Random Matrix Theory

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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
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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Compact and Infinite-Order Error Analysis for Null-Space SVD Estimation

Xin Li, Jonathan Cohen, Rami Puzis

Random Matrix Theory Singular Value Decomposition

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We study null-space estimation from a noisy matrix. For a simple left null space, we first derive an exact compact expression for the error of the smallest left singular vector. We then give an all-order series for the SVD vector and projector, followed by compact and consistently truncated series forms for the fixed-realization empirical risk and conditional population generalization risk. The recursion extends to a multiple-dimensional null space by following the complete invariant subspace. The convergence radius is not inferred from an error plot: it is computed independently from the nearest complex exceptional point that joins a retained eigenvalue branch to its complement. A reduced-nullity experiment shows that moving this spectral boundary can increase the radius, although the improvement is not monotone in the retained nullity. For individually ordered null directions under Gaussian training with

τ\geq m

, we prove that the Wishart splitting matrix

W

gives a strict second-order empirical ranking. Gaussian averaging equalizes the leading generalization risks at both small and very large noise, while a column-swap theorem proves strict expected generalization ranking for an isotropic signal subspace. For unequal spikes, an exact population-overlap criterion and a simultaneous

99\%

Monte Carlo confidence certificate explain the observed intermediate ranking. A sixth-order risk correction improves the lower-crossover estimate in the reported experiment. This equal--ranked--equal phenomenon is a finite-sample diagnostic related to spectral mixing, but its tolerance crossings, the exceptional-point radius, and the asymptotic BBP threshold are three distinct quantities.

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