Badrinarayanan, Davor Davidovic, Edoardo Di Napoli +4

Jülich Supercomputing Centre, Forschungszentrum Jülich, Germany · CEA / IRIG / MEM / L_Sim, France

Quantum Chemistry Density Functional Theory Spectral Representation Method High-Performance Computing Spectral Decomposition Scalability+1

May 24, 2026cs.DS

Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications

The

2 \rightarrow q

norm of a matrix

X \in \mathbb{R}^{n \times d}

is defined as

\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q

. We give polynomial-time multiplicative approximation algorithms for this norm when

q > 2

(i.e. in the hypercontractive setting). This problem either directly captures or is closely related to long-standing open problems in combinatorial optimization and hardness of approximation (e.g. Small Set Expansion), quantum information (e.g. Best Separable State), and algorithmic statistics. Very little is known about what approximation factors we can achieve for this problem in polynomial time, even though such approximations have significant downstream consequences. Barak, Brandão, Harrow, Kelner, Steurer, and Zhou showed that no polynomial-time algorithm can achieve an approximation factor better than

2^{\sqrt{\log n}}

, assuming the Exponential Time Hypothesis (FOCS'12). On the other hand, a simple spectral algorithm gives a

d^{1/4}

-approximation as a baseline. We give, to the best of our knowledge, the first polynomial-time approximation algorithm beating this baseline by polynomial factors. For the important special case of

q = 4

it achieves a

d^{1/8}

-approximation. All previous algorithms required additional assumptions on

X

, or only surpassed the baseline for small values of

n

. Moreover, we construct sum-of-squares certificates for the

2 \rightarrow q

norm. This directly implies improved algorithms for robust mean and covariance estimation, robust regression, and clustering, when the data only satisfies a bound on its

q

-th moment.

Samuel B. Hopkins, Stefan Tiegel

MIT

Spectral Norm Approximation Algorithms Spectral Representation Method Approximation Convex Relaxation

May 18, 2026stat.ML

Geometric Dictionary Learning of Dynamical Systems with Optimal Transport

Learning dynamical systems through operator-theoretic representations provides a powerful framework for analyzing complex dynamics, as spectral quantities such as eigenvalues and invariant structures encode characteristic time scales and long-term behavior. However, dynamical operators are typically estimated independently for each system, preventing the discovery of shared structure across related dynamics. To address this limitation, we posit that related dynamical systems lie near a low-dimensional manifold in spectral operator space. Based on this hypothesis, we introduce DOODL (Dynamical OperatOr Dictionary Learning), a framework that learns a dictionary of characteristic spectral dynamics whose combinations approximate this manifold and yield compact, interpretable embeddings of individual systems. Beyond representation learning, DOODL enables fast and interpretable operator estimation from short and partially observed trajectories by constraining the estimation to the learned operator manifold. Experiments on metastable Langevin dynamics and turbulent plasma simulations demonstrate that DOODL scales to highly complex multiscale regimes while capturing characteristic spectral structure governing the dynamics rather than merely fitting trajectories, achieving errors one to two orders of magnitude lower than independent operator estimation methods in challenging low-data regimes.

Thibaut Germain, Sami Chemlal, Rémi Flamary +2

CMAP, Ecole Polytechnique · Istituto Italiano di Tecnologia & University of Novi Sad

Dynamical Systems Learning Dynamics Spectral Representation Method Spectral Geometric Machine Learning Langevin Dynamics

May 18, 2026math.ST

Self-Distillation is Optimal Among Spectral Shrinkage Estimators in Spiked Covariance Models

Self-distillation has emerged as a promising technique for improving model performance in modern machine learning systems. We develop the statistical foundations of self-distillation in spiked covariance models, by introducing and analyzing a broad class of estimators, namely spectral shrinkage estimators. We establish that for spiked covariance matrices with

s

spikes,

s

-step self-distillation achieves optimal performance among spectral shrinkage estimators, outperforming well-known estimators in statistics and machine learning. Moreover, we show that

s

steps are necessary for optimality: any

(s-k)

-step distilled estimator is strictly suboptimal for

1 \leq k \leq s

. For the special subclass of isotropic covariances, we show that optimally tuned Ridge regression performs best among spectral shrinkage estimators. We also study a federated approach where multiple data centers share spectral shrinkage estimators and a common server seeks to aggregate them to achieve optimal performance. In this case, we find that the best local rule again takes the form of self-distillation, though it differs from the optimal rule when data are hosted centrally on a single server. Together, our results elucidate why self-distillation improves predictive performance and provide a broader statistical framework connecting it with classical shrinkage-based methods.

