Spectral Methods

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Spectral methods study problems by expressing data or operators in terms of eigenvalues, eigenvectors, or frequency modes. Classical approaches include Fourier and graph-Laplacian transforms, PCA and low-rank approximation, kernel methods, and polynomial filters; they are used for tasks such as clustering networks, estimating dimensions, and solving or learning partial differential equations. Newer work also uses spectral structure to design neural operators and optimizers, diagnose model behavior, and speed up large matrix computations. A central challenge is balancing efficient, robust approximations with the assumptions and information lost when choosing a fixed basis, truncating modes, or relying on noisy spectra.

Latest papers

238 papers21 papers in the last four weeks, down 30% on the four weeks before. 0.1% of all new papers.

Mar 20, 2026

Model Selection and Parameter Estimation for Multidimensional Gaussian Mixture Models with a Common Covariance Matrix

We study model-order selection and component-mean estimation for multidimensional Gaussian mixture models with a known common covariance matrix. Using empirical characteristic-function measurements, we construct Fourier covariance matrices whose population counterparts have rank equal to the number of mixture components. We establish a minimax lower bound showing that distinguishing a separated

k

-component mixture from the class of

(k-1)

-component mixtures requires

Ω(Δ^{-(4k-4)})

samples. We then develop an oracle spectral-thresholding estimator with a sufficient sample size of order

Δ^{-(8k-8)}

for fixed

k

, together with a practical singular-value-ratio estimator. Given the model order, we estimate the component means by score-initialized gradient descent on a MUSIC-type projection objective. Under an explicit sample-size condition, a qualifying sample initialization lies in a certified attraction region with high probability, after which the iterates converge linearly. For fixed positive component separation, the resulting mean estimates achieve the parametric rate

\mathcal{O}_p(n^{-1/2})

. Numerical experiments demonstrate competitive accuracy and lower computational cost than expectation-maximization across a range of multidimensional settings.

Xinyu Liu, Hai Zhang

Department of Mathematics, Hong Kong University of Science and Technology (HKUST) · HKUST-Shenzhen-Hong Kong Collaborative Innovation Research Institute

Gaussian Mixture Models Spectral Methods

Mar 19, 2026

Subspace Projection Methods for Fast Spectral Embeddings of Evolving Graphs

Several graph data mining, signal processing, and machine learning downstream tasks rely on information related to the eigenvectors of the associated adjacency or Laplacian matrix. Classical eigendecomposition methods are powerful when the matrix remains static but cannot be applied to problems where the matrix entries are updated or the number of rows and columns increases frequently. Such scenarios occur routinely in graph analytics when the graph is changing dynamically and either edges and/or nodes are being added and removed. This paper puts forth a new algorithmic framework to update the eigenvectors associated with the leading eigenvalues of an initial adjacency or Laplacian matrix as the graph evolves dynamically. The proposed algorithm is based on Rayleigh-Ritz projections, in which the original eigenvalue problem is projected onto a restricted subspace which ideally encapsulates the invariant subspace associated with the sought eigenvectors. Following ideas from eigenvector perturbation analysis, we present a new methodology to build the projection subspace. The proposed framework features lower computational and memory complexity with respect to competitive alternatives while empirical results show strong qualitative performance, both in terms of eigenvector approximation and accuracy of downstream learning tasks of central node identification and node clustering.

