Fundamental limits of symmetric low-rank matrix estimation

Marc Lelarge 1, 2 Léo Miolane 2
2 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique : UMR 8548, Inria de Paris
Abstract : We consider the high-dimensional inference problem where the signal is a low-rank symmetric matrix which is corrupted by an additive Gaussian noise. Given a probabilistic model for the low-rank matrix, we compute the limit in the large dimension setting for the mutual information between the signal and the observations, as well as the matrix minimum mean square error, while the rank of the signal remains constant. We also show that our model extends beyond the particular case of additive Gaussian noise and we prove an universality result connecting the community detection problem to our Gaussian framework. We unify and generalize a number of recent works on PCA, sparse PCA, submatrix localization or community detection by computing the information-theoretic limits for these problems in the high noise regime. In addition, we show that the posterior distribution of the signal given the observations is characterized by a parameter of the same dimension as the square of the rank of the signal (i.e. scalar in the case of rank one). Finally, we connect our work with the hard but detectable conjecture in statistical physics.
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https://hal.archives-ouvertes.fr/hal-01648368
Contributor : Léo Miolane <>
Submitted on : Saturday, November 25, 2017 - 4:20:52 PM
Last modification on : Wednesday, January 30, 2019 - 11:07:32 AM

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  • HAL Id : hal-01648368, version 1
  • ARXIV : 1611.03888

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Marc Lelarge, Léo Miolane. Fundamental limits of symmetric low-rank matrix estimation. 2017. ⟨hal-01648368⟩

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