Radu Lecoiu, Debarghya Mukherjee, Pragya Sur

Department of Statistics, Harvard University · Department of Mathematics & Statistics, Boston University

Covariance Self-Distillation Framework Sparsity Spectral Representation Method Optimal

May 14, 2026stat.ML

Scaling Laws from Sequential Feature Recovery: A Solvable Hierarchical Model

We propose a simple mechanism by which scaling laws emerge from feature learning in multi-layer networks. We study a high-dimensional hierarchical target that is a globally high-degree function, but that can be represented by a combination of latent compositional features whose weights decrease as a power law. We show that a layer-wise spectral algorithm adapted to this compositional structure achieves improved scaling relative to shallow, non-adaptive methods, and recovers the latent directions sequentially: strong features become detectable at small sample sizes, while weaker features require more data. We prove sharp feature-wise recovery thresholds and show that aggregating these transitions yields an explicit power-law decay of the prediction error. Technically, the analysis relies on random matrix methods and a resolvent-based perturbation argument, which gives matching upper and lower bounds for individual eigenvector recovery beyond what standard gap-based perturbation bounds provide. Numerical experiments confirm the predicted sequential recovery, finite-size smoothing of the thresholds, and separation from non-hierarchical kernel baselines. Together, these results show how smooth scaling laws can emerge from a cascade of sharp feature-learning transitions.

Arie Wortsman-Zurich, Hugo Tabanelli, Yatin Dandi +2

Département d’Informatique, Ecole Normale Supérieure, PSL & CNRS · Information Learning and Physics Laboratory, École Polytechnique Fédérale de Lausanne (EPFL) · Statistical Physics of Computation Laboratory, École Polytechnique Fédérale de Lausanne (EPFL)

Feature Learning Scaling Laws Two-Layer Neural Networks Spectral Representation Method

May 11, 2026cs.LG

Don't Fix the Basis -- Learn It: Spectral Representation with Adaptive Basis Learning for PDEs

Spectral neural operators achieve strong performance for PDE learning, but rely on fixed global bases that limit their ability to represent spatially heterogeneous and multiscale dynamics. We propose Adaptive Basis Learning (ABLE), a framework that learns data-dependent spectral representations instead of relying on predefined bases. ABLE constructs a spatially adaptive Parseval frame via a learned ancillary density, enabling the operator to act in a lifted spectral space while preserving invertibility and maintaining

O(N\log N)

complexity through FFT-based implementation. This shifts the source of expressivity from spectral coefficients to the representation itself, allowing the model to capture localized structures and non-translation-invariant interactions more efficiently. ABLE integrates seamlessly into existing neural operator architectures as a drop-in replacement for spectral layers. Across a range of benchmarks ABLE improves accuracy over strong baselines, with the largest gains in regimes characterized by sharp gradients and multiscale behavior. Moreover, augmenting existing models (e.g., U-FNO, HPM) with ABLE further enhances their performance, demonstrating its role as a general and complementary spectral refinement. Our results highlight that the data-driven choice of representation, rather than operator complexity alone, is a key bottleneck in neural operator design. By learning the basis itself, ABLE provides a principled and efficient framework for improving spectral methods in PDE learning.

Xuxiang Zhao, Angelica I. Aviles-Rivero

Qiuzhen College, Tsinghua University · YMSC, Tsinghua University

Neural Operators Neural Partial Differential Equation Solvers Spectral Representation Method Partial Differential Equations Multiscale Dynamics Optimal Sample Complexity

May 9, 2026cs.LG

Bridging Spectral Operator Learning and U-Net Hierarchies: SpectraNet for Stable Autoregressive PDE Surrogates

