The Role of Rank in Mismatched Low-Rank Symmetric Matrix Estimation

TL;DR

This paper analyzes how mismatches in assumed parameters affect the performance of Bayesian estimators in recovering low-rank matrices from noisy data, providing explicit formulas for the asymptotic MSE.

Contribution

It derives an explicit formula for the asymptotic MSE of Bayesian estimators under rank, signal power, and SNR mismatches in low-rank matrix estimation.

Findings

Rank mismatch significantly impacts the estimator's MSE.

Explicit asymptotic MSE formulas are provided for various mismatches.

Analysis uses GOE spectrum and spherical integrals techniques.

Abstract

We investigate the performance of a Bayesian statistician tasked with recovering a rank-\(k\) signal matrix \(\bS \bS^{\top} \in \mathbb{R}^{n \times n}\), corrupted by element-wise additive Gaussian noise. This problem lies at the core of numerous applications in machine learning, signal processing, and statistics. We derive an analytic expression for the asymptotic mean-square error (MSE) of the Bayesian estimator under mismatches in the assumed signal rank, signal power, and signal-to-noise ratio (SNR), considering both sphere and Gaussian signals. Additionally, we conduct a rigorous analysis of how rank mismatch influences the asymptotic MSE. Our primary technical tools include the spectrum of Gaussian orthogonal ensembles (GOE) with low-rank perturbations and asymptotic behavior of \(k\)-dimensional spherical integrals.