Mohammad Eini, Abdullah Karaaslanli, Vassilis Kalantzis +1

IBM Research

Spectral Methods Dynamic Graph Learning

Mar 17, 2026

GIST: Gauge-Invariant Spectral Transformers for Scalable Graph Neural Operators

Neural operators on irregular meshes face a fundamental tension. Spectral positional encodings, the natural choice for capturing geometry, require cubic-complexity eigendecomposition and inadvertently break gauge invariance through numerical solver artifacts; existing efficient approximations sacrifice gauge symmetry by design. Both failure modes break discretization invariance: models fail to transfer across mesh resolutions of the same domain, and similarly across different graphs of related structure in inductive settings. We propose GIST (Gauge-Invariant Spectral Transformer), a scalable neural operator that resolves this tension by restricting attention to pairwise inner products of efficient approximate spectral embeddings. We prove these inner products estimate an exactly gauge-invariant graph kernel at end-to-end

\mathcal{O}(N)

complexity, and establish a formal connection between gauge invariance and discretization-invariant learning with bounded mismatch error. To our knowledge, GIST is the first scalable graph neural operator with a provable discretization-mismatch bound. Empirically, GIST sets state-of-the-art on the AirfRANS, ShapeNet-Car, DrivAerNet, and DrivAerNet++ mesh benchmarks (up to 750K nodes), and additionally matches strong baselines on standard graph benchmarks (e.g., 99.50% micro-F1 on PPI).

Mattia Rigotti, Nicholas Thumiger, Thomas Frick

IBM Research

Graph Neural Operators Neural Operators Spectral Methods Invariant Representation Learning

Mar 16, 2026

Spectral Hierarchy of the Cosmic Web

We introduce a spectral hierarchy of cosmic-web classifications obtained by applying simple scale-weighting kernels to the density field before performing a standard eigenvalue-based web classification. This unifies and extends several widely used web definitions within a single framework: the familiar potential/tidal web (large-scale, nonlocal), a curvature-based web (more local, peak- and ridge-sensitive), and additional higher-derivative levels that progressively emphasize smaller-scale structure. Because the classification is built from second derivatives of the filtered field, successive hierarchy levels align naturally with operator families that appear in renormalised bias and effective descriptions of large-scale structure, providing an explicit bridge between cosmic-web environments and long- and short-range nonlocal bias ingredients. We quantify the information content of the hierarchy with a compact statistic: we map each cell to one of four ordered web types (void, sheet, filament, knot), construct a corresponding ``web contrast'' field, and measure its cross-correlation with halos from the AbacusSummit simulation suite on a coarse mesh with

ΔL\simeq 5.5\,h^{-1}\mathrm{Mpc}

. We find that the hierarchy retains significant tracer-relevant information from very large scales down to the mesh Nyquist limit, with the more local (curvature/higher-derivative) levels dominating toward nonlinear scales. This makes the spectral hierarchy a practical, interpretable conditioning basis for fast mock-galaxy production and field-level modelling, and a flexible tool for studying environment-dependent clustering and assembly bias.

Francisco-Shu Kitaura, Francesco Sinigaglia

Instituto de Astrofísica de Canarias, s/n, E-38205, La Laguna, Tenerife, Spain · Departamento de Astrofísica, Universidad de La Laguna, E-38206, La Laguna, Tenerife, Spain · Institute for Fundamental Physics of the Universe, Via Beirut 2, I-34151 Trieste, Italy +3

Hierarchical Classification Spectral Methods Astrophysics

Mar 14, 2026

Spectral Edge Dynamics of Training Trajectories: Signal--Noise Geometry Across Scales

Despite hundreds of millions of parameters, transformer training trajectories evolve within only a few coherent directions. We introduce Spectral Edge Dynamics (SED) to quantify this structure: a rolling-window SVD of parameter updates reveals a sharp boundary -- the spectral edge -- between coherent optimization directions and stochastic noise, identified via the maximum consecutive singular value ratio

σ_k / σ_{k+1}

. Across a 51M-parameter TinyStories model (4 seeds) and GPT-2 124M under distribution shift, the spectral edge exhibits a universal three-phase pattern (rise, plateau, collapse). The effective signal rank adapts to task complexity (

k^* = 2

at 51M,

k^* = 3

at 124M), and the directional coupling between spectral geometry and validation loss reverses with window size -- a lag flip reflecting the timescale of trajectory integration. Johnson--Lindenstrauss projection to

d = 10W

dimensions (e.g.,

d = 100

for

W = 10

) preserves the spectral gap within

5.7\%

, making the framework applicable to models of arbitrary scale. In companion work, the same spectral geometry provides early-warning signals of grokking -- predicting generalization

600

--

1{,}700

steps before it occurs across modular arithmetic, Dyck languages, and the SCAN benchmark.

