Joint Sparse Estimation with Cardinality Constraint via Mixed-Integer Semidefinite Programming

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.