  The multiple measurement vectors (MMV) problem refers to the joint estimation
of a row-sparse signal matrix from multiple realizations of mixtures with a
known dictionary. As a generalization of the standard sparse representation
problem for a single measurement, this problem is fundamental in various
applications in signal processing, e.g., spectral analysis and
direction-of-arrival (DOA) estimation. In this paper, we consider the maximum a
posteriori (MAP) estimation for the MMV problem, which is classically
formulated as a regularized least-squares (LS) problem with an
$\ell_{2,0}$-norm constraint, and derive an equivalent mixed-integer
semidefinite program (MISDP) reformulation. The proposed MISDP reformulation
can be exactly solved by a generic MISDP solver, which, however, becomes
computationally demanding for problems of extremely large dimensions. To
further reduce the computation time in such scenarios, a relaxation-based
approach can be employed to obtain an approximate solution of the MISDP
reformulation, at the expense of a reduced estimation performance. Numerical
simulations in the context of DOA estimation demonstrate the improved error
performance of our proposed method in comparison to several popular DOA
estimation methods. In particular, compared to the deterministic maximum
likelihood (DML) estimator, which is often used as a benchmark, the proposed
method applied with a state-of-the-art MISDP solver exhibits a superior
estimation performance at a significantly reduced running time. Moreover,
unlike other nonconvex approaches for the MMV problem, including the greedy
methods and the sparse Bayesian learning, the proposed MISDP-based method
offers a guarantee of finding a global optimum.