Yongzhong Xu

Transformer Training Dynamics Neural Network Training Dynamics Gradient Descent Dynamics Spectral Methods

Mar 13, 2026

Filtered Spectral Projection for Quantum Principal Component Analysis

Quantum principal component analysis (qPCA) is commonly formulated as the extraction of eigenvalues and eigenvectors of a covariance-encoded density operator. Yet in many qPCA settings the practical goal is simpler: projection onto the dominant spectral subspace. Here we introduce a projection-first framework, the Filtered Spectral Projection Algorithm (FSPA), which bypasses explicit eigenvalue estimation while preserving the relevant spectral structure. FSPA amplifies any nonzero warm-start overlap with the leading subspace and remains robust in small-gap and near-degenerate regimes, without artificial symmetry breaking in the absence of bias. We show that FSPA achieves an oracle complexity

\mathcal{O}((\log(1/ε)+\log(1/|a_1|^2))/\log(λ_1/λ_2))

,which is tight by a matching lower bound, establishing it as anoptimal projection primitive. We derive a convergence rate for degenerate spectra, give a circuit resource analysis with

n+\mathcal{O}(1)

qubit overhead independent of system dimension, and extend the method to threshold spectral projection, Threshold-FSPA, which converges in

\mathcal{O}(\log(1/ε))

calls when the threshold lies between eigenvalues. In the density matrix exponentiation access model, FSPA gives an exponential copy-complexity advantage over classical methods. For classical datasets, we show that for amplitude-encoded centered data the ensemble density matrix

ρ=\sum_i p_i|ψ_i\rangle\langleψ_i|

equals the covariance matrix. Numerical tests on chemistry density matrices, noisy circuit outputs, Breast Cancer Wisconsin, handwritten Digits, and 1--4-qubit scalability confirm the theory. A minimal Qiskit implementation validates magnitude invariance, signal amplification, and no spurious symmetry breaking. These results establish FSPA as an optimal and deployable quantum spectral projection primitive.

Sk Mujaffar Hossain, Satadeep Bhattacharjee

Indo-Korea Science and Technology Center (IKST), Bengaluru 560064, India

Principal Component Analysis Quantum Machine Learning Spectral Methods

Mar 12, 2026

Uncovering Locally Low-dimensional Structure in Networks by Locally Optimal Spectral Embedding

Standard Adjacency Spectral Embedding (ASE) relies on a global low-rank assumption often incompatible with the sparse, transitive structure of real-world networks, causing local geometric features to be 'smeared'. To address this, we introduce Local Adjacency Spectral Embedding (LASE), which uncovers locally low-dimensional structure via weighted spectral decomposition. Under a latent position model with a kernel feature map, we treat the image of latent positions as a locally low-dimensional set in infinite-dimensional feature space. We establish finite-sample bounds quantifying the trade-off between the statistical cost of localisation and the reduced truncation error achieved by targeting a locally low-dimensional region of the embedding. Furthermore, we prove that sufficient localisation induces rapid spectral decay and the emergence of a distinct spectral gap, theoretically justifying low-dimensional local embeddings. Experiments on synthetic and real networks show that LASE improves local reconstruction and visualisation over global and subgraph baselines, and we introduce UMAP-LASE for assembling overlapping local embeddings into high-fidelity global visualisations.

