Random Matrix Theory

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Latest papers 13

Aug 31, 2026math.ST

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

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.

Xin Li, Jonathan Cohen, Rami Puzis

Random Matrix Theory Singular Value Decomposition

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Jun 29, 2026cs.LG

I-BBS: Coordinate-Free Inference of Latent Sub-Manifolds Using Random Distance Matrix Theory

Bogomolny, Bohigas and Schmit (BBS) found that the spectrum of the pairwise distance matrix on N points sampled from a smooth d-dimensional manifold encodes a signature of the underlying geometry. We develop I-BBS (Inference-BBS), a coordinate-free method that identifies a low-dimensional latent sub-manifold embedded in a high-dimensional ambient distance matrix alone, without accessing an ambient high-dimensional vector space. It therefore applies even when that space is only partly observable or undefined. We model the ambient embedding by two classes of generative noise, model-based and model-free. The noise mixes the latent signal with off-manifold components, so the eigenvalues reorganise collectively and the latent geometry cannot be read off eigenvalue by eigenvalue. We recover it instead from two integer-stable signatures that survive the noise: the multiplicity of the top non-Perron multiplet, which fixes

d

, and a parameter-free law for how the multiplet positions shrink as the noise grows. On synthetic spheres

S^1

S^2

and

S^3

these integer signatures are far more stable under noise than the continuous spectral slope, and a blind test recovers both the manifold and the noise model from a single distance matrix. Applications to neural-network representations and to the dynamic training regime are developed in two companion papers.

Igor Halperin

Intrinsic Dimensionality Random Matrix Theory Manifold Learning

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Jun 26, 2026cond-mat.dis-nn

Spectral phase transitions and trainability in neural network learning dynamics

The emergence of low-dimensional structures in the spectra of neural network weight matrices is a common empirical feature of trained models, but the dynamical origin of this phenomenon during learning remains an open problem. We formulate neural network training as the stochastic evolution of an initially random matrix ensemble, driven by stochastic gradient descent (SGD) updates that reshape the spectral bulk while amplifying signal strength. This induces a Baik-Ben Arous-Péché (BBP) transition during training, where isolated eigenvalues detach from the random bulk distribution, providing a dynamical framework for representation formation in high-dimensional learning dynamics. We demonstrate this in a solvable linear teacher-student model, where spectral evolution is analytically tractable and a phase diagram of trainability governed by the step size (or learning rate) and initial weight variance is obtained, and subsequently extend our formalism beyond the linear regime to nonlinear and stochastic settings. Numerical simulations in realistic settings support this picture, showing robust emergence of spectral alignment during training. Our results suggest that spectral analysis may provide a unified perspective of stochastic learning dynamics, linking trainability, optimisation hyperparameters, spectral phase transitions, and representation learning in neural networks.

Chanju Park, Dario Bocchi, Francesco D'Amico +2

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

Random Matrix Theory Neural Network Training Dynamics Stochastic Gradient Descent

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Jun 23, 2026math.PR

Uniform Sampling from High-dimensional Spectral Norm Balls

Motivated by an application in machine learning optimization, this paper focuses on the challenges of sampling a matrix uniformly from the unit spectral norm ball. It is proven that all singular values of sampled matrices converge to 1 almost surely as the matrix dimensions increase. This result provides the theoretical justification for a proposed simple sampling method applicable for large dimension sizes matching matrices found in modern large language models. Experimental results demonstrate both the convergence of the singular values, as well as the exact and proposed approximate sampling methods.

Michael R. Metel

Random Matrix Theory

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Jun 9, 2026math.ST

Geometric bias in eigenspace perturbation under random heterogeneous noise

Spectral methods rely fundamentally on the stability of principal eigenspaces under random perturbations. Classically, this stability is quantified by the Davis-Kahan and Wedin theorems, which bound the eigenspace error using the operator norm of the noise and the relevant spectral gaps. While these worst-case bounds are sharp for arbitrary deterministic perturbations, they can be wasteful in the low-rank signal-plus-random-noise setting, as they fail to capture the fine-grained interaction between the signal geometry and the noise distribution. In this paper, we study the spectral perturbation of signal-plus-noise matrices corrupted by sparse, random noise with an arbitrary, inhomogeneous variance profile. We demonstrate that under heterogeneous noise variances, the empirical eigenvectors suffer a systematic, deterministic geometric bias that is entirely invisible to classical perturbation bounds. By leveraging the Quadratic Vector Equation (QVE) and establishing fine-grained isotropic local laws, we derive near-optimal, non-asymptotic perturbation bounds for the leading eigenspaces in the operator and

