Matrix denoising: Bayes-optimal estimators via low-degree polynomials
<|reference_start|>Matrix denoising: Bayes-optimal estimators via low-degree polynomials: 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.<|reference_end|>
@article{semerjian2024matrix,
    title={Matrix denoising: Bayes-optimal estimators via low-degree polynomials},
    author={Guilhem Semerjian},
    journal={J. Stat. Phys. 191, 139 (2024)},
    year={2024},
    doi={10.1007/s10955-024-03359-9},
    archivePrefix={arXiv},
    eprint={2402.16719},
    primaryClass={cond-mat.dis-nn cond-mat.stat-mech cs.IT math.IT math.PR math.ST stat.TH}
}