Hannah Sansford, Nick Whiteley, Patrick Rubin-Delanchy

School of Mathematics, University of Bristol, UK · School of Mathematics, University of Edinburgh, UK

Spectral Graph Embedding Graph Embedding Spectral Methods Manifold Learning

Mar 9, 2026

First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation

Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces. In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient, revealing how constraints induce a distorted ascent geometry. We further demonstrate that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression, and establish a compatibility principle that characterizes the existence of common admissible directions across multiple objectives. The resulting framework unifies gradient projection, spectral truncation, and multi-objective feasibility within a single geometric structure.

Changkai Li

Constrained Optimization Multi-Objective Optimization Spectral Methods

Mar 7, 2026

Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

Continuous symmetries are fundamental to many scientific and learning problems, yet they are often unknown a priori. Existing symmetry discovery approaches typically search directly in the space of transformation generators or rely on learned augmentation schemes. We propose a fundamentally different perspective based on spectral structure. We introduce a framework for discovering continuous one-parameter subgroups using the Generalized Fourier Transform (GFT). Our central observation is that invariance to a subgroup induces structured sparsity in the spectral decomposition of a function across irreducible representations. Instead of optimizing over generators, we detect symmetries by identifying this induced sparsity pattern in the spectral domain. We develop symmetry detection procedures on maximal tori, where the GFT reduces to multi-dimensional Fourier analysis through their irreducible representations. Across structured tasks, including the double pendulum and top quark tagging, we demonstrate that spectral sparsity reliably reveals one-parameter symmetries. These results position spectral analysis as a principled and interpretable alternative to generator-based symmetry discovery.

Pavan Karjol, Kumar Shubham, Prathosh AP

Department of Electrical Communication Engineering , Indian Institute of Science , Bengaluru, Karnataka

Symmetry Discovery Spectral Methods Lie Group Methods

Feb 26, 2026

Regular Fourier Features for Nonstationary Gaussian Processes

Simulating a Gaussian process requires sampling from a high-dimensional Gaussian distribution, which scales cubically with the number of sample locations. Spectral methods address this challenge by exploiting the Fourier representation and treating the spectral density as a probability distribution suitable for Monte Carlo approximation. Although this probabilistic interpretation is valid for stationary processes, it is overly restrictive for the nonstationary case, where spectral densities are generally not probability measures. To avoid this limitation, we propose regular Fourier features for harmonizable processes with one-dimensional inputs. Our method discretizes the spectral representation directly, preserving the correlation structure among spectral weights without requiring probability assumptions. Assuming finite spectral support, this yields an efficient low-rank approximation that is positive semi-definite by construction and consistent under mild regularity conditions. When the spectral density is unknown, the framework also extends to kernel learning from data, which we explore as a proof of concept. We demonstrate the approximation on locally stationary and harmonizable mixture kernels, the latter with a complex-valued spectral density. As a feasibility study, we then apply the kernel-learning extension to real and synthetic data, where it matches competitive baselines.

Arsalan Jawaid, Abdullah Karatas, Jörg Seewig

Institute of Measurement and Sensor Technology, University of Kaiserslautern-Landau, Kaiserslautern, Germany · Independent Researcher, Mannheim, Germany

Fourier Feature Embeddings Gaussian Processes Low-Rank Approximation Kernel Methods Spectral Methods

Feb 21, 2026

Limits of Convergence-Rate Control for Open-Weight Safety

Open-weight foundation models can be fine-tuned for harmful purposes after release, yet no existing training resistance methods provide theoretical guarantees. Treating these interventions as convergence-rate control problems allows us to connect optimization speed to the spectral structure of model weights. We leverage this insight to develop a novel understanding of convergence rate control through spectral reparameterization and derive an algorithm, SpecDef, that can both provably and empirically slow first- and second-order optimization in non-adversarial settings. In adversarial settings, we establish a fundamental limit on a broad class of convergence rate control methods including our own: an attacker with sufficient knowledge can restore fast convergence at a linear increase in model size. In order to overcome this limitation, future works will need to investigate methods that are not equivalent to controlling convergence rate.

