Rank-sparsity incoherence for matrix decomposition

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Rank-sparsity incoherence for matrix decomposition (2010)

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

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230 - 21 self

BibTeX

@TECHREPORT{Chandrasekaran10rank-sparsityincoherence,
    author = {Venkat Chandrasekaran and Sujay Sanghavi and Pablo A. Parrilo and Alan S. Willsky},
    title = {Rank-sparsity incoherence for matrix decomposition},
    institution = {},
    year = {2010}
}

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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.

Keyphrases

rank-sparsity incoherence matrix decomposition low-rank matrix sufficient condition exact recovery convex optimization formulation unknown sparse matrix low-rank component synthetic matrix decomposition problem sparsity pattern certain natural random ensemble column space fundamental identifiability algebraic variety system identification high probability nuclear norm unknown low-rank matrix linear combination prominent role tangent space simulation result uncertainty principle

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