Improving M-SBL for Joint Sparse Recovery Using a Subspace Penalty

A multiple measurement vector problem (MMV) is a generalization of the compressed sensing problem that addresses the recovery of a set of jointly sparse signal vectors. One of the important contributions of this paper is to show that the seemingly least related state-of-the-art MMV joint sparse recovery algorithms - the M-SBL (multiple sparse Bayesian learning) and subspace-based hybrid greedy algorithms - have a very important link. More specifically, we show that replacing the log det(·) term in the M-SBL by a rank surrogate that exploits the spark reduction property discovered in the subspace-based joint sparse recovery algorithms provides significant improvements. In particular, if we use the Schatten-p quasi-norm as the corresponding rank surrogate, the global minimizer of the cost function in the proposed algorithm becomes identical to the true solution as p → 0. Furthermore, under regularity conditions, we show that convergence to a local minimizer is guaranteed using an alternating minimization algorithm that has closed form expressions for each of the minimization steps, which are convex. Numerical simulations under a variety of scenarios in terms of SNR and the condition number of the signal amplitude matrix show that the proposed algorithm consistently outperformed the M-SBL and other state-of-the art algorithms.