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

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

Spectral Braids: A Topological Descriptor for Dynamic Directed Networks
Oct 27, 2026Mehmet Emin Aktas, Esra AkbasDynamic Graph Learning Spectral Methods
School of Data Science and Analytics, Kennesaw State University, Kennesaw, GA, USA · Department of Computer Science, Georgia State University, Atlanta, GA, USA
Neural Eigenfluid Bases: Spectral Objectives, Gauge, and Topology
Oct 24, 2026Baiming Zhang, Shiying Xiong, Zhecheng WangReduced-Order Modeling Computational Fluid Dynamics
Zhejiang University · University of Toronto
Topological Equivalence of Spectral and Density Cluster Hierarchies
Oct 22, 2026Sajjad HashemianHierarchical Clustering Density-Based Clustering
Random Projections for Spectral Algorithms: Optimal Rates in Sobolev Norms
Oct 22, 2026Caixing Wang, Yue WangGeneralization Bounds Spectral Methods
School of Statistics and Data Science, Southeast University, Nanjing, 210000, Jiangsu, China · Faculty of Business for Science and Technology, School of Management, University of Science and Technology of China, Hefei, 230000, Anhui, China
Wavenumber-Aware Deep Learning for Robust Analysis of Raman Spectra
Oct 22, 2026Jihwan Kim, Jaeseong Heo, Yujin Kim +7Neural Network Robustness Positional Encoding
Department of Chemistry, Kookmin University, 77 Jeongneung-ro, Seongbuk-gu, Seoul, 02707, Republic of Korea. · BB21+, Department of Chemistry, Pukyong National University, Busan, 48513, Republic of Korea. · Department of Chemical Engineering, Myongji University, Yongin, 17058, Republic of Korea. +4
WaveSpec: Multi-Spectral Iris Presentation Attack Detection via Wavelength-Conditioned Spectral Coherence Adaptation
Oct 20, 2026Raghavendra RamachandraSpectral Methods
SAFE Centre, Norwegian University of Science and Technology (NTNU), Norway
Minimax estimation of the pseudo-spectral gap of continuous Markov chains, with applications to empirical concentration bounds
Oct 14, 2026Vahe Karagulyan, Mikołaj Kasprzak, Pierre AlquierMarkovian Bandits Minimax Estimation
ESSEC Business School
Global-in-time Spectral Duhamel-Picard Method for Post-Exposure Bake Simulation and Physics-Informed Neural Operators for EUV Lithography
Oct 13, 2026Doyun Kim, Mircea Dusa, Werner Gilijns +1Reaction-Diffusion Systems Semiconductor Manufacturing
IMEC, Kapeldreef 75, 3001 Leuven, Belgium
Which Geometry on Which Layer? Measured Sign-Spectral Assignment in Mixed-Optimizer Training
Oct 12, 2026Hiroki Naganuma, Laura Gomezjurado, Mahdi Ghaznavi +5Spectral Optimization Deep Learning Optimization
Mila · Université de Montréal · Stanford University +4
Spectral Costs of Multinomial and Residual Resampling in High-Dimensional Particle Ensembles: A Free-Probability Analysis
Oct 12, 2026Deming Song, Chi Kin LamSpectral Methods Random Matrix Theory
the Faculty of Applied Sciences, Macao Polytechnic University, Macao, China
Mesh-Free Convolution: Learned Spectral Attenuation and Transport
Oct 11, 2026Niloufar Zakariaei, Manav Manav, Eldad HaberPDE Operator Learning Spectral Methods
University of British Columbia · Computational Geoscience Inc.
Spectrally Targeted Muon
Oct 7, 2026Vishrut Goyal, Rohan RamkumarSpectral Optimization Deep Learning Optimization
Department of Computer Science, Stanford University · Department of Electrical Engineering, Stanford University.
"Spectral Harmony" Towards the Unification of Timbre and Speech: Resolution of the Young Mahler Schoenberg Dream
Oct 3, 2026Yusei TAMURA, Shigekazu ISHIHARA, Ken ITOSpectral Methods
The University of Tokyo 7-3-1 Hongo, Bunkyo Ward, Tokyo, JAPAN · Hiroshima International University 555-36 Kurose Gakuen-dai, Higashi-Hiroshima, Hiroshima JAPAN
Stochastic Adaptive Fourier Decomposition for Operator Learning
Oct 3, 2026Pengqing Shi, Liming Zhang, Tao Qian +3PDE Operator Learning Neural PDE Solvers
University of Sydney · University of Macau · Macau University of Science and Technology
Invariant Shape Analysis of Surfaces with Spherical Topology
Sep 30, 2026T. Shaska, M. -R. SiadatSpherical Harmonics Spectral Methods
Bounded Channel-Adaptive Spectral Learning for Forward-Consistent Inverse Flapping-Wing Aerodynamics
Sep 30, 2026Haichuan LiGated Recurrent Unit Recurrent Neural Networks
University of Turku
A Spectral Theory of Distortion in LLM Graph Reconstruction: Sharp Bounds and Empirical Characterization
Sep 29, 2026Jianru ShenLLMs for Graph Learning Wasserstein Distance
Columbia University New York, NY, USA
A Finslerian Approach for Embedding Directed Data
Sep 29, 2026Gwendal Debaussart-Joniec, Théau Blanchard, Argyris KalogeratosFinsler Geometry Graph Embedding
CB, ENS Paris Saclay · UMR 1346 HeKA, Universit´e Paris Cit´e, Inria, Inserm GE Healthcare F-75015 Paris, France · HeKA | U1346, GE Healthcare
Flattening the Connectome Spectrum: A Spectral Filter for FC Induces a Pretraining Target for fMRI Encoders
Sep 29, 2026Giovanni Marraffini, Victoria Shevchenko, Carlo Alberto Barbano +1Kernel Ridge Regression Medical Imaging Foundation Models
UNITO · MIND
Graph-Spectral Flow Matching for Multivariate Time Series Anomaly Detection
Sep 29, 2026Zepeng Zhang, Jhony H. Giraldo, Wenbin Wang +1Time-Series Anomaly Detection Flow Matching
IMOS, EPFL · LTCI, T´el´ecom Paris, IP Paris · Independent Researcher
Learning Conditional Expectation Operators via Functional Newton Updates
Sep 28, 2026Thiago Ramos, Alek Fröhlich, Daniel Perazzo +1Density Ratio Estimation Low-Rank Approximation
Federal University of São Carlos · CSML, Istituto Italiano di Tecnologia and University of Genoa · CSML, Istituto Italiano di Tecnologia and University College London
Hierarchical Clustering and Signal Denoising on Digraphs
Sep 28, 2026Yi Wang, Sippanon Kitimoon, Hrushikesh N. Mhaskar +1Graph Signal Processing Hierarchical Clustering
Department of Mathematics, City University of Hong Kong, Hong Kong, SAR China · Data Science Research Center, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand · Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, USA
TNF based Spectral Embedding for Effective Application of Supervised Machine Learning Techniques in Automobile Insurance Fraud Detection
Sep 27, 2026Rohan Yashraj Gupta, Lalith Srikanth Chintalapati, Satya Sai Mudigonda +2Financial Fraud Detection Spectral Methods
Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning, Vidyagiri, Puttaparthi, Andhra Pradesh, India.
Cost-free Spectral Estimation for Adaptive Newton--Schulz in Matrix Optimizers
Sep 27, 2026Kristi Topollai, Anna ChoromanskaMatrix Optimization Newton-Schulz Iteration
New York University
Graph Memory: Spectral Associative Memory via Dirichlet Energy
Sep 26, 2026Zhaoyang ShiAssociative Memory Graph-Based Retrieval
Center for Applied Mathematics, Fudan University Shanghai, China
Spectral Reversal: Counteracting Singular Value Bias for Graph Prompting
Sep 26, 2026Hanxu Yang, Yuhuan Zhao, Xiaodong He +1Graph Neural Networks Spectral Bias
School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, China
NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures
Sep 25, 2026Márcio Marques, Leonardo Mendonça, Leonardo M. Moreira +6Neural PDE Solvers PDE Solving
Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brasil · Universidade do Estado do Rio de Janeiro (UERJ), Rio de Janeiro, Brasil · Universidade Federal Fluminense (UFF), Niterói, Brasil
Moment-guided edge sampling
Sep 24, 2026Weibin Cai, Reza ZafaraniSpectral Methods
Data Lab, Department of EECS Syracuse University Syracuse, NY, USA
Adaptive Random Matrices in Gaussian Bandits: Spectral Universality and Selection-Induced Outliers
Sep 23, 2026Sudarshan Manikantan, Abhishek BhattacharjeeGaussian Process Bandits Multi-Armed Bandits
Abstract Math Institute
Disentangling Heterogeneous Traffic Dynamics for Multi-Step Traffic Forecasting via Adaptive Spectral Decomposition
Sep 22, 2026Zijun Huang, Chenrui Fu, Wenhao Wang +3Time Series Decomposition Spatiotemporal Forecasting
Beijing Normal-Hong Kong Baptist University · Dalian Maritime University · National Chung Hsing University