Neural operators for time-dependent PDEs face a structural tension: spectral architectures (FNO and descendants) inherit exponential rollout-error growth from their one-step Lipschitz constant, while hierarchical U-Net operators trade resolution invariance for multi-scale detail. We introduce SpectraNet, an autoregressive neural operator that composes truncated spectral convolutions inside a U-Net hierarchy with a Residual-Target Spectral Block trained under a Semigroup-Consistency Loss. The residual-target parametrization replaces L^T stability blow-up with linear T*delta drift, and the spectral path's parameter count is Theta(L w^2 M^2), independent of grid N. Under a single unified protocol against 16 published neural-operator baselines on Navier-Stokes nu=1e-5 at 64x64, SpectraNet reaches test relative L2 = 0.0822 at 2.04M parameters -- 2.33x fewer than canonical FNO at ~20% lower error -- and wins five of six rows in a cross-PDE comparison against FNO (NS at nu in {1e-4, 1e-3}, PDEBench Shallow-Water 2D and Diffusion-Reaction, with the Active-Matter row going to FNO inside its seed spread). Trained from scratch at native 128^2 under the same protocol, SpectraNet improves to 0.0724 while FNO regresses to 0.3080. Free rollout stays bounded for T=100 where FNO diverges across all 200 test trajectories. On consumer CPU at B=1, SpectraNet runs sub-200ms while the full-attention Transformer that wins raw L2 pays ~60x latency; we do not claim to beat that Transformer on raw L2, only to dominate the lightweight (<=5M parameter, sub-200ms CPU) Pareto frontier. Source code: https://github.com/Enrikkk/spectranet

Enrique Hernández Noguera, Md Meftahul Ferdaus, Elias Ioup +2

University of New Orleans, New Orleans, LA, USA · Naval Research Laboratory, Stennis Space Center, MS, USA

Neural Operators Partial Differential Equations Spectral Representation Method U-Net

May 7, 2026cs.LG

Orth-Dion: Eliminating Geometric Mismatch in Distributed Low-Rank Spectral Optimization

Low-rank gradient compression reduces communication in distributed training by representing updates with rank-

r

factors. Dion is a recent method that approximates Muon, a spectral optimizer that orthogonalizes momentum, using one step of power iteration followed by column normalization (rescaling each column of the right factor to unit length). This makes it compatible with fully sharded data parallel training, but it converges more slowly than full-rank spectral methods. We show that this gap is geometric: column normalization does not yield the rank-

r

polar factor that Muon implicitly targets, so the resulting direction violates the dual-norm constraint of the low-rank spectral geometry, and the rate picks up an extra factor of

\sqrt{r}

even though the low-rank approximation of the gradient itself is accurate. The same mismatch enters the smoothness term and the error-feedback recursion in the analysis, which has a knock-on effect on empirical performance. We propose Orth-Dion, which replaces column normalization with QR orthogonalization of the right factor. Under non-Euclidean smoothness, with

L_r

the curvature constant along rank-

r

directions, Orth-Dion attains rate

O(\sqrt{L_r/T})

, matching exact spectral methods at the same per-step communication cost as Dion. The proof removes the bounded-drift assumption common in prior error-feedback analyses via a self-consistent fixed-point argument, and uses a time-averaged contraction that only requires the error sequence to contract on average rather than at every step. Experiments on large-scale language model pre-training validate the predicted

\sqrt{r}

scaling and show that Orth-Dion closes the convergence gap to Muon at Dion's communication cost.

Tatsuhiro Nakamori, Laura Gomezjurado Gonzalez, Ganesh Talluri +5

1Keio University · 2Stanford University · 3Midwestern University +3

Low-Rank Structure Spectral Representation Method Decentralized Optimization Riemannian Optimization Muon Mismatch

May 6, 2026math.NA

Neural-Guided Domain Restriction to Accelerate Pseudospectra Computation for Structured Non-normal Banded Matrices

Computing pseudospectra of non-normal matrices is essential for understanding the stability and transient behavior of dynamical systems. Such analysis is critical in applications including fluid dynamics, control systems, and differential operators, where non-normality can lead to significant transient amplification and sensitivity to perturbations that are not captured by eigenvalue analysis alone. At large scales, commonly used numerical approaches for pseudospectra computation can become computationally demanding, as they require repeated auxiliary computations to identify spectrally sensitive regions in the complex plane. We present a neural network-based approach that predicts sensitive regions directly from matrix features, thereby avoiding exhaustive pseudospectra evaluation across the entire complex plane. We calibrate the prediction threshold on validation data to ensure reliable coverage of sensitive regions. The trained neural network guides the selection of grid points requiring full computation, enabling focused computation only where necessary. The approach provides a practical preprocessing strategy for efficient pseudospectra computation. Numerical experiments on non-normal banded matrices demonstrate substantial speedup compared to full grid-based numerical evaluation while maintaining high accuracy in identifying sensitive regions.

