Alternating Direction Method of Multipliers Based on $\ell_{2,0}$-Norm for Multiple Measurement Vector Problem

The multiple measurement vector (MMV) problem is an extension of the single measurement vector (SMV) problem, and it has many applications. Nowadays, most studies of the MMV problem are based on the
$\boldsymbol{\ell}_{\mathbf{2,1}}$
-norm relaxation, which will fail in recovery under some adverse conditions. We propose an alternating direction method of multipliers (ADMM)-based optimization algorithm to achieve a larger undersampling rate for the MMV problem. The key innovation is the introduction of an
$\boldsymbol{\ell}_{\mathbf{2,0}}$
-norm sparsity constraint to describe the joint-sparsity of the MMV problem; this differs from the
$\boldsymbol{\ell}_{\mathbf{2,1}}$
-norm constraint that has been widely used in previous studies. To illustrate the advantages of the
$\boldsymbol{\ell}_{\mathbf{2,0}}$
-norm, we first prove the equivalence of the sparsity of the row support set of a matrix and its
$\boldsymbol{\ell}_{\mathbf{2,0}}$
-norm. Then, the MMV problem based on the
$\boldsymbol{\ell}_{\mathbf{2,0}}$
-norm is proposed. Next, we give our algorithm called MMV-ADMM-
$\boldsymbol{\ell}_{\mathbf{2,0}}$
by applying ADMM to the reformulated problem. Moreover, based on the Kurdyka-Lojasiewicz property of objective functions, we prove that the iteration generated by the proposed algorithm globally converges to the optimal solution of the MMV problem. Finally, the performance of the proposed algorithm and comparisons with other algorithms under different conditions are studied with simulated examples. The results show that the proposed algorithm can solve a larger range of MMV problems even under adverse conditions.