Convex-structured covariance estimation via the entropy loss under the majorization-minimization algorithm framework

We estimated convex-structured covariance/correlation matrices by minimizing the entropy loss corresponding to the given matrix. We first considered the estimation of the Weighted sum of known Rank-one matrices with unknown Weights (W-Rank1-W) structural covariance matrices, which appeared commonly in array signal processing tasks, e.g., direction-of-arrival (DOA) estimation. The associated minimization problem is convex and can be solved using the primal-dual interior-point algorithm. However, the objective functions (the entropy loss function) can be bounded above by a sequence of separable functions—we proposed a novel estimation algorithm based on this property under the Majorization-Minimization (MM) algorithmic framework. The proposed MM algorithm exhibited very low computational complexity in each iteration, and its convergence was demonstrated theoretically. Subsequently, we focused on the estimation of Toeplitz autocorrelation matrices, which appeared frequently in time-series analysis. In particular, we considered cases in which the autocorrelation coefficient decreased as the time lag increased. We transformed the Toeplitz structure into a W-Rank1-W structure via special variable substitution, and proposed an MM algorithm similar to that for the W-Rank1-W covariance estimation. However, each MM iteration involved a second-order cone programming SOCP problem that must be resolved. Our numerical experiments demonstrated the high computational efficiency and satisfactory estimation accuracy of the proposed MM algorithms in DOA and autocorrelation matrix estimation.