Title: Fundamental limits of low-rank matrix estimation: the non-symmetric case,
Abstract: We consider the high-dimensional inference problem where the signal is a
low-rank 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. This allows to locate the information-theoretic
threshold for this estimation problem, i.e. the critical value of the signal
intensity below which it is impossible to recover the low-rank matrix.