Spectral Methods

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Overview

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

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

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

    Mar 20, 2026Xinyu Liu, Hai ZhangGaussian Mixture ModelsSpectral Methods

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

  2. Subspace Projection Methods for Fast Spectral Embeddings of Evolving Graphs

    Mar 19, 2026Mohammad Eini, Abdullah Karaaslanli, Vassilis Kalantzis +1Spectral MethodsDynamic Graph Learning

    IBM Research

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

    Mar 17, 2026Mattia Rigotti, Nicholas Thumiger, Thomas FrickGraph Neural OperatorsNeural Operators

    IBM Research

  4. Spectral Hierarchy of the Cosmic Web

    Mar 16, 2026Francisco-Shu Kitaura, Francesco SinigagliaHierarchical ClassificationSpectral Methods

    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

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

    Mar 14, 2026Yongzhong XuTransformer Training DynamicsNeural Network Training Dynamics

  6. Filtered Spectral Projection for Quantum Principal Component Analysis

    Mar 13, 2026Sk Mujaffar Hossain, Satadeep BhattacharjeePrincipal Component AnalysisQuantum Machine Learning

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

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

    Mar 12, 2026Hannah Sansford, Nick Whiteley, Patrick Rubin-DelanchySpectral Graph EmbeddingGraph Embedding

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

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

    Mar 9, 2026Changkai LiConstrained OptimizationMulti-Objective Optimization

  9. Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

    Mar 7, 2026Pavan Karjol, Kumar Shubham, Prathosh APSymmetry DiscoverySpectral Methods

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

  10. Regular Fourier Features for Nonstationary Gaussian Processes

    Feb 26, 2026Arsalan Jawaid, Abdullah Karatas, Jörg SeewigFourier Feature EmbeddingsGaussian Processes

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

  11. Limits of Convergence-Rate Control for Open-Weight Safety

    Feb 21, 2026Domenic Rosati, Xijie Zeng, Hong Huang +4Neural Network OptimizationSpectral Methods

    Dalhousie University · Vector Institute · Indian Institute of Management Bangalore

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

    Feb 19, 2026Sie Hendrata Dharmawan, Peter ChinStochastic Block ModelGraph Clustering

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

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

    Feb 10, 2026David S. Duque-Castaño, Lauren Flor-Torres, Jorge I. ZuluagaDenoising AutoencodersAutoencoders

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

  14. Learning to Discover Iterative Spectral Algorithms

    Feb 10, 2026Zihang Liu, Oleg Balabanov, Yaoqing Yang +1Automated Algorithm DiscoveryLearning to Optimize

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

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

    Jan 29, 2026Ji-Xuan He, Guohang Zhuang, Bo Junge +5Hyperspectral Image Super-ResolutionHyperspectral Imaging

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

  16. SFO: Learning PDE Operators via Spectral Filtering

    Jan 23, 2026Noam Koren, Rafael Moschopoulos, Kira Radinsky +1PDE Surrogate ModelingPDE Operator Learning

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

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

    Dec 17, 2025Federico Ottomano, Yingzhen Li, Alex M. GanoseMultimodal GenerationMolecular Generation

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

  18. Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

    Dec 9, 2025Muhammad Abid, Omer SanPDE Surrogate ModelingPDE Operator Learning

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

  19. Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization

    Oct 2, 2025Dhruv Kohli, Sawyer J. Robertson, Gal Mishne +1Manifold LearningSpectral Methods

  20. Universal Learning of Nonlinear Dynamics

    Aug 16, 2025Evan Dogariu, Anand Brahmbhatt, Elad HazanDynamical SystemsNonlinear System Identification

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

  21. On the Upper Bounds for the Matrix Spectral Norm

    Jun 18, 2025Alexey Naumov, Maxim Rakhuba, Denis Ryapolov +1Spectral Methods

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

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

    Feb 16, 2025Nan-Hong Kuo, Renata WongSupport Vector MachinesKernel Methods

    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

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

    Jan 18, 2025Haotian Lin, Matthew ReimherrSpectral RegularizationConcept Drift

    Department of Statistics, The Pennsylvania State University.

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

    Jul 8, 2024Matthew J. Colbrook, Igor Mezić, Alexei StepanenkoKoopman Operator LearningSpectral Methods

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

  25. Hearing the shape of an arena with spectral swarm robotics

    Mar 25, 2024Leo Cazenille, Nicolas Lobato-Dauzier, Alessia Loi +7Swarm RoboticsSpectral Methods

    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

  26. Semidefinite programming relaxations and debiasing for MAXCUT-based clustering

    Jan 16, 2024Shuheng ZhouSemidefinite ProgrammingUnsupervised Clustering

    University of California, Riverside, CA 92521

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

    Nov 29, 2022Adarsh Ganeshram, Valentin Duruisseaux, Haydn Maust +6Physics-Informed Neural OperatorsPDE Solving

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

  28. Robust Streaming PCA

    Feb 8, 2019Daniel Bienstock, Minchan Jeong, Apurv Shukla +1Principal Component AnalysisStreaming Algorithms

    Graduate School of AI KAIST