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Freeness over the diagonal and outliers detection in deformed random matrices with a variance profile

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arxiv 1907.07753 v4 pith:TZ5FSNC6 submitted 2019-07-17 math.ST math.PRstat.TH

Freeness over the diagonal and outliers detection in deformed random matrices with a variance profile

classification math.ST math.PRstat.TH
keywords matrixvarianceprofiledistributionrandomadditiveapproachdeformed
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We study the eigenvalue distribution of a GUE matrix with a variance profile that is perturbed by an additive random matrix that may possess spikes. Our approach is guided by Voiculescu's notion of freeness with amalgamation over the diagonal and by the notion of deterministic equivalent. This allows to derive a fixed point equation to approximate the spectral distribution of certain deformed GUE matrices with a variance profile and to characterize the location of potential outliers in such models in a non-asymptotic setting. We also consider the singular values distribution of a rectangular Gaussian random matrix with a variance profile in a similar setting of additive perturbation. We discuss the application of this approach to the study of low-rank matrix denoising models in the presence of heteroscedastic noise, that is when the amount of variance in the observed data matrix may change from entry to entry. Numerical experiments are used to illustrate our results.

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