  In this work, a Bayesian approximate message passing algorithm is proposed
for solving the multiple measurement vector (MMV) problem in compressive
sensing, in which a collection of sparse signal vectors that share a common
support are recovered from undersampled noisy measurements. The algorithm,
AMP-MMV, is capable of exploiting temporal correlations in the amplitudes of
non-zero coefficients, and provides soft estimates of the signal vectors as
well as the underlying support. Central to the proposed approach is an
extension of recently developed approximate message passing techniques to the
amplitude-correlated MMV setting. Aided by these techniques, AMP-MMV offers a
computational complexity that is linear in all problem dimensions. In order to
allow for automatic parameter tuning, an expectation-maximization algorithm
that complements AMP-MMV is described. Finally, a detailed numerical study
demonstrates the power of the proposed approach and its particular suitability
for application to high-dimensional problems.
