Sparse and low-rank matrix decompositions

Authors:
Venkat Chandrasekaran;Sujay Sanghavi;Pablo A. Parrilo;Alan S. Willsky
Affiliations:
Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA;Department of Electrical and Computer Engineering, University of Texas-Austin, Austin, TX;Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA;Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA
Venue:
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Year:
2009

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Abstract

We consider the following fundamental problem: given a matrix that is the sum of an unknown sparse matrix and an unknown low-rank matrix, is it possible to exactly recover the two components? Such a capability enables a considerable number of applications, but the goal is both ill-posed and NP-hard in general. In this paper we develop (a) a new uncertainty principle for matrices, and (b) a simple method for exact decomposition based on convex optimization. Our uncertainty principle is a quantification of the notion that a matrix cannot be sparse while having diffuse row/column spaces. It characterizes when the decomposition problem is ill-posed, and forms the basis for our decomposition method and its analysis. We provide deterministic conditions - on the sparse and low-rank components - under which our method guarantees exact recovery.