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

Guilhem Semerjian
Condensed Matter, Disordered Systems and Neural Networks, Disordered Systems and Neural Networks (cond-mat.dis-nn), Statistical Mechanics (cond-mat.stat-mech), Information Theory (cs.IT), Probability (math.PR), Statistics Theory (math.ST)
2024-02-26 00:00:00
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 {\bf 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.
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