Amit Punia, Rakesh Kumar, Madan Lal

1Jai Narain Vyas University, Jodhpur, Rajasthan, India · 2SRM University-AP, Amaravati, Andhra Pradesh, India · 3Jai Narain Vyas University, Jodhpur, Rajasthan, India

Spectral Representation Method Matrix Fluid Dynamics

Apr 25, 2026stat.ML

Learning Curves and Benign Overfitting of Spectral Algorithms in Large Dimensions

Existing large-dimensional theory for spectral algorithms resolves either the optimally tuned point or the interpolation limit, but leaves the under-regularized regime unexplored. We study the learning curve and benign overfitting of spectral algorithms in the large-dimensional setting where the sample size and dimension are of comparable order, i.e.,

n \asymp d^γ

for some

γ>0

. We first consider inner-product kernels on the sphere

\mathbb{S}^{d-1}

and establish a sharp asymptotic characterization of the excess risk across the full regularization path under various source conditions

s \geq 0

, where

s

measures the relative smoothness of the regression function. Our results reveal that the learning curve is not simply U-shaped but instead consists of three distinct regimes: over-regularized, under-regularized, and interpolation regimes. This characterization allows us to fully capture the benign overfitting phenomenon, demonstrating that benign overfitting arises consistently across both the under-regularized and interpolation regimes whenever

s

is positive but no larger than a critical threshold. We further show that, in the sufficiently regularized regime, the kernel learning curve is recovered by an associated sequence model. Finally, we extend the learning-curve analysis to large-dimensional KRR for a class of kernels on general domains in

\mathbb{R}^d

whose low-degree eigenspaces satisfy spectral-scaling and hyper-contractivity conditions.

Weihao Lu, Qian Lin, Yingcun Xia +1

Department of Statistics and Data Science, National University of Singapore. · Department of Statistics and Data Science, Tsinghua University.

Kernel Ridge Regression Spectral Representation Method Overfitting Regularization Optimal Sample Complexity Finite Time Analysis

Apr 22, 2026cs.LG

SMART: A Spectral Transfer Approach to Multi-Task Learning

Multi-task learning is effective for related applications, but its performance can deteriorate when the target sample size is small. Transfer learning can borrow strength from related studies; yet, many existing methods rely on restrictive bounded-difference assumptions between the source and target models. We propose SMART, a spectral transfer method for multi-task linear regression that instead assumes spectral similarity: the target left and right singular subspaces lie within the corresponding source subspaces and are sparsely aligned with the source singular bases. Such an assumption is natural when studies share latent structures and enables transfer beyond the bounded-difference settings. SMART estimates the target coefficient matrix through structured regularization that incorporates spectral information from a source study. Importantly, it requires only a fitted source model rather than the raw source data, making it useful when data sharing is limited. Although the optimization problem is nonconvex, we develop a practical ADMM-based algorithm. We establish general, non-asymptotic error bounds and a minimax lower bound in the noiseless-source regime. Under additional regularity conditions, these results yield near-minimax Frobenius error rates up to logarithmic factors. Simulations confirm improved estimation accuracy and robustness to negative transfer, and analysis of multi-modal single-cell data demonstrates better predictive performance. The Python implementation of SMART, along with the code to reproduce all experiments in this paper, is publicly available at https://github.com/boxinz17/smart.

Boxin Zhao, Mladen Kolar, Jinchi Lv

University of Chicago · University of Southern California

Transfer Learning Multi--Task Learning Semi-Supervised Learning Spectral Representation Method

Apr 22, 2026cs.LG

Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identification

We introduce Fourier Weak SINDy, a minimal noise-robust and interpretable derivative-free equation learning method that combines weak-form sparse equation learning with spectral density estimation for data-driven test function selection. By using orthogonal sinusoidal test functions inspired by their prevalence in Modulating Function-based system identification, the weak-form sparse regression problem reduces to a regression over Fourier coefficients. Dominant frequencies are then selected via multitaper estimation of the frequency spectrum of the data. This formulation unifies weak-form learning and spectral estimation within a compact and flexible framework. We illustrate the effectiveness of this approach in numerical experiments across multiple chaotic and hyperchaotic ODE benchmarks.

