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.

翻译：在多测量向量（MMV）模型中，测量向量通过采用单一已知测量矩阵（或字典）的线性回归模型与未知的联合稀疏信号向量相关联。通常，字典原子（字典的列）数量多于测量数量，稀疏信号恢复问题通常是不适定的。本文中，我们将信号和测量噪声分别视为具有未知信号协方差矩阵和噪声方差的高斯随机向量，并通过固定点方程刻画似然方程的解，从而利用似然方程的固定点特性，通过块坐标下降（BCD）算法实现稀疏信号支撑（具有非零方差的源）的恢复。此外，本文还引入了一种类似于经典同步正交匹配追踪（OMP）的贪婪追踪方法。数值实验表明，所提出的协方差学习（CL）算法在经典稀疏信号恢复和波达方向（DOA）估计问题中均表现出有效性，在多种设置下其性能优于当前最先进的算法。