Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods

TL;DR

This paper establishes the fundamental limits and proposes optimal spectral methods for matrix denoising in the presence of doubly heteroscedastic noise, advancing understanding beyond previous one-sided heteroscedastic models.

Contribution

It characterizes the exact asymptotic MMSE and introduces a spectral estimator that achieves optimality under doubly heteroscedastic noise conditions.

Findings

The spectral estimator attains positive correlation with signals when information-theoretically possible.

Under one-sided heteroscedasticity, it achieves the Bayes-optimal error.

Numerical experiments show significant improvements over existing methods.

Abstract

We study the matrix denoising problem of estimating the singular vectors of a rank-$1$ signal corrupted by noise with both column and row correlations. Existing works are either unable to pinpoint the exact asymptotic estimation error or, when they do so, the resulting approaches (e.g., based on whitening or singular value shrinkage) remain vastly suboptimal. On top of this, most of the literature has focused on the special case of estimating the left singular vector of the signal when the noise only possesses row correlation (one-sided heteroscedasticity). In contrast, our work establishes the information-theoretic and algorithmic limits of matrix denoising with doubly heteroscedastic noise. We characterize the exact asymptotic minimum mean square error, and design a novel spectral estimator with rigorous optimality guarantees: under a technical condition, it attains positive correlation with the signals whenever information-theoretically possible and, for one-sided heteroscedasticity, it also achieves the Bayes-optimal error. Numerical experiments demonstrate the significant advantage of our theoretically principled method with the state of the art. The proofs draw connections with statistical physics and approximate message passing, departing drastically from standard random matrix theory techniques.