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Published June 30, 2011 | Submitted + Published
Journal Article Open

Rank-Sparsity Incoherence for Matrix Decomposition


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 ℓ_1 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 we 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.

Additional Information

© 2011 Society for Industrial and Applied Mathematics. Received by the editors June 11, 2009; accepted for publication (in revised form) November 8, 2010; published electronically June 30, 2011. This work was supported by MURI AFOSR grant FA9550-06-1-0324, MURI AFOSR grant FA9550-06-1-0303, and NSF FRG 0757207. A preliminary version of this work appeared in Sparse and low-rank matrix decompositions, in Proceedings of the 15th IFAC Symposium on System Identification, 2009.

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Published - cspw_slr_siopt11.pdf

Submitted - 0906.2220.pdf


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