Domenic Rosati, Xijie Zeng, Hong Huang +4

Dalhousie University · Vector Institute · Indian Institute of Management Bangalore

Neural Network Optimization Spectral Methods

Feb 19, 2026

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

Stochastic Block Model Graph Clustering Spectral Methods

Feb 10, 2026

Efficient reduction of stellar contamination and noise in planetary transmission spectra using neural networks

The characterization of exoplanetary atmospheres has been transformed by the James Webb Space Telescope (JWST), whose infrared sensitivity enables transmission spectroscopy at unprecedented precision. However, stellar heterogeneities (e.g., spots and faculae) remain a dominant source of contamination that can bias atmospheric retrievals if not properly corrected. We present a methodology for reducing stellar contamination and instrument-specific noise from exoplanet transmission spectra using neural networks, in particular the so-called Denoising AutoEncoders (DAEs). Our goals are to enable fast, accurate corrections that improve the reliability of atmospheric parameter retrievals and to promote the use of unsupervised algorithms for efficient data processing. We designed and trained DAE architectures using large synthetic datasets of terrestrial (TRAPPIST-1e analogues) and sub-Neptune (K2-18b analogues) planets. Atmospheric retrieval experiments were then performed on contaminated spectra in order to compare our deep-learning approach against standard correction methods in terms of accuracy and computational cost. Our autoencoders successfully reconstruct uncontaminated spectra, preserving essential molecular features even in low-S/N regimes. In retrieval tests, the denoising autoencoder pre-processing yields atmospheric parameter estimates broadly comparable to those obtained with simultaneous stellar-contamination fitting. Notably, our method maintains a much lower computational cost, approximately one order of magnitude smaller. These results demonstrate that DAEs outperform conventional correction methods in computational efficiency while maintaining high accuracy, paving the way for their integration into future atmospheric characterization pipelines for both rocky and sub-Neptune exoplanets.

David S. Duque-Castaño, Lauren Flor-Torres, Jorge I. Zuluaga

SEAP/FACom, Instituto de Física - FCEN, Universidad de Antioquia, Calle 70 No. 52-21, Medellín, Colombia

Denoising Autoencoders Autoencoders Spectral Methods Data Contamination

Feb 10, 2026

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

Automated Algorithm Discovery Learning to Optimize Spectral Methods

Jan 29, 2026

Correcting Spectra Outside the Backbone: A Model-Agnostic Rectifier for Hyperspectral Image Super-Resolution

Hyperspectral image super-resolution (HSI-SR) aims to recover spatial detail while preserving the spectral shape on which quantitative analysis relies. Recent HSI-SR methods, from repurposed RGB super-resolution backbones to dedicated spectral-spatial architectures, have greatly improved spatial reconstruction. However, overlooking the compact spectral structure of hyperspectral data leaves residual spectral errors, while binding the spectral treatment to each architecture forces it to be rebuilt for every new backbone. Yet the low-dimensional structure of spectra belongs to the data, not to any backbone. Backbones differ in the errors they leave, but not in the structure of the true spectra. One rectifier design can therefore serve any backbone. Building on this insight, we propose the Spectral Rectification Super-Resolution Network (\textbf{SR

^{2}

-Net}), a model-agnostic rectifier that needs nothing from the backbone but its output, leaves its internal architecture untouched, and is trained per backbone. SR

^{2}

-Net follows an enhance-then-rectify pipeline in which Hierarchical Spectral-Spatial Synergy Attention (\textbf{H-S

^{3}

A}) reinforces cross-band interactions, while Mode-Constrained Rectification (MCR) confines the correction to a learned compact spectral subspace. A degradation-consistency constraint further ties the output to the observed low-resolution input. Experiments with five backbones spanning CNN, Transformer, and diffusion families show that one fixed configuration improves spectral fidelity in every reported setting while preserving or improving spatial quality. Averaged over thirty in-domain settings, SR

^{2}

-Net removes 22.6% of the residual spectral error at a backbone-independent cost of 0.048M parameters.

