Alternating direction and Taylor expansion minimization algorithms for unconstrained nuclear norm optimization

In the past decade, robust principal component analysis (RPCA) and low-rank matrix completion (LRMC), as two very important optimization problems with the view of recovering original low-rank matrix from sparsely and highly corrupted observations or a subset of its entries, have already been successfully adopted in image denoising, video processing, web search, biological information, etc. This paper proposes an efficient and effective algorithm, named the alternating direction and step size minimization (ADSM) algorithm, which employs the alternating direction minimization idea to solve the general relaxed model that can describe small noise (e.g., Gaussian noise). The coupling of sparse noise and small noise makes low-rank matrix recovery more challenging than that of RPCA. We make use of Taylor expansion, singular value decomposition and shrinkage operator as the alternating direction minimization method to deduce iterative direction matrices. A continuous technology is incorporated into ADSM to accelerate convergence. Similarly, the Taylor expansion and step size minimization (TESM) algorithm for LRMC is designed by the above way, but the alternating direction minimization idea needs to be ruled out since there is not a sparse matrix in it. Theoretically, it is proved that the two algorithms globally converge to their respective optimal points based on some conditions. The numerical results are reported, illustrating that ADSM and TESM are quite efficient and effective for recovering large-scale low-rank matrix problems at many cases.