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 unify and generalize a number of recent works on PCA, sparse PCA, submatrix localization or community detection by computing the informationtheoretic limits for these problems in the high noise regime. This allows to locate precisely the information-theoretic thresholds for the above mentioned problems.1