Sparse and low-rank matrix decompositions
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...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Institute of Electrical and Electronics Engineers
2010
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| Online Access: | http://hdl.handle.net/1721.1/58946 https://orcid.org/0000-0003-1132-8477 https://orcid.org/0000-0003-0149-5888 |
| Summary: | 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. |
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