The Power of Convex Relaxation: Near-Optimal Matrix Completion
- Creators
- Candès, Emmanuel J.
- Tao, Terence
Abstract
This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number of entries is impossible, but the knowledge that the unknown matrix has low rank radically changes this premise, making the search for solutions meaningful. This paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). This convex program simply finds, among all matrices consistent with the observed entries, that with minimum nuclear norm. As an example, we show that on the order of nr log(n) samples are needed to recover a random n x n matrix of rank r by any method, and to be sure, nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog(n).
Additional Information
© 2010 IEEE. Manuscript received March 11, 2009; revised August 12, 2009. Current version published April 21, 2010. E. J. Candès was supported in part by ONR grants N00014-09-1-0469 and N00014-08-1-0749 and in part by the NSF Waterman Award. T. Tao was supported in part by a grant from the MacArthur Foundation, in part by NSF grant DMS-0649473, and in that part by the NSF Waterman Award. E. J. Candès would like to thank X. Li and C. Sabatti for helpful conversations related to this project. The authors would also like to thank S. Gandy and the anonymous referees for a very careful reading and for suggesting corrections.Attached Files
Published - Candes2010p10336Ieee_T_Inform_Theory.pdf
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Additional details
- Eprint ID
- 18696
- Resolver ID
- CaltechAUTHORS:20100615-154111770
- N00014-09-1-0469
- Office of Naval Research (ONR)
- N00014-08-1-0749
- Office of Naval Research (ONR)
- MacArthur Foundation
- DMS-0649473
- NSF
- Created
-
2010-06-17Created from EPrint's datestamp field
- Updated
-
2021-11-08Created from EPrint's last_modified field
- Other Numbering System Name
- INSPEC Accession Number
- Other Numbering System Identifier
- 11256628