Sparse and Low-Rank Matrix Decomposition Via Alternating Direction Method

The problem of recovering sparse and low-rank components of a given matrix captures a broad spectrum of applications. However, this recovery problem is NP-hard and thus not tractable in general. Recently, it was shown in [3, 6] that this recovery problem can be well approached by solving a convex relaxation problem where the l1-norm and the nuclear norm are used to induce sparse and low-rank structures, respectively. Generic interior-point solvers can be applied to solve a semi-definite programming reformulation of the convex relaxation problem. However, the interior-point approach can only handle small cases on personal computers where the dimension of the matrix to be decomposed is less than 100. In this paper, we propose the use of the alternating direction method (ADM) to solve the convex relaxation problem. We show by numerical results that the ADM approach is easily implementable and computationally efficient for solving the sparse and low-rank matrix decomposition problem.