Rank-Sparsity Incoherence for Matrix Decomposition

TL;DR

This paper introduces a convex optimization approach for decomposing a matrix into sparse and low-rank components, using a new incoherence measure to characterize when exact recovery is possible, supported by theoretical analysis and simulations.

Contribution

It proposes the rank-sparsity incoherence concept and provides deterministic and probabilistic conditions for exact matrix decomposition.

Findings

Sufficient conditions for exact recovery are satisfied with high probability under certain random models.

The geometric analysis involves tangent spaces of algebraic varieties of sparse and low-rank matrices.

Simulation results validate the theoretical findings on synthetic data.

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, and is NP-hard in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components, by minimizing a linear combination of the $\ell_1$ norm and the nuclear norm of the components. We develop a notion of \emph{rank-sparsity incoherence}, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature, with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems.