A non-convex algorithm framework based on DC programming and DCA for matrix completion

Matrix completion aims to recover an unknown low-rank or approximately low-rank matrix from a sampling set of its entries. It is shown that this problem can be solved via its tightest convex relaxation obtained by minimizing the nuclear norm instead of the rank function. Recent studies have also shown that some non-convex penalties like M (p) minimization and weighted nuclear norm minimization algorithms are able to recover low-rank matrix in a more efficient way. In this paper, we propose a unified framework based on Difference of Convex functions (DC) programming and DC Algorithms (DCA), by which M (p) minimization and weighted nuclear norm minimization algorithms can be obtained as special cases of the general framework. In addition, we give another non-convex penalty-exponential type penalty. We make some comparison between numerical tests of our algorithms and the state-of-the-art method APGL and NIHT on randomly generated matrices and real matrix completion problems, the results suggest that our methods are more effective and promising. Moreover, for the application on low-rank image recovery, these non-convex algorithms we proposed also perform well and the results are more satisfactory and reasonable.