Ji-Xuan He, Guohang Zhuang, Bo Junge +5

School of Computer Science and Technology, Xi’an Jiaotong University · School of Computer Science and Information Engineering, Hefei University of Technology

Hyperspectral Image Super-Resolution Hyperspectral Imaging Spectral Methods Hyperspectral Image Reconstruction

Jan 23, 2026

SFO: Learning PDE Operators via Spectral Filtering

Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels using the Universal Spectral Basis (USB), a fixed, global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. Motivated by our theoretical finding that the discrete Green's functions of shift-invariant PDE discretizations exhibit spatial Linear Dynamical System (LDS) structure, we prove that these kernels admit compact approximations in the USB. By learning only the spectral coefficients of rapidly decaying eigenvalues, SFO achieves a highly efficient representation. Across six benchmarks, including reaction-diffusion, fluid dynamics, and 3D electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters.

Noam Koren, Rafael Moschopoulos, Kira Radinsky +1

Department of Computer Science Technion - Israel Institute of Technology · Department of Computer Science Princeton University

PDE Surrogate Modeling PDE Operator Learning Spectral Methods Neural Operators

Dec 17, 2025

NMIRacle: Multi-modal Generative Molecular Elucidation from IR and NMR Spectra

Molecular structure elucidation from spectroscopic data is a long-standing challenge in Chemistry, traditionally requiring expert interpretation. We introduce NMIRacle, a two-stage generative framework that builds upon recent paradigms in AI-driven spectroscopy with minimal assumptions. In the first stage, NMIRacle learns to reconstruct molecular structures from count-aware fragment representations, capturing both fragment identities and their occurrences. In the second stage, a spectral encoder maps input spectra (IR, 1H-NMR, 13C-NMR) into a latent embedding used to condition the pre-trained generator, which is fine-tuned for direct spectra-to-molecule generation. This formulation bridges fragment-level chemical modeling with spectral evidence, yielding accurate molecular predictions. Empirical results demonstrate that NMIRacle outperforms existing baselines on molecular elucidation, while maintaining robust performance across increasing levels of molecular complexity.

Federico Ottomano, Yingzhen Li, Alex M. Ganose

Department of Chemistry, Imperial College London · Department of Computing, Imperial College London

Multimodal Generation Molecular Generation Spectral Methods Conditional Generative Modeling

Dec 9, 2025

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.

PDE Surrogate Modeling PDE Operator Learning Spectral Methods Deep Operator Networks

Oct 2, 2025

Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization

Estimating the tangent spaces of a data manifold is a fundamental problem in geometric data analysis. The standard approach, Local Principal Component Analysis (LPCA), struggles in high-noise settings due to a critical trade-off in choosing the neighborhood size. Selecting an optimal size requires prior knowledge of the geometric and noise characteristics of the data that are often unavailable. In this paper, we propose a spectral method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation. Instead of relying solely on local neighborhoods, LEGO estimates the tangent space at each data point by orthogonalizing the gradients of low-frequency eigenvectors of the graph Laplacian. We provide two theoretical justifications of our method. First, a differential geometric analysis on the tubular neighborhood of a manifold shows that gradients of the low-frequency Neumann eigenfunctions of the tube align closely with the manifold's tangent bundle, while an eigenfunction with high gradient in directions orthogonal to the manifold lies deeper in the spectrum. Second, a random matrix theoretic analysis also demonstrates that low-frequency eigenvectors are robust to sub-Gaussian noise. These results allow us to derive the asymptotic scaling and stability of the estimated eigenvector gradients. Numerical experiments demonstrate that LEGO yields tangent space estimates that are significantly more robust to noise than those from LPCA, resulting in marked improvements in downstream tasks such as manifold learning, boundary detection, and local intrinsic dimension estimation.

