Random Matrix Theory

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Random Matrix Theory explains how eigenvalues and eigenvectors behave in large random systems, including when spectral patterns reveal hidden signals or transitions. Researchers use tools such as limiting spectral laws, resolvents and local laws, free probability, and approximate message passing to analyze matrices from statistics, machine learning, and physics. Applications include detecting signal spikes, understanding neural-network training, and characterizing high-dimensional covariance and kernel matrices. A central challenge is extending results from idealized independent-entry models to structured, nonlinear, heterogeneous, or adaptive data without losing tractable predictions or universality.

Latest papers

20 papers2 papers in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

Distinguishability and Recovery for a Doubly Sparse Spiked Wigner Model
Oct 16, 2026Ioana Dumitriu, JD Flynn, Zhichao WangSparse Recovery Random Matrix Theory
University of California, San Diego · University of California, Berkeley
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
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
Compact and Infinite-Order Error Analysis for Null-Space SVD Estimation
Aug 31, 2026Xin Li, Jonathan Cohen, Rami PuzisRandom Matrix Theory Singular Value Decomposition
I-BBS: Coordinate-Free Inference of Latent Sub-Manifolds Using Random Distance Matrix Theory
Jun 29, 2026Igor HalperinIntrinsic Dimensionality Random Matrix Theory
Spectral phase transitions and trainability in neural network learning dynamics
Jun 26, 2026Chanju Park, Dario Bocchi, Francesco D'Amico +2Random Matrix Theory Neural Network Training Dynamics
Centre for Quantum Fields and Gravity, Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom · Physics Department, University of Rome Sapienza, Piazzale Aldo Moro 5, 00185 Rome, Italy · Institute of Nanotechnology, CNR-NANOTEC, Piazzale Aldo Moro 5, 00185 Rome, Italy +1
Uniform Sampling from High-dimensional Spectral Norm Balls
Jun 23, 2026Michael R. MetelRandom Matrix Theory
Geometric bias in eigenspace perturbation under random heterogeneous noise
Jun 9, 2026Fengkai Liu, Ke Wang, Wanjie WangRandom Matrix Theory Spectral Methods
Department of Mathematics, Hong Kong University of Science and Technology · Department of Statistics and Data Science, National University of Singapore
Easy, robust approximate message passing for planted spike models
May 30, 2026Misha Ivkov, Tselil SchrammRandom Matrix Theory Principal Component Analysis
*Stanford University. · †Stanford University.
Eigen-Spike Emergence and Quadratic Equivalents for Conjugate Kernels on Nonlinearly Separable Data
May 28, 2026Collin Cranston, Zhichao Wang, Todd Kemp +1Random Matrix Theory Kernel Methods
Department of Mathematics University of California, San Diego, USA · ICSI and Department of Statistics University of California, Berkeley, USA · ICSI, LBNL and Department of Statistics University of California, Berkeley, USA
Free Decompression with Algebraic Spectral Curves
May 5, 2026Siavash Ameli, Chris van der Heide, Liam Hodgkinson +1Random Matrix Theory Spectral Methods
1ICSI and Department of Statistics, University of California, Berkeley · 2Dept. of Electrical and Electronic Engineering, University of Melbourne · School of Mathematics and Statistics, University of Melbourne +1
Well-Conditioned Oblivious Perturbations in Linear Space
Apr 25, 2026Shabarish Chenakkod, Michał Dereziński, Xiaoyu Dong +1Random Matrix Theory
University of Michigan, Ann Arbor, MI, USA · National University of Singapore, Singapore
Random Matrix Theory of Early-Stopped Gradient Flow: A Transient BBP Scenario
Apr 20, 2026Florentin Coeurdoux, Grégoire Ferré, Jean-Philippe BouchaudRandom Matrix Theory Early Stopping
1Capital Fund Management, Paris, France · 2X-CFM Chair of EconophysiX, Ecole polytechnique, Palaiseau, France · 3Académie des Sciences, Paris, France
Universality of first-order methods on random and deterministic matrices
Apr 13, 2026Nicola Gorini, Chris Jones, Dmitriy Kunisky +1Approximate Message Passing Random Matrix Theory
Bocconi University. · UC Davis. · Johns Hopkins University. +1
Persistence diagrams of random matrices via Morse theory: universality and a new spectral diagnostic
Mar 29, 2026Matthew LoftusRandom Matrix Theory Persistent Homology
Cedar Loop LLC, Washington, USA
Structure and Redundancy in Large Language Models: A Spectral Study via Random Matrix Theory
Feb 25, 2026Davide EttoriLLM Compression Representation Geometry in Language Models
Beyond Accuracy: A Unified Random Matrix Theory Diagnostic Framework for Crash Classification Models
Feb 23, 2026Ibne Farabi Shihab, Sanjeda Akter, Anuj SharmaModel Selection Classification
Department of Computer Science, Iowa State University, Ames, IA 50010 · Department of Civil, Construction and Environmental Engineering, Iowa State University, Ames, IA 50010
Random matrix theory of sparse neuronal networks with heterogeneous timescales
Dec 14, 2025Thiparat Chotibut, Oleg Evnin, Weerawit HorinouchiRandom Matrix Theory Recurrent Neural Networks
Chula Intelligent and Complex Systems, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, Thailand · High Energy Physics Research Unit, Faculty of Science, Chulalongkorn University, Bangkok, Thailand · Theoretische Natuurkunde, Vrije Universiteit Brussel (VUB) and International Solvay Institutes, Brussels, Belgium +2
Beyond the Semicircle: Free Diffusion Models with Prescribed Equilibria
Oct 26, 2025Swagatam DasRandom Matrix Theory Diffusion Models
Random Matrix Theory for Deep Learning: Beyond Eigenvalues of Linear Models
Jun 16, 2025Zhenyu Liao, Michael W. MahoneyRandom Matrix Theory Neural Network Generalization
School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, China · ICSI, LBNL, and Department of Statistics, University of California, Berkeley, USA

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Distinguishability and Recovery for a Doubly Sparse Spiked Wigner Model

Ioana Dumitriu, JD Flynn, Zhichao Wang

University of California, San Diego · University of California, Berkeley

Sparse Recovery Random Matrix Theory

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We study distinguishability and recovery of a sparse rank-one signal in a spiked Wigner model with independently sparse noise through a variety of algorithms and parameter regimes. The signal and noise have separate sparsity parameters, while the signal-to-noise ratio (SNR) remains of constant order as the matrix size grows. For sub-Gaussian noise and spikes whose nonzero coordinates are bounded away from zero, a leave-one-out quadratic thresholding procedure (a generalization of Ding et al., 2024) identifies the support. Sign synchronization yields exact recovery for a Rademacher spike, and spectral refinement yields strong recovery. The quadratic thresholding search parameter gives an explicit tradeoff between signal sparsity, noise sparsity, and computational cost. We also study extensions to sub-Gaussian spike priors through diagonal and quadratic thresholding. For Rademacher spikes with continuous nonzero noise, we develop an identical search algorithm that exploits unblurred signal entries, the existence of which permits distinguishability; connectivity of the graph they reveal provides a sufficient condition for exact recovery. These guarantees do not require moment assumptions on the continuous noise distribution. Together with our previous PCA results (Dumitriu et al., 2026), our analysis covers almost the entire space of parameters and regimes for distinguishing and recovering in our doubly sparse spiked Wigner model.