Probability Theory and Related Fields

, Volume 173, Issue 3–4, pp 859–929 | Cite as

Fundamental limits of symmetric low-rank matrix estimation

  • Marc Lelarge
  • Léo MiolaneEmail author


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 squared 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). This allows to locate precisely the information-theoretic thresholds for the above mentioned problems. Finally, we connect our work with the hard but detectable conjecture in statistical physics.

Mathematics Subject Classification

60B20 62B10 60G57 62M15 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département d’informatique de l’ENS, École normale supérieure, CNRSPSL Research UniversityParisFrance
  2. 2.InriaParisFrance

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