Linearly Involved Generalized Moreau Enhanced Models and Their Proximal Splitting Algorithm under Overall Convexity Condition

Abstract: The convex envelopes of the direct discrete measures, for the sparsity of vectors or for the low-rankness of matrices, have been utilized extensively as practical penalties in order to compute a globally optimal solution of the corresponding regularized least-squares models. Motivated mainly by the ideas in [Zhang'10, Selesnick'17, Yin, Parekh, Selesnick'19] to exploit nonconvex penalties in the regularized least-squares models without losing their overall convexities, this paper presents the Linearly involved Generalized Moreau Enhanced (LiGME) model as a unified extension of such utilizations of nonconvex penalties. The proposed model can admit multiple nonconvex penalties without losing its overall convexity and thus is applicable to much broader scenarios in the sparsity-rank-aware signal processing. Under the general overall-convexity condition of the LiGME model, we also present a novel proximal splitting type algorithm of guaranteed convergence to a globally optimal solution. Numerical experiments in typical examples of the sparsity-rank-aware signal processing demonstrate the effectiveness of the LiGME models and the proposed proximal splitting algorithm.