Sparse and Low-Rank Matrix Decompositions

Abstract

Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification, but obtaining an exact solution is NP-hard in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components; in fact our approach reduces to solving a semidefinite program. We provide sufficient conditions that guarantee exact recovery of the components by solving the semidefinite program. We also show that when the sparse and low-rank matrices are drawn from certain natural random ensembles, these sufficient conditions are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems.