Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements

IEEE Transactions on Information Theory

Volumn 57, Issue 4, 2011, Pages 2342-2359

  Candès, Emmanuel J ^a   Plan, Yaniv ^a,b  

^a STANFORD UNIVERSITY   (United States)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Convex optimization; Dantzig selector; matrix completion; norm of random matrices; oracle inequalities and semidefinite programming

Indexed keywords

COLUMN SPACE; DANTZIG SELECTOR; DEGREE OF FREEDOM; ERROR BOUND; HIGH PROBABILITY; LINEAR COMBINATIONS; LOW-RANK MATRICES; MATRIX; MATRIX COMPLETION; MINIMAX RISK; NOISY DATA; NOISY MEASUREMENTS; NORM OF RANDOM MATRICES; ORACLE INEQUALITIES AND SEMIDEFINITE PROGRAMMING; RANDOM MEASUREMENT; SINGULAR VALUES; THEORETICAL RESULT;

CONVEX OPTIMIZATION; ERROR ANALYSIS; OPTIMIZATION; RECOVERY;

MATRIX ALGEBRA;

EID: 79952823272     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2011.2111771     Document Type: Article

References (33)

1. E. J. Candès and T. Tao, "The Dantzig selector: Statistical estimation when n is much larger than p," Ann. Statist., vol. 35, no. 6, pp. 2313-2351, 2007.
2. P. Bickel, Y. Ritov, and A. Tsybakov, "Simultaneous analysis of Lasso and Dantzig selector," Ann. Statist., vol. 37, no. 4, pp. 1705-1732, Aug. 2009.
3. B. Recht, M. Fazel, and P. Parrilo, "Guaranteed minimum rank solutions of matrix equations via nuclear norm minimization," SIAM Rev., 2007, submitted for publication.
4. E. J. Candes and B. Recht, "Exact matrix completion via convex optimization," Found. Comput. Math., vol. 9, no. 6, pp. 717-772, Dec. 2009.
5. E. J. Candès and T. Tao, The Power of Convex Relaxation: Nearoptimal Matrix Completion, 2009 [Online]. Available: http://arxiv.org/ abs/0903.1476
6. E. Candès and Y. Plan, "Matrix completion with noise," Proc. IEEE, 2009.
7. R. Keshavan, S. Oh, and A. Montanari, "Matrix Completion From a Few Entries," in Proc. Int. Symp. Information Theory, 2009, vol. 9.
8. R. Keshavan, A. Montanari, and S. Oh, Matrix completion from noisy entries, 2009 [Online]. Available: http://arxiv.org/abs/0906.2027
9. K. Lee et al., Admira: Atomic Decomposition for Minimum Rank Approximation, 2009 [Online]. Available: http://arxiv.org/PS-cache/arxiv/pdf/ 0905/0905.0044v2.pdf
10. E. J. Candès, X. Li, Y. Ma, and J. Wright, Robust Principal Component Analysis? Stanford Univ., Stanford, CA, 2009 [Online]. Available: http://statistics.stanford.edu/~ckirby/techreports/ GEN/2009/2009-13.pdf
11. R. Jain and I. Dhillon, Guaranteed Rank Minimization via Singular Value Projection, 2009 [Online]. Available: http://arxiv.org/PS-cache/ arxiv/pdf/0909/0909.5457v3.pdf
12. D. Gross, Y. Liu, S. Flammia, S. Becker, and J. Eisert, Quantum State Tomography via Compressed Sensing, 2009 [Online].Available: http:// arxiv.org/PS-cache/arxiv/pdf/0909/ 0909.3304v2.pdf
13. R. Basri and D. Jacobs, "Lambertian reflectance and linear subspaces," IEEETrans.Pattern Anal.Mach. Intell., vol. 25, no. 2, pp. 218-233, Feb. 2003.
14. S. Ma, D. Goldfarb, and L. Chen, "Fixed point and bregman iterative methods for matrix rank minimization," Math. Programm., 2009.
15. J.-F. Cai, E. J. Candès, and Z. Shen, A singular value thresholding algorithm for matrix completion, 2008.
16. W. Dai and O. Milenkovic, SET: An Algorithm for Consistent Matrix Completion, 2009 [Online]. Available: http://arxiv.org/abs/0909.2705
17. N. Srebro and A. Shraibman, "Rank, trace-norm and max-norm," Learning Theory, pp. 545-560, 2005.
18. N. Srebro, J. Rennie, and T. Jaakkola, "Maximum-margin matrix factorization," Adv. Neural Inf. Process. Syst., vol. 17, pp. 1329-1336, 2005.
19. F. Bach, "Consistency of trace norm minimization," J. Mach. Learn. Res., vol. 9, pp. 1019-1048, 2008.
20. D. Gross, Recovering Low-Rank Matrices From Few Coefficients in any Basis, 2009 [Online]. Available: http://arxiv.org/PS-cache/arxiv/ pdf/0910/0910.1879v4.pdf
21. S. Negahban, P. Ravikumar, M. Wainwright, and B. Yu, "A unified framework for high-dimensional analysis of M-estimators with decomposable regularizers," presented at the Advances in Neural Information Processing Systems 2009.
22. S. Negahban and M. Wainwright, Estimation of (Near) Lowrank Matrices With Noise and High-Dimensional Scaling, 2009 [Online]. Available: http://arxiv.org/PS-cache/arxiv/pdf/0912/0912.5100v1.pdf
23. A. Rohde and A. Tsybakov, Estimation of High-Dimensional Low- Rank Matrices, 2009 [Online]. Available: http://arxiv.org/PS-cache/ arxiv/pdf/0912/0912.5338v2.pdf
24. D. Wallace, "Bounds on normal approximations to student's and the chi-square distributions," Ann. Math. Statist., vol. 30, no. 4, pp. 1121-1130, 1959.
25. B. Laurent and P. Massart, "Adaptive estimation of a quadratic functional by model selection," Ann. Statist., vol. 28, no. 5, pp. 1302-1338, 2000. (Pubitemid 33244915)
26. D. Achlioptas, "Database-friendly random projections: Johnson-Lindenstrauss with binary coins," J. Comput. Syst. Sci., vol. 66, no. 4, pp. 671-687, Jun. 2003.
27. R. Vershynin, "On large random almost Euclidean bases," Acta Mathematica Universitatis Comenianae, vol. 69, no. 2, pp. 137-144, 2000.
28. P. Wojtaszczyk, "Stability and instance optimality for Gaussian measurements in compressed sensing," Found. Comput. Math., 2009.
29. R. Vershynin, Math 280 Lecture Notes, 2007 [Online]. Available: http://www-personal.umich.edu/~romanv/teaching/2006-07/280/lec6. pdf
30. E. J. Candès and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203-4215, Dec. 2005. (Pubitemid 41800353)
31. M. Fazel, E. Candes, B. Recht, and P. Parrilo, "Compressed sensing and robust recovery of low rank matrices," in Proc. IEEE 42nd Asilomar Conf. Signals, Systems and Computers, Oct. 2008, pp. 1043-1047.
32. E. J. Candès, J. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math., vol. 59, no. 8, pp. 1207-1207, 2006.
33. E. Lehmann and G. Casella, Theory of Point Estimation. New York: Springer Verlag, 1998.