Matrix denoising: Bayes-optimal estimators via low-degree polynomials
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929556099694592 |
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| author | Semerjian, Guilhem |
| author_facet | Semerjian, Guilhem |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16719 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matrix denoising: Bayes-optimal estimators via low-degree polynomials Semerjian, Guilhem Disordered Systems and Neural Networks Statistical Mechanics Information Theory Probability Statistics Theory 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. |
| title | Matrix denoising: Bayes-optimal estimators via low-degree polynomials |
| topic | Disordered Systems and Neural Networks Statistical Mechanics Information Theory Probability Statistics Theory |
| url | https://arxiv.org/abs/2402.16719 |