Fundamental limits of symmetric low-rank matrix estimation

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

Phase transitions in spiked matrix estimation: information-theoretic analysis

We study here the so-called spiked Wigner and Wishart models, where one observes a low-rank matrix perturbed by some Gaussian noise. These models encompass many classical statistical tasks such as sparse PCA, submatrix localization, community detection or Gaussian mixture clustering. The goal of these notes is to present in a unified manner recent results (as well as new developments) on the information-theoretic limits of these spiked matrix models. We compute the minimal mean squared error for the estimation of the low-rank signal and compare it to the performance of spectral estimators and message passing algorithms. Phase transition phenomena are observed: depending on the noise level it is either impossible, easy (i.e. using polynomial-time estimators) or hard (information-theoretically possible, but no efficient algorithm is known to succeed) to recover the signal.Comment: These notes present in a unified manner recent results (as well as new developments) on the information-theoretic limits in spiked matrix estimatio