Zhiheng Chen, Urban Fasel, Anastasia Bizyaeva

Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York, USA, 14850 · Department of Aeronautics, Imperial College London, London, United Kingdom

System Identification Spectral Representation Method Fourier Least Squares

Apr 20, 2026cs.LG

Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms

Data-driven discovery of governing equations has advanced significantly in recent years; however, existing methods often struggle in multiscale systems where dynamically significant terms may have small coefficients. Therefore, we propose Balance-Guided SINDy (BG-SINDy) inspired by the principle of dominant balance, which reformulates

\ell_0

-constrained sparse regression as a term-level

\ell_{2,0}

-regularized problem and solves it using a progressive pruning strategy. Terms are ranked according to their relative contributions to the governing equation balance rather than their absolute coefficient magnitudes. Based on this criterion, BG-SINDy alternates between least-squares regression and elimination of negligible terms, thereby preserving dynamically significant terms even when their coefficients are small. Numerical experiments on the Korteweg--de Vries equation with a small dispersion coefficient, a modified Burgers equation with vanishing hyperviscosity, a modified Kuramoto--Sivashinsky equation with multiple small-coefficient terms, and a two-dimensional reaction--diffusion system demonstrate the validity of BG-SINDy in discovering small-coefficient terms. The proposed method thus provides an efficient approach for discovering governing equations that contain small-coefficient terms.

Zhenhua Dang, Lei Zhang, Long Wang +1

Coefficients System Identification Multiscale Dynamics Spectral Representation Method Balance

Apr 7, 2026math.OC

Optimal Centered Active Excitation in Linear System Identification

We propose an active learning algorithm for linear system identification with optimal centered noise excitation. Notably, our algorithm, based on ordinary least squares and semidefinite programming, attains the minimal sample complexity while allowing for efficient computation of an estimate of a system matrix. More specifically, we first establish lower bounds of the sample complexity for any active learning algorithm to attain the prescribed accuracy and confidence levels. Next, we derive a sample complexity upper bound of the proposed algorithm, which matches the lower bound for any algorithm up to universal factors. Our tight bounds are easy to interpret and explicitly show their dependence on the system parameters such as the state dimension.

Kaito Ito, Alexandre Proutiere

Department of Information Physics and Computing, The University of Tokyo, Tokyo 113-8654, Japan · Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm 114 28, Sweden

System Identification Least Squares Active Learning Sample Complexity Single Excitation Time History Spectral Representation Method+1

Feb 19, 2026cs.SI

Simplify to Amplify: Achieving Information-Theoretic Bounds with Fewer Steps in Spectral Community Detection

We propose a streamlined spectral algorithm for community detection in the two-community stochastic block model (SBM) under constant edge density assumptions. By reducing algorithmic complexity through the elimination of non-essential preprocessing steps, our method directly leverages the spectral properties of the adjacency matrix. We demonstrate that our algorithm exploits specific characteristics of the second eigenvector to achieve improved error bounds that approach information-theoretic limits, representing a significant improvement over existing methods. Theoretical analysis establishes that our error rates are tighter than previously reported bounds in the literature. Comprehensive experimental validation confirms our theoretical findings and demonstrates the practical effectiveness of the simplified approach. Our results suggest that algorithmic simplification, rather than increasing complexity, can lead to both computational efficiency and enhanced performance in spectral community detection.

