Sparse and low-rank matrix decompositions

Abstract

We consider the following fundamental problem: given a matrix that is the sum of an unknown sparse matrix and an unknown low-rank matrix, is it possible to exactly recover the two components? Such a capability enables a considerable number of applications, but the goal is both ill-posed and NP-hard in general. In this paper we develop (a) a new uncertainty principle for matrices, and (b) a simple method for exact decomposition based on convex optimization. Our uncertainty principle is a quantification of the notion that a matrix cannot be sparse while having diffuse row/column spaces. It characterizes when the decomposition problem is ill-posed, and forms the basis for our decomposition method and its analysis. We provide deterministic conditions — on the sparse and low-rank components — under which our method guarantees exact recovery.