Statistical limits of dictionary learning: random matrix theory and the spectral replica method

TL;DR

This paper develops a novel spectral replica method combining random matrix theory and the replica approach to analyze the fundamental limits of matrix denoising and dictionary learning in high-rank regimes, extending beyond traditional low-rank assumptions.

Contribution

It introduces the spectral replica method, a new analytical framework that simplifies the analysis of high-rank matrix models in dictionary learning using eigenvalue-based representations.

Findings

Derives variational formulas for mutual information in high-rank models.

Provides Coulomb gas representations of mutual information.

Extends analysis to regimes where matrix rank grows linearly with system size.

Abstract

We consider increasingly complex models of matrix denoising and dictionary learning in the Bayes-optimal setting, in the challenging regime where the matrices to infer have a rank growing linearly with the system size. This is in contrast with most existing literature concerned with the low-rank (i.e., constant-rank) regime. We first consider a class of rotationally invariant matrix denoising problems whose mutual information and minimum mean-square error are computable using techniques from random matrix theory. Next, we analyze the more challenging models of dictionary learning. To do so we introduce a novel combination of the replica method from statistical mechanics together with random matrix theory, coined spectral replica method. This allows us to derive variational formulas for the mutual information between hidden representations and the noisy data of the dictionary learning problem, as well as for the overlaps quantifying the optimal reconstruction error. The proposed method reduces the number of degrees of freedom from $\Theta(N^2)$ matrix entries to $\Theta(N)$ eigenvalues (or singular values), and yields Coulomb gas representations of the mutual information which are reminiscent of matrix models in physics. The main ingredients are a combination of large deviation results for random matrices together with a new replica symmetric decoupling ansatz at the level of the probability distributions of eigenvalues (or singular values) of certain overlap matrices and the use of HarishChandra-Itzykson-Zuber spherical integrals.