  Real-world applications such as magnetic resonance imaging with multiple
coils, multi-user communication, and diffuse optical tomography often assume a
linear model where several sparse signals sharing common sparse supports are
acquired by several measurement matrices and then contaminated by noise.
Multi-measurement vector (MMV) problems consider the estimation or
reconstruction of such signals. In different applications, the estimation error
that we want to minimize could be the mean squared error or other metrics such
as the mean absolute error and the support set error. Seeing that minimizing
different error metrics is useful in MMV problems, we study
information-theoretic performance limits for MMV signal estimation with
arbitrary additive error metrics. We also propose a message passing algorithmic
framework that achieves the optimal performance, and rigorously prove the
optimality of our algorithm for a special case. We further conjecture the
optimality of our algorithm for some general cases, and back it up through
numerical examples. As an application of our MMV algorithm, we propose a novel
setup for active user detection in multi-user communication and demonstrate the
promise of our proposed setup.
