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In this paper, we present a novel approach to the low rank matrix recovery(LRMR) problem by casting it as a group sparsity problem. Specifically, wepropose a flexible group sparse regularizer (FLGSR) that can group any numberof matrix columns as a unit, whereas existing methods group each column as aunit. We prove the equivalence between the matrix rank and the FLGSR under somemild conditions, and show that the LRMR problem with either of them has thesame global minimizers. We also establish the equivalence between the relaxedand the penalty formulations of the LRMR problem with FLGSR. We then propose aninexact restarted augmented Lagrangian method, which solves each subproblem byan extrapolated linearized alternating minimization method. We analyze theconvergence of our method. Remarkably, our method linearizes each group of thevariable separately and uses the information of the previous groups to solvethe current group within the same iteration step. This strategy enables ouralgorithm to achieve fast convergence and high performance, which are furtherimproved by the restart technique. Finally, we conduct numerical experiments onboth grayscale images and high altitude aerial images to confirm thesuperiority of the proposed FLGSR and algorithm.

Efficient Low Rank Matrix Recovery With Flexible Group Sparse Regularization