2\to\infty

norms. The bounds separate the usual signal-to-noise contribution, stochastic fluctuations, and structured geometric bias terms determined by the alignment between the signal eigenspaces and the row-wise variance profile.

Fengkai Liu, Ke Wang, Wanjie Wang

Department of Mathematics, Hong Kong University of Science and Technology · Department of Statistics and Data Science, National University of Singapore

Random Matrix Theory Spectral Methods

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May 30, 2026cs.DS

Easy, robust approximate message passing for planted spike models

We present a simple and efficient algorithm for robust approximate message passing (AMP) in the spiked matrix setting. In particular, let

\varepsilon

be a sufficiently small constant, and suppose that

X \in \mathbb R^{n \times n}

is a Gaussian matrix with a planted rank-

1

spike, and

E \in \mathbb R^{n \times n}

is an adversarially chosen matrix supported on an

\varepsilon n \times \varepsilon n

principal minor. Let

v_{\mathrm{AMP}}(X)

be the output of an AMP iteration on the uncorrupted matrix

X

. We give a procedure that, given access only to the corrupted matrix

Y = X + E

, computes a vector

v_{\mathrm{ALG}}(Y)

which is

\tilde{O}(\sqrt{\varepsilon})

-close to

v_{\mathrm{AMP}}(X)

, for any of a class of AMP iterations which includes sparse Principal Component Analysis (PCA), non-negative PCA, and

\mathbb Z_2

synchronization. Our algorithm consists of a spectral pre-processing step combined with a robust spectral initialization procedure; given these inputs, we prove that (perhaps surprisingly) AMP is robust out-of-the-box.

Misha Ivkov, Tselil Schramm

*Stanford University. · †Stanford University.

Random Matrix Theory Principal Component Analysis Adversarial Robustness Spectral Methods

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May 28, 2026stat.ML

Eigen-Spike Emergence and Quadratic Equivalents for Conjugate Kernels on Nonlinearly Separable Data

Recent work in random matrix theory (RMT) has developed the notion of deterministic equivalents: typically linear surrogate models that approximate the spectral behavior of large nonlinear random matrices, such as nonlinear feature maps in neural networks (NNs). Such equivalents make theoretical predictions tractable by reducing a complex model to a simpler one with properties that fall under the umbrella of classical RMT tools. However, this leaves open the question of whether this idealized linear equivalence remains meaningful for classification of high-dimensional nonlinearly separable data. Motivated by this, we consider the conjugate kernel (CK), which is the nonlinear feature map of a one-layer feedforward NN, under a canonical nonlinearly separable dataset for the XOR problem; and we use the study of informative outlier eigenvalues in the CK and whether their corresponding eigenvectors asymptotically align with XOR labels as a proxy for nonlinear learnability. We develop a robust quadratic equivalent of the CK matrix that enables a precise analysis of emergent informative spikes, as one modifies various knobs common in ML practice: sample complexity, signal-to-noise ratio (SNR), nonlinear activation choice, and pretrained features. We identify regimes in which these knobs move the CK beyond the linear equivalent and produce BBP-type transitions to label-aligned outlier eigenspaces. Our analysis helps bring deterministic-equivalence tools from RMT to bear on problems of practical relevance in ML.

