Sparse and low-rank matrix decompositions
Name
Chandrasekaran-2009-Sparse and low-rank matrix decompositions.pdf
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223.35 KB
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Author(s)
Chandrasekaran, Venkat
Sanghavi, Sujay
Parrilo, Pablo A.
Willsky, Alan S.
Date Issued
January 2010
Journal
47th Annual Allerton Conference on Communication, Control, and Computing, 2009
Publisher
Institute of Electrical and Electronics Engineers
Citation
Chandrasekaran, V. et al. “Sparse and low-rank matrix decompositions.” Communication, Control, and Computing, 2009. Allerton 2009. 47th Annual Allerton Conference on. 2009. 962-967. © 2009 IEEE
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1109/ALLERTON.2009.5394889
