Rank-Sparsity Incoherence for Matrix Decomposition

Abstract

Suppose we are given a matrix that is formed by adding an unknown sparse\nmatrix to an unknown low-rank matrix. Our goal is to decompose the given matrix\ninto its sparse and low-rank components. Such a problem arises in a number of\napplications in model and system identification, and is NP-hard in general. In\nthis paper we consider a convex optimization formulation to splitting the\nspecified matrix into its components, by minimizing a linear combination of the\n$\\ell_1$ norm and the nuclear norm of the components. We develop a notion of\n\\emph{rank-sparsity incoherence}, expressed as an uncertainty principle between\nthe sparsity pattern of a matrix and its row and column spaces, and use it to\ncharacterize both fundamental identifiability as well as (deterministic)\nsufficient conditions for exact recovery. Our analysis is geometric in nature,\nwith the tangent spaces to the algebraic varieties of sparse and low-rank\nmatrices playing a prominent role. When the sparse and low-rank matrices are\ndrawn from certain natural random ensembles, we show that the sufficient\nconditions for exact recovery are satisfied with high probability. We conclude\nwith simulation results on synthetic matrix decomposition problems.\n