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

We investigate the performance of a Bayesian statistician tasked with recovering a rank-\ (k\) signal matrix \ (^ R^n 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.