  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.
