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Rank-sparsity incoherence for matrix decomposition

By Venkat Chandrasekaran, Sujay Sanghavi, Pablo A. Parrilo and Alan S. Willsky

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 intractable to solve 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 ℓ1 norm and the nuclear norm of the components. We develop a notion of 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

Topics: Key words. matrix decomposition, convex relaxation, ℓ1 norm minimization, nuclear norm minimization
Year: 2010
OAI identifier: oai:CiteSeerX.psu:10.1.1.190.2637
Provided by: CiteSeerX
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