Collin Cranston, Zhichao Wang, Todd Kemp +1

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

Random Matrix Theory Kernel Methods

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May 5, 2026stat.ML

Free Decompression with Algebraic Spectral Curves

Tools from random matrix theory have become central to deep learning theory, using spectral information to provide mechanisms for modeling generalization, robustness, scaling, and failure modes. While often capable of modeling empirical behavior, practical computations are limited by matrix size, often imposing a restriction to models that are too small to be realistic. This motivates the inference of properties of larger models from the behavior of smaller ones. Free decompression (FD) is a recently proposed method for extrapolating spectral information across matrix sizes, but its utility is currently limited by strong assumptions that preclude its implementation on more realistic machine learning (ML) models. We use algebraic spectral curve theory to provide a general FD methodology for spectral densities whose Stieltjes transform satisfies an algebraic relation, a modeling assumption that is more likely to hold in practice. This recasts FD as an evolution along spectral curves which can be readily integrated. Our framework enables the expansion of spectral densities that have multiple or multi-modal bulks, that exist at multiple scales, and that contain atoms, all characteristic of real-world data and popular ML models. We demonstrate the efficacy of our framework on models of interest in modern ML, including Hessian and activation matrices associated with neural networks and large-scale diffusion models.

Siavash Ameli, Chris van der Heide, Liam Hodgkinson +1

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

Random Matrix Theory Spectral Methods

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Apr 25, 2026cs.DS

Well-Conditioned Oblivious Perturbations in Linear Space

Perturbing a deterministic

n

-dimensional matrix with small Gaussian noise is a cornerstone of smoothed analysis of algorithms [Spielman and Teng, JACM 2004], as it reduces the condition number of the input to

O(n)

, and with it the complexity of many matrix algorithms. However, when deployed algorithmically, these perturbations are expensive due to the cost of generating and storing

n^2

Gaussian random variables. We propose a perturbation that requires generating and storing

O(n)

random numbers in

O(\log n)

bits of precision, and reduces the condition number of any deterministic matrix to

O(n)

, matching Gaussian perturbations. Our result in particular implies a better complexity for the perturbed conjugate gradient algorithm, showing that we can solve an

n\times n

linear system in linear space to within an arbitrarily small constant backward error using

O(n)

matrix-vector products. In our construction, we introduce the concept of a pattern matrix, which is a dense deterministic matrix that maps all sparse vectors into dense vectors, and we combine it with a sparse perturbation whose entries are dependent and located in a non-uniform fashion. In order to analyze this construction, we develop new techniques for lower bounding the smallest singular value of a random matrix with dependent entries.

Shabarish Chenakkod, Michał Dereziński, Xiaoyu Dong +1

University of Michigan, Ann Arbor, MI, USA · National University of Singapore, Singapore

Random Matrix Theory

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Apr 20, 2026stat.ML

Random Matrix Theory of Early-Stopped Gradient Flow: A Transient BBP Scenario

Empirical studies of trained models often report a transient regime in which signal is detectable in a finite gradient descent time window before overfitting dominates. We provide an analytically tractable random-matrix model that reproduces this phenomenon for gradient flow in a linear teacher--student setting. In this framework, learning occurs when an isolated eigenvalue separates from a noisy bulk, before eventually disappearing in the overfitting regime. The key ingredient is anisotropy in the input covariance, which induces fast and slow directions in the learning dynamics. In a two-block covariance model, we derive the full time-dependent bulk spectrum of the symmetrized weight matrix through a

2\times 2

Dyson equation, and we obtain an explicit outlier condition for a rank-one teacher via a rank-two determinant formula. This yields a transient Baik-Ben Arous-Péché (BBP) transition: depending on signal strength and covariance anisotropy, the teacher spike may never emerge, emerge and persist, or emerge only during an intermediate time interval before being reabsorbed into the bulk. We map the corresponding phase diagrams and validate the theory against finite-size simulations. Our results provide a minimal solvable mechanism for early stopping as a transient spectral effect driven by anisotropy and noise.

Florentin Coeurdoux, Grégoire Ferré, Jean-Philippe Bouchaud

1Capital Fund Management, Paris, France · 2X-CFM Chair of EconophysiX, Ecole polytechnique, Palaiseau, France · 3Académie des Sciences, Paris, France

Random Matrix Theory Early Stopping Phase Transitions Teacher-Student Learning Gradient Descent Dynamics

