Matrix denoising: Bayes-optimal estimators via low-degree polynomials

TL;DR

This paper derives the Bayes-optimal polynomial estimators for symmetric matrix denoising under orthogonally invariant priors, showing they converge to a known estimator and conjecturing broader universality.

Contribution

It introduces asymptotically optimal polynomial estimators for matrix denoising using free probability, extending previous results and conjecturing universality beyond orthogonally invariant priors.

Findings

Optimal polynomial estimators converge to a known estimator as degree increases.

The approach uses free probability theory for asymptotic analysis.

Partial evidence suggests universality beyond invariant priors.

Abstract

We consider the additive version of the matrix denoising problem, where a random symmetric matrix $S$ of size $n$ has to be inferred from the observation of $Y=S+Z$, with $Z$ an independent random matrix modeling a noise. For prior distributions of $S$ and $Z$ that are invariant under conjugation by orthogonal matrices we determine, using results from first and second order free probability theory, the Bayes-optimal (in terms of the mean square error) polynomial estimators of degree at most $D$, asymptotically in $n$, and show that as $D$ increases they converge towards the estimator introduced by Bun, Allez, Bouchaud and Potters in [IEEE Transactions on Information Theory 62, 7475 (2016)]. We conjecture that this optimality holds beyond strictly orthogonally invariant priors, and provide partial evidences of this universality phenomenon when $S$ is an arbitrary Wishart matrix and $Z$ is drawn from the Gaussian Orthogonal Ensemble, a case motivated by the related extensive rank matrix factorization problem.