Dhruv Kohli, Sawyer J. Robertson, Gal Mishne +1

Manifold Learning Spectral Methods

Aug 16, 2025

Universal Learning of Nonlinear Dynamics

We study the fundamental problem of one-step prediction of a marginally stable unknown nonlinear dynamical system. We describe an algorithm for this problem, based on the technique of spectral filtering, which learns a mapping from past observations to the next based on a spectral representation of the system. Using techniques from online convex optimization, we prove vanishing prediction error for any nonexpansive nonlinear dynamical system with finitely many marginally stable modes, with rates governed by a novel quantitative control-theoretic notion of learnability. The main technical component of our method is a new spectral filtering algorithm for linear dynamical systems, which incorporates past observations and applies to general noisy and marginally stable systems. This generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.

Evan Dogariu, Anand Brahmbhatt, Elad Hazan

Courant Institute of Mathematical Sciences, New York University · Princeton University

Dynamical Systems Nonlinear System Identification Spectral Methods

Jun 18, 2025

On the Upper Bounds for the Matrix Spectral Norm

We consider the problem of estimating the spectral norm of a matrix using only matrix-vector products. We propose a new Counterbalance estimator that provides upper bounds on the norm and derive probabilistic guarantees on its underestimation. Compared to standard approaches such as the power method, the proposed estimator produces significantly tighter upper bounds in both synthetic and real-world settings. Our method is especially effective for matrices with fast-decaying spectra, such as those arising in deep learning and inverse problems.

Alexey Naumov, Maxim Rakhuba, Denis Ryapolov +1

HSE University, Russia · Steklov Mathematical Institute of Russian Academy of Sciences, Russia

Spectral Methods

Feb 16, 2025

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

Support Vector Machines Kernel Methods Physics-Informed ML Spectral Methods

Jan 18, 2025

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.

Spectral Regularization Concept Drift Transfer Learning Kernel Methods Spectral Methods Statistical Learning Theory

Jul 8, 2024

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 Learning Spectral Methods Statistical Learning Theory Chaotic Time-Series Prediction

Mar 25, 2024

Hearing the shape of an arena with spectral swarm robotics

Swarm robotics relies on robots with minimal sensing and computation that achieve self-organization at the level of the collective to address a variety of tasks. Yet swarm robotics still struggles with global tasks that require understanding the broader spatial context in which robots operate, such as identifying the shape of the arena in which they are embedded. Here, we introduce spectral swarm robotics, a framework in which robots recover global spectral descriptors of the swarm's geometric structure through strictly local interactions. To do so, robots emulate diffusion to locally estimate Laplacian spectral invariants, quantities normally obtained only through centralized computation. These spectral values serve as compact descriptors of the environment. In the present proof of concept, robots estimated the second Laplacian eigenvalue and used it to classify arenas from a predefined set of shapes with distinct spectral signatures. We show that estimating a single eigenvalue is sufficient for one-shot classification among selected confined environments with distinct spectral signatures. We uncovered a scaling law linking swarm size (a global parameter) and interaction radius (a local parameter) that ensures robust estimation. We validated the approach experimentally on two distinct hardware platforms, Kilobots and Pogobots, achieving reliable classification of arena shapes under realistic noise and sparse-connectivity conditions. Last, we discuss how locally recovering Laplacian spectral invariants opens avenues for swarm robotics in settings where producing efficient collective behaviors strongly depends on the global structure of the swarm.

Leo Cazenille, Nicolas Lobato-Dauzier, Alessia Loi +7

Université Paris Cité, CNRS, LIED UMR 8236, F-75006 Paris, France · LIMMS (IRL2820)/CNRS-IIS, University of Tokyo, Tokyo, Japan · Sorbonne Université, CNRS, Institut de Biologie Paris-Seine, Laboratoire Jean Perrin, F-75005, Paris, France +3

