Exact matrix completion via convex optimization

Communications of the ACM

Volumn 55, Issue 6, 2012, Pages 111-119

  Candès, Emmanuel ^a   Recht, Benjamin ^b  

^a STANFORD UNIVERSITY   (United States)

^b University of Wisconsin Madison   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPRESSIVE SENSING; CONVEX PROGRAMMING PROBLEMS; LIMITED INFORMATION; LOW-RANK MATRICES; MATRIX COMPLETION; TECHNIQUES USED;

COMPUTER APPLICATIONS; COMPUTER SCIENCE;

CONVEX OPTIMIZATION;

EID: 84863765049     PISSN: 00010782     EISSN: 15577317     Source Type: Journal    
DOI: 10.1145/2184319.2184343     Document Type: Article

References (35)

1. Amit, Y., Fink, M., Srebro, N., Ullman, S. Uncovering shared structures in multiclass classifcation. In Proceedings of the Twenty-fourth International Conference on Machine Learning (2007).
2. Argyriou, A., Evgeniou, T., Pontil, M. Multi-task feature learning. In Neural Information Processing Systems, 2007.
3. Beck, C., D'andrea, R. Computational study and comparisons of lFt reducibility methods. In Proceedings of the American Control Conference (1998).
4. Cai, J.F., Candès, E.J., Shen, Z. A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 4 (2008), 1956-1982.
5. Candes, E.J., Li, X., Ma, Y., Wright, J. Robust principal component analysis? submitted for publication. Preprint available at http://www-stat. stanford. edu/∼candes/papers/robustPCa.pdf, 2009.
6. Candès, E.J., Plan, Y. Matrix completion with noise. Proc. IEEE 98, 6 (2009), 925-936.
7. Candès, E.J., Romberg, J. Sparsity and incoherence in compressive sampling. Inverse Probl. 23, 3 (2007), 969-985. (Pubitemid 46838836)
8. Candès, E.J., Romberg, J., Tao, T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theor. 52, 2 (2006), 489-509. (Pubitemid 43174093)
9. Candès, E.J., Tao, T. The power of convex relaxation: near-optimal matrix completion. IEEE Trans. Inform. Theor. 56, 5 (2009), 2053-2080.
10. Chandrasekaran, V., Parrilo, P.A., Willsky, A.S. Latent variable graphical model selection via convex optimization. submitted for publication. Preprint available at http://ssg.mit.edu/∼venkatc/cpw- lgm-preprint10.pdf, 2010.
11. Chandrasekaran, V., Recht, B., Parrilo, P.A., Willsky, A. The convex geometry of linear inverse problems. submitted for publication. Preprint available at arxiv. org/1012.0621, 2010.
12. Chandrasekaran, V., Sanghavi, S., Parrilo, P.A., Willsky, A.S. Rank-sparsity incoherence for matrix decomposition. SIAM J. Optim. 21, 2 (2011), 572-596.
13. Donoho, D.L. Compressed sensing. IEEE Trans. Inform. Theor. 52, 4 (2006), 1289-1306.
14. Fazel, M. Matrix Rank Minimization with Applications. Phd thesis, stanford university, 2002.
15. Frieze, A., Kannan, R., Vempala, S. Fast Monte-Carlo algorithms for fnding low-rank approximations. J. ACM 51, 6 (2004).
16. Funk, S. Netfix update: try this at home, december 2006. http://sifter.org/∼simon/journal/20061211. html.
17. Gross, D. Recovering low-rank matrices from few coeffcients in any basis. IEEE Trans. Inform. Theor. 57 (2011), 1548-1566.
18. Gross, D., Liu, Y.K., Flammia, S.T., Becker, S., Eisert, J. Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 15 (2010), 150401.
19. Halko, N., Martinsson, P.G., Tropp, J.A. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53, 2 (2011), 217-288.
20. Ji, S., Ye, J. An accelerated gradient method for trace norm minimization. In Proceedings of the ICML (2009).
21. Keshavan, R.H., Montanari, A., Oh, S. Matrix completion from a few entries. IEEE Trans. Inform. Theor. 56, 6 (2009), 2980-2998.
22. Keshavan, R.H., Montanari, A., Oh, S. Matrix completion from noisy entries. J. Mach. Learn. Res. 11 (2009), 2057-2078.
23. Koltchinskii, V., Tsybakov, A.B., Lounici, K. Nuclear norm penalization and optimal rates for noisy low rank matrix completion. submitted for publication. Preprint available at arxiv.org/abs/1011.6256, 2010.
24. Koren, Y., Bell, R., Volinsky, C. Matrix factorization techniques for recommender systems. Computer 42, 8 (2009), 30-37.
25. Liberty, E., Woolfe, F., Martinsson, P.G., Rokhlin, V., Tygert, M. Randomized algorithms for the low-rank approximation of matrices. Proc. Nat. Acad. Sci. 104, 51 (2007), 20167-20172.
26. Linial, N., London, E., Rabinovich, Y. The geometry of graphs and some of its algorithmic applications. Combinatorica 15 (1995), 215-245,.
27. Ma, S., Goldfarb, D., Chen, L. Fixed point and bregman iterative methods for matrix rank minimization. Math. Program. 128, 1 (2011), 321-353.
28. Negahban, S., Wainwright, M.J. Restricted strong convexity and weighted matrix completion: optimal bounds with noise. submitted for publication. Preprint available at arxiv.org/abs/1009.2118, 2010.
29. Recht, B. A simpler approach to matrix completion. J. Mach. Learn. Res. (2009). to appear 2012.
30. Preprint available at http://arxiv.org/abs/0910.0651.
31. Recht, B., Fazel, M., Parrilo, P. Guaranteed minimum rank solutions of matrix equations via nuclear norm minimization. SIAM Rev. 52, 3 (2010), 471-501.
32. Rennie, J.D.M., Srebro, N. Fast maximum margin matrix factorization for collaborative prediction. In Proceedings of the International Conference of Machine Learning (2005).
33. So, A.M.C., Ye, Y. Theory of semidefnite programming for sensor network localization. Math. Program., Ser. B 109 (2007), 367-384. (Pubitemid 46197861)
34. Sturm, J.F. Using seduMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Meth. Software 11-12 (1999), 625-653.
35. Toh, K.-C., Yun, S. An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. Pac. J. Math. 6 (2010), 615-640.