Fundamental limits of symmetric low-rank matrix estimation

TL;DR

This paper investigates the fundamental limits of estimating a symmetric low-rank matrix from noisy observations, providing a unified information-theoretic framework that extends to various high-dimensional inference problems.

Contribution

It derives the mutual information and MMSE limits for low-rank symmetric matrix estimation, generalizes to non-Gaussian noise, and connects multiple problems under a universal framework.

Findings

Computed the mutual information and MMSE limits in high dimensions.

Extended results beyond Gaussian noise to broader models.

Unified analysis of PCA, sparse PCA, community detection, and submatrix localization.

Abstract

We consider the high-dimensional inference problem where the signal is a low-rank symmetric matrix which is corrupted by an additive Gaussian noise. Given a probabilistic model for the low-rank matrix, we compute the limit in the large dimension setting for the mutual information between the signal and the observations, as well as the matrix minimum mean square error, while the rank of the signal remains constant. We also show that our model extends beyond the particular case of additive Gaussian noise and we prove an universality result connecting the community detection problem to our Gaussian framework. We unify and generalize a number of recent works on PCA, sparse PCA, submatrix localization or community detection by computing the information-theoretic limits for these problems in the high noise regime. In addition, we show that the posterior distribution of the signal given the observations is characterized by a parameter of the same dimension as the square of the rank of the signal (i.e. scalar in the case of rank one). Finally, we connect our work with the hard but detectable conjecture in statistical physics.