Sparse and Low-rank Matrix Decomposition via Alternating Direction Methods

The problem of recovering the sparse and low-rank components of a matrix captures a broad spectrum of applications. Authors in [4] proposed the concept of ”rank-sparsity incoherence” to characterize the fundamental identifiability of the recovery, and derived practical sufficient conditions to ensure the high possibility of recovery. This exact recovery is achieved via solving a convex relaxation problem where the l1 norm and the nuclear norm are utilized for being surrogates of the sparsity and low-rank. Numerically, this convex relaxation problem was reformulated into a semi-definite programming (SDP) problem whose dimension is considerably enlarged, and this SDP reformulation was proposed to be solved by generic interior-point solvers in [4]. This paper focuses on the algorithmic improvement for the sparse and low-rank recovery. In particular, we observe that the convex relaxation problem generated by the approach of [4] is actually well-structured in both the objective function and constraint, and it fits perfectly the applicable range of the classical alternating direction method (ADM). Hence, we propose the ADM approach for accomplishing the sparse and low-rank recovery, by taking full exploitation to the high-level separable structure of the convex relaxation problem. Preliminary numerical results are reported to verify the attractive efficiency of the ADM approach for recovering sparse and low-rank components of matrices.