An efficient semi-proximal ADMM algorithm for low-rank and sparse regularized matrix minimization problems with real-world applications

Abstract

With the development of technology and the arrival of the era of big data, a large number of complex data structures have been generated, which makes matrix minimization become important and necessary. However, in the process of analyzing those data, there exist some latent structures such as low-rank and sparse decomposition. To address this issue, in this paper, we propose and study a novel low-rank and sparse regularized matrix minimization. Then an efficient semi-proximal alternating direction method of multipliers (sPADMM) is developed from the dual perspective. In theoretical analysis, it has proved that the sequence generated by the sPADMM converges to a global minimizer. Furthermore, the non-asymptotic statistical property is derived which suggests that the gap between the corresponding estimators and the true values is bounded with high probability and their consistency is guaranteed under rather weak regularity conditions. We also prove that the sequence generated by the proposed algorithm converges to the true value with high probability. Finally, extensive numerical experiments are conducted to verify the superiority of the proposed method in wide applications such as signal processing, image reconstruction, and video denoising.