Augmented Lagrangian alternating direction method for low-rank minimization via non-convex approximation

This paper concerns the low-rank minimization problems which consist of finding a matrix of minimum rank subject to linear constraints. Many existing approaches, which used the nuclear norm as a convex surrogate of the rank function, usually result in a suboptimal solution. To seek a tighter rank approximation, we develop a non-convex surrogate to approximate the rank function based on the Laplace function. An iterative algorithm based on the augmented Lagrangian multipliers method is developed. Empirical studies for practical applications including robust principal component analysis and low-rank representation demonstrate that our proposed algorithm outperforms many other state-of-the-art convex and non-convex methods developed recently in the literature.