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 derive fixed point (FP) equation for solving the likelihood equation for signal powers, thereby enabling the recovery of the sparse signal support (sources with non-zero variances). Two practical algorithms, a block coordinate descent (BCD) and a cyclic coordinate descent (CCD) algorithms, that leverage on the FP characterization of the likelihood equation are then proposed. 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，测量向量与未知的联合稀疏信号向量通过采用单一已知测量矩阵（或字典）的线性回归模型相关联。通常，原子（字典列）的数量大于测量次数，因此稀疏信号恢复问题通常是不适定的。本文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独立高斯随机向量，推导出用于求解信号功率似然方程的定点（FP）方程，从而恢复稀疏信号支持（具有非零方差的源）。随后提出了两种实用算法：基于似然方程FP特征的块坐标下降（BCD）算法和循环坐标下降（CCD）算法。此外，还引入了一种类似于流行的同步正交匹配追踪（OMP）的贪婪追踪方法。我们的数值算例表明，所提出的协方差学习（CL）算法在经典稀疏信号恢复以及波达方向（DOA）估计问题中均具有有效性，在多种设置下其性能优于现有主流算法。