Swarm Robotics Spectral Methods Robotic Perception

Jan 16, 2024

Semidefinite programming relaxations and debiasing for MAXCUT-based clustering

In this paper, we consider the problem of partitioning a small data sample of size

n

drawn from a mixture of

2

sub-gaussian distributions in

\mathbb{R}^p

. We consider semidefinite programming relaxations of an integer quadratic program that is formulated essentially as finding the maximum cut on a graph, where edge weights in the cut represent dissimilarity scores between two nodes based on their

p

features. We define the signal-to-noise ratio (SNR) as

s^2 := \min\{n p γ^2, Δ^2\}

, where

Δ^2 := p γ

denotes the

\ell_2^2

distance between the two cluster centers. Our contributions are twofold. First, we provide a unified framework for analyzing three computationally efficient algorithms: SDP1, BalancedSDP, and Spectral clustering, yielding universal polynomial-rate misclassification guarantees for all three algorithms. Moreover, our theory allows for partial recovery (success rate

< 100\%

) as long as

s^2

is lower bounded by a constant. Second, we prove that the misclassification errors for SDP1 and BalancedSDP decay exponentially with respect to the SNR

s^2

and the BalancedSDP requires no explicit debiasing when the two clusters have equal sizes. To our knowledge, this is the first time such results are obtained for semidefinite relaxations of MAX CUT in population clustering. We provide simulation evidence illuminating the theoretical predictions.

Shuheng Zhou

University of California, Riverside, CA 92521

Semidefinite Programming Unsupervised Clustering Graph Clustering Spectral Methods Max-Cut

Nov 29, 2022

FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation

The physics-informed neural operator (PINO) is a machine learning paradigm that has demonstrated promising results for learning solutions to partial differential equations (PDEs). It leverages the Fourier Neural Operator to learn solution operators in function spaces and leverages physics losses during training to penalize deviations from known physics laws. Spectral differentiation provides an efficient way to compute derivatives for the physics losses, but it inherently assumes periodicity. When applied to non-periodic functions, this assumption can lead to significant errors, including Gibbs phenomena near domain boundaries which degrade the accuracy of both function representations and derivative computations. To overcome this limitation, we introduce the FC-PINO (Fourier-Continuation-based Physics-Informed Neural Operator) architecture which extends the accuracy and efficiency of PINO and spectral differentiation to non-periodic and non-smooth PDEs. In FC-PINO, we propose integrating Fourier continuation into the PINO framework, and test two different continuation approaches: FC-Legendre and FC-Gram. By transforming non-periodic signals into periodic functions on extended domains in a well-conditioned manner, Fourier continuation enables fast and accurate derivative computations. This approach avoids the discretization sensitivity of finite differences and the memory overhead of automatic differentiation. We demonstrate that standard PINO fails (without padding) or struggles (even with padding) to solve non-periodic and non-smooth PDEs with high precision, across challenging benchmarks. In contrast, the proposed FC-PINO provides accurate, robust, and scalable solutions, substantially outperforming PINO alternatives, and demonstrating that Fourier continuation is critical for extending PINO to a wider range of PDE problems when high-precision solutions are needed.

Adarsh Ganeshram, Valentin Duruisseaux, Haydn Maust +6

Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA

Physics-Informed Neural Operators PDE Solving Fourier Neural Operator Spectral Methods

Feb 8, 2019

Robust Streaming PCA

We consider streaming principal component analysis when the stochastic data generating model is subject to perturbations. While existing models assume a fixed covariance, we adopt a robust perspective where the covariance matrix belongs to a temporal uncertainty set. Under this setting, we provide fundamental limits on convergence of any algorithm recovering principal components. We analyze the convergence of the noisy power method and Oja's algorithm, both studied for the stationary data generating model, and argue that the noisy power method is rate-optimal in our setting. Finally, we demonstrate the validity of our analysis through numerical experiments on synthetic and real-world datasets.

Daniel Bienstock, Minchan Jeong, Apurv Shukla +1

Graduate School of AI KAIST

Principal Component Analysis Streaming Algorithms Spectral Methods Non-Stationary Time Series