An efficient matrix bi-factorization alternative optimization method for low-rank matrix recovery and completion.

In recent years, matrix rank minimization problems have aroused considerable interests from machine learning, data mining and computer vision communities. All of these problems can be solved via their convex relaxations which minimize the trace norm instead of the rank of the matrix, and have to be solved iteratively and involve singular value decomposition (SVD) at each iteration. Therefore, those algorithms for trace norm minimization problems suffer from high computation cost of multiple SVDs. In this paper, we propose an efficient Matrix Bi-Factorization (MBF) method to approximate the original trace norm minimization problem and mitigate the computation cost of performing SVDs. The proposed MBF method can be used to address a wide range of low-rank matrix recovery and completion problems such as low-rank and sparse matrix decomposition (LRSD), low-rank representation (LRR) and low-rank matrix completion (MC). We also present three small scale matrix trace norm models for LRSD, LRR and MC problems, respectively. Moreover, we develop two concrete linearized proximal alternative optimization algorithms for solving the above three problems. Experimental results on a variety of synthetic and real-world data sets validate the efficiency, robustness and effectiveness of our MBF method comparing with the state-of-the-art trace norm minimization algorithms.