Sie Hendrata Dharmawan, Peter Chin

Thayer School Of Engineering · Dartmouth College · Hanover, NH, USA

Community Detection Spectral Representation Method Simplification Information-Theoretic Limits

Feb 10, 2026cs.LG

Learning to Discover Iterative Spectral Algorithms

We introduce AutoSpec, a neural network framework for discovering iterative spectral algorithms for large-scale numerical linear algebra and numerical optimization. Our self-supervised models adapt to input operators using coarse spectral information (e.g., eigenvalue estimates and residual norms), and predict recurrence coefficients for computing or applying a matrix polynomial tailored to a downstream task. The effectiveness of AutoSpec relies on three ingredients: an architecture whose inference pass implements short, executable numerical linear algebra recurrences; efficient training on small synthetic problems with transfer to large-scale real-world operators; and task-defined objectives that enforce the desired approximation or preconditioning behavior across the range of spectral profiles represented in the training set. We apply AutoSpec to discovering algorithms for representative tasks on spd matrices: accelerating matrix function approximation; accelerating sparse linear solvers; and spectral filtering/preconditioning for eigenvalue computations. On real-world matrices, the learned procedures deliver up to order-of-magnitude improvements in accuracy and/or reductions in iteration count, relative to spectrum-agnostic baselines. We find clear connections to classical theory: the induced polynomials may exhibit equioscillation behavior characteristic of Chebyshev polynomial approximation. The code is available at: https://github.com/zihanghliu/AutoSpec .

Zihang Liu, Oleg Balabanov, Yaoqing Yang +1

1International Computer Science Institute · University of California, Berkeley · 4Lawrence Berkeley National Laboratory +1

Spectral Representation Method Self-Supervised Learning

Dec 9, 2025cs.LG

Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

Deep Operator Networks (DeepONets) have emerged as a powerful framework for data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk network, which operates directly on raw spatial or spatiotemporal coordinates through fully connected layers, often struggles to represent sharp gradients, boundary layers, and other non-periodic solution structures on bounded domains. To address these limitations, we introduce the Spectral-Embedded Deep Operator Network (SEDONet), a novel DeepONet architecture in which the trunk is driven by a fixed Chebyshev spectral dictionary instead of coordinate inputs. This non-periodic spectral embedding provides a principled inductive bias for bounded domains, enabling the learned operator to capture fine-scale features that are difficult for Fourier-based or MLP-only trunks to represent. SEDONet is evaluated on the 2-D Poisson equation, 1-D Burgers' equation, 1-D advection-diffusion equation, Allen-Cahn equation, Lorenz-96 chaotic system, and Darcy flow, covering elliptic, hyperbolic, parabolic, chaotic, and multiscale problems. Across all benchmarks, SEDONet consistently achieves the lowest or statistically comparable relative

L^2

errors among DeepONet, FEDONet, and SEDONet, with improvements of up to 54% over the baseline DeepONet and consistent gains over Fourier-embedded variants on bounded, non-periodic problems. Energy spectrum analyses further demonstrate that SEDONet more accurately preserves intermediate- and high-frequency solution structures. The proposed framework provides a simple, parameter-neutral modification to DeepONets, offering a robust and computationally efficient spectral approach for surrogate modeling of nonlinear operators in scientific computing.

Muhammad Abid, Omer San

Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

Fourier Neural Operators Spectral Representation Method Fourier Spectral Approximation

Sep 14, 2025stat.ML

A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning

We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) operator-valued kernels whose associated integral operators are compact and hence admit discrete spectral decompositions, and (ii) separable kernels of the form

K(x,x')=k(x,x')T

, where

k

is a scalar-valued kernel and

T

is a positive operator on the output space. This broad setting induces expressive vector-valued reproducing kernel Hilbert spaces (RKHSs) that generalize the classical

K=kI

paradigm, thereby enabling rich structural modeling with rigorous theoretical guarantees. To address target operators lying outside the RKHS, we introduce vector-valued interpolation spaces to precisely quantify misspecification error. Within this framework, we establish non-asymptotic convergence rates for prediction, estimation, and misspecification errors in the online and finite-horizon settings. Importantly, the framework also accommodates a range of operator learning settings, from Fredholm integral operators to encoder--decoder architectures. Numerical experiments on the two-dimensional Navier--Stokes equations illustrate the proposed approach.

Jia-Qi Yang, Lei Shi

School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China.

