Sparse signal recovery and source localization via covariance learning

In the Multiple Measurements Vector (MMV) model, measurement vectors are connected to unknown, jointly sparse signal vectors through a linear regression model employing a single known measurement matrix (or dictionary). Typically, the number of atoms (columns of the dictionary) is greater than the number measurements and the sparse signal recovery problem is generally ill-posed. In this paper, we treat the signals and measurement noise as independent Gaussian random vectors with unknown signal covariance matrix and noise variance, respectively, and characterize the solution of the likelihood equation in terms of fixed point equation, thereby enabling the recovery of the sparse signal support (sources with non-zero variances) via a block coordinate descent (BCD) algorithm that leverage the FP characterization of the likelihood equation. Additionally, a greedy pursuit method, analogous to popular simultaneous orthogonal matching pursuit (OMP), is introduced. Our numerical examples demonstrate effectiveness of the proposed covariance learning (CL) algorithms both in classic sparse signal recovery as well as in direction-of-arrival (DOA) estimation problems where they perform favourably compared to the state-of-the-art algorithms under a broad variety of settings.