Fundamental limits of low-rank matrix estimation: the non-symmetric case

TL;DR

This paper analyzes the fundamental limits of estimating a low-rank matrix from noisy observations in high dimensions, identifying the threshold below which accurate recovery is impossible.

Contribution

It computes the mutual information and MMSE limits for non-symmetric low-rank matrix estimation, establishing the information-theoretic threshold in high dimensions.

Findings

Identifies the critical signal strength threshold for recovery

Calculates the mutual information limit in the large dimension limit

Determines the MMSE behavior at the threshold

Abstract

We consider the high-dimensional inference problem where the signal is a low-rank 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. This allows to locate the information-theoretic threshold for this estimation problem, i.e. the critical value of the signal intensity below which it is impossible to recover the low-rank matrix.