Rotational invariant estimator for general noisy matrices

TL;DR

This paper develops a rotational invariant estimator for symmetric matrices corrupted by arbitrary rotationally invariant noise, providing asymptotic laws, eigenvector deviations, and practical eigenvalue cleaning methods.

Contribution

It extends existing results by deriving the asymptotic behavior of noisy symmetric matrices under general rotational invariant perturbations using the Replica method.

Findings

Asymptotic global law estimates for noisy matrices

Exact eigenvector deviation (overlap) results

Practical eigenvalue cleaning techniques

Abstract

We investigate the problem of estimating a given real symmetric signal matrix $\textbf{C}$ from a noisy observation matrix $\textbf{M}$ in the limit of large dimension. We consider the case where the noisy measurement $\textbf{M}$ comes either from an arbitrary additive or multiplicative rotational invariant perturbation. We establish, using the Replica method, the asymptotic global law estimate for three general classes of noisy matrices, significantly extending previously obtained results. We give exact results concerning the asymptotic deviations (called overlaps) of the perturbed eigenvectors away from the true ones, and we explain how to use these overlaps to "clean" the noisy eigenvalues of $\textbf{M}$. We provide some numerical checks for the different estimators proposed in this paper and we also make the connection with some well known results of Bayesian statistics.