Mismatched estimation of non-symmetric rank-one matrices corrupted by structured noise

TL;DR

This paper analyzes the performance of mismatched Bayesian and AMP estimators in estimating a rank-one matrix corrupted by structured noise, revealing a performance gap due to incorrect signal norm estimation.

Contribution

It provides the exact error formula for the mismatched Bayes estimator and analyzes an AMP algorithm in a non-symmetric noise setting.

Findings

Exact error expression for mismatched Bayes estimator

Performance gap identified between AMP and Bayes estimators

Performance gap attributed to incorrect signal norm estimation

Abstract

We study the performance of a Bayesian statistician who estimates a rank-one signal corrupted by non-symmetric rotationally invariant noise with a generic distribution of singular values. As the signal-to-noise ratio and the noise structure are unknown, a Gaussian setup is incorrectly assumed. We derive the exact analytic expression for the error of the mismatched Bayes estimator and also provide the analysis of an approximate message passing (AMP) algorithm. The first result exploits the asymptotic behavior of spherical integrals for rectangular matrices and of low-rank matrix perturbations; the second one relies on the design and analysis of an auxiliary AMP. The numerical experiments show that there is a performance gap between the AMP and Bayes estimators, which is due to the incorrect estimation of the signal norm.