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Dec 14, 2025q-bio.NC

Random matrix theory of sparse neuronal networks with heterogeneous timescales

Training recurrent neuronal networks consisting of excitatory (E) and inhibitory (I) units with additive noise for working memory computation slows and diversifies inhibitory timescales, leading to improved task performance that is attributed to emergent marginally stable equilibria [PNAS 122 (2025) e2316745122]. Yet the link between trained network characteristics and their roles in shaping desirable dynamical landscapes remains unexplored. Here, we investigate the Jacobian matrices describing the dynamics near these equilibria and show that they are sparse, non-Hermitian rectangular-block matrices modified by heterogeneous synaptic decay timescales and activation-function gains. We specify a random matrix ensemble that faithfully captures the spectra of trained Jacobian matrices, arising from the inhibitory core - excitatory periphery network motif (pruned E weights, broadly distributed I weights) observed post-training. An analytic theory of this ensemble is developed using statistical field theory methods: a Hermitized resolvent representation of the spectral density is processed with a supersymmetry-based treatment in the style of Fyodorov and Mirlin. In this manner, an analytic description of the spectral edge is obtained, relating statistical parameters of the Jacobians (sparsity, weight variances, E/I ratio, and the distributions of timescales and gains) to near-critical features of the equilibria essential for robust working memory computation.

Thiparat Chotibut, Oleg Evnin, Weerawit Horinouchi

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

Random Matrix Theory Recurrent Neural Networks Neural Network Training Dynamics

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Oct 26, 2025math.PR

Beyond the Semicircle: Free Diffusion Models with Prescribed Equilibria

A growing class of machine-learning objects -- covariance and Gram matrices, kernel and attention matrices, MIMO channel matrices, density operators -- are naturally spectra rather than coordinate vectors. Building a denoising diffusion model for such data by corrupting eigenvalues coordinatewise is not merely elegant: it converges to the wrong limit, because eigenvalues repel rather than move independently. Free probability theory supplies the correct forward process, with free convolution replacing classical convolution and Voiculescu's conjugate variable replacing the score, but constant-coefficient free diffusions have a hidden limitation of their own: their only possible equilibrium is the semicircular law, whatever the target distribution looks like. We show that state-dependent free volatility removes this restriction. For any sufficiently regular compactly supported target law, we explicitly construct, through a closed-form Hilbert-transform drift, a free Fokker--Planck flow having that law as a stationary spectral distribution. Under an additional convex-potential condition, the construction admits a globally relaxing diffusion interpretation, reverse-time dynamics, and a trainable score. The generalization is genuine rather than cosmetic: no change of spectral variable reduces it to the constant-coefficient case, and it is a Wasserstein gradient flow only for an explicitly characterized family of coefficients that contains no bounded non-constant member. We instantiate the construction on the Marchenko--Pastur law, the limiting spectrum of isotropic sample covariance matrices, and validate it numerically: the free-versus-coordinatewise gap, an exact dimension-independent score driving reverse-time matrix dynamics at several matrix sizes, the designed Marchenko--Pastur equilibrium, and end-to-end generation with a learned score.

Swagatam Das

Random Matrix Theory Diffusion Models

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Jun 16, 2025stat.ML

Random Matrix Theory for Deep Learning: Beyond Eigenvalues of Linear Models

Modern Machine Learning (ML) and Deep Neural Networks (DNNs) often operate on high-dimensional data and rely on overparameterized models, where classical low-dimensional intuitions break down. In particular, the proportional regime where the data dimension, sample size, and number of model parameters are all large and comparable, gives rise to novel and sometimes counterintuitive behaviors. This paper extends traditional Random Matrix Theory (RMT) beyond eigenvalue-based analysis of linear models to address the challenges posed by nonlinear ML models such as DNNs in this regime. We introduce the concept of High-dimensional Equivalent, which unifies and generalizes both Deterministic Equivalent and Linear Equivalent, to systematically address three technical challenges: high dimensionality, nonlinearity, and the need to analyze generic eigenspectral functionals. Leveraging this framework, we provide precise characterizations of the training and generalization performance of linear models, nonlinear shallow networks, and deep networks. Our results capture rich phenomena, including scaling laws, double descent, and nonlinear learning dynamics, offering a unified perspective on the theoretical understanding of deep learning in high dimensions.

Zhenyu Liao, Michael W. Mahoney

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

Random Matrix Theory Neural Network Generalization Neural Network Training Dynamics Neural Scaling Laws

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