Signal recovery using expectation consistent approximation for linear observations

A signal recovery scheme is developed for linear observation systems based on expectation consistent (EC) mean field approximation. Approximate message passing (AMP) is known to be consistent with the results obtained using the replica theory, which is supposed to be exact in the large system limit, when each entry of the observation matrix is independently generated from an identical distribution. However, this is not necessarily the case for general matrices. We show that EC recovery exhibits consistency with the replica theory for a wider class of random observation matrices. This is numerically confirmed by experiments for the Bayesian optimal signal recovery of compressed sensing using random row-orthogonal matrices.