Probabilistic Operator Learning Nonlinear Operators Reproducing Kernel Hilbert Spaces Kernel Hilbert Spaces Kernel Method Stochastic Approximation+1

Feb 16, 2025quant-ph

Physics-Informed Support Vector Kernels via Green-Function Analogies and Jackson-Chebyshev Spectral Design

Kernel selection for regression of physical observables is often heuristic. We investigate a physics-informed strategy in which functional forms and spectral structures associated with Green's functions motivate kernel selection without requiring an exact identification between a machine-learning kernel and a physical propagator. The principal construction is a Jackson-damped Chebyshev kernel inspired by the kernel polynomial method (KPM); its explicit feature map yields a positive-semidefinite Gram matrix by construction and provides an inspectable spectral prior for structured observables. We evaluate standard and custom SVR models on copper-conductivity proxies, local Dirac-like band dispersion, quartic-oscillator energy levels, photonic-crystal transmission, and Fibonacci-chain transmission using repeated nested validation, learning curves, random-forest and multilayer-perceptron baselines, and low-rank Nyström tests where relevant. The framework is intended for finite-data regression of precomputed observables while boundary conditions remain part of the physical model that generates those observables.

Nan-Hong Kuo, Renata Wong

Department of Physics, National Taiwan University, Taipei, Taiwan · Department of Artificial Intelligence, Chang Gung University, Taoyuan, Taiwan · Department of Neurology, Chang Gung Memorial Hospital, Keelung, Taiwan +1

Kernel Method Physics-Informed Learning Spectral Representation Method Low-Rank Structure Analogy

Jan 18, 2025stat.ML

Fixed-Gaussian Spectral Algorithms: Minimax Optimal Rates for Misspecified Learning and Transfer

The principal objective of this work is twofold within nonparametric regression settings: (1) to establish the minimax optimal convergence rates for fixed-bandwidth Gaussian kernel spectral algorithms when the true regression function resides in a Sobolev space, and (2) to apply Gaussian spectral algorithms for achieving robust and adaptive transfer learning under concept shift. While minimax optimality of misspecified spectral algorithms has been established, existing guarantees are typically restricted to the non-saturation regime. We demonstrate that the infinite smoothness of fixed-bandwidth Gaussian kernels provides universal robustness to model misspecification by showing that this kernel choice enables any spectral algorithm to attain minimax optimal rates, provided the regularization parameter decays exponentially. This result effectively decouples optimality from the algorithm's inherent qualification. Building on this, we then advocate Gaussian spectral algorithms as powerful components in a learning framework for robust and adaptive transfer. Specifically, we derive the adaptive convergence rate of the excess risk for this framework and show that the rates are optimal up to logarithmic factors. Our results also reveal the impact of the magnitude of the concept shift and the sample size on the generalization error.

Haotian Lin, Matthew Reimherr

Department of Statistics, The Pennsylvania State University.

Minimax Rate Kernel Ridge Regression Spectral Representation Method Kernel Method Transfer Learning Regularization

Jul 8, 2024math.DS

Adversarial dynamical systems characterize when data-driven learning succeeds or fails

Many systems resist analytical modeling, making data-driven inference of dynamics important. Yet data-driven methods can fail to converge or generalize, leaving open a central question: When can system behavior be learned reliably from data, and when is such learning impossible? We answer this question using adversarial dynamical systems to identify the boundary between accessible and inaccessible regimes. In Koopman operator learning, a leading framework for representing nonlinear dynamics through linear spectral objects, we design optimal data-driven spectral algorithms with convergence and certification guarantees under conditions arising broadly in physical systems. This yields a convergence theory for Koopman-operator approximations and resolves a longstanding open problem in Koopman spectral analysis. Conversely, by constructing adversarial systems, we prove matching impossibility results: without these conditions, no single-sequence limiting procedure can guarantee learning, regardless of data quality. These results sharply characterize when data-driven spectral learning can succeed and when it must fail. We validate the framework on oscillators, chaotic fluid flows and Arctic sea ice concentration forecasting. In the latter, we uncover hidden modes of Arctic sea ice decline, deliver long-range forecasts with geographic error bounds, and outperform state-of-the-art dynamical and deep learning models at substantially lower computational cost, enabling real-time deployment on standard CPUs.

Matthew J. Colbrook, Igor Mezić, Alexei Stepanenko

DAMTP, University of Cambridge, Cambridge, CB3 0WA, UK. · University of California, Santa Barbara, CA 93106, USA.

Koopman Operator Spectral Representation Method Data-Driven Data Assimilation Convergence Sea-Surface Temperature Data