Oct 27, 2026J/K move · Enter open · S save

Spectral Braids: A Topological Descriptor for Dynamic Directed Networks

Mehmet Emin Aktas, Esra Akbas

School of Data Science and Analytics, Kennesaw State University, Kennesaw, GA, USA · Department of Computer Science, Georgia State University, Atlanta, GA, USA

Dynamic Graph Learning Spectral Methods

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Eigenvalues are a workhorse of graph machine learning: spectral clustering, spectral graph neural networks, and graph kernels all summarize a network by the eigenvalues of a matrix. These eigenvalues are almost always treated as a static list of numbers. When a graph changes, however, its eigenvalues move, and when the matrix is not symmetric, they move in the complex plane, where they can wind around each other. The pattern of this winding is a braid, a classical object from topology. We propose spectral braids, a methodology for dynamic directed networks: each snapshot is aggregated into a community-level transition matrix, its complex eigenvalues are tracked over time, and the braid they trace is read off as a word. The word is then mapped to a matrix by the Burau representation, which gives a fixed-size feature vector for any classifier. The braid has a concrete meaning: each eigenvalue describes how a group of communities behaves, and each crossing records one group overtaking another. We evaluate the method on dynamic network classification, using synthetic networks of online communities whose labels describe how the network evolved between the same starting and ending states. Our experimental results show that the proposed features reach 97.1% accuracy on four classes, against 44.5% for the best feature set that does not track eigenvalues, and that the Burau features also separate networks containing the same events in a different order, which simple counts of crossings cannot do. To the best of our knowledge, this is the first study that defines braids and braid monodromy on graph data and uses them as features for learning on dynamic networks.