Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion

Annals of Statistics

Volumn 39, Issue 5, 2011, Pages 2302-2329

  Koltchinskii, Vladimir ^a   Lounici, Karim ^a   Tsybakov, Alexandre B ^b  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b CREST   (France)

Author keywords

Lasso; Low rank matrix estimation; Matrix completion; Noncommutative Bernstein inequality; Optimal rate of convergence; Recovery of the rank; Statistical learning

Indexed keywords

EID: 82655171609     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AOS894     Document Type: Article

References (28)

1. AHLSWEDE, R. and WINTER, A. (2002). Strong converse for identification via quantum channels. IEEE Trans. Inform. Theory 48 569-579. MR1889969 (Pubitemid 34239788)
2. ARGYRIOU, A., EVGENIOU, T. and PONTIL, M. (2008). Convex multi-task feature learning. Machine Learning 73 243-272.
3. ARGYRIOU, A., MICCHELLI, C. A. and PONTIL, M. (2010). On spectral learning. J. Mach. Learn. Res. 11 935-953. MR2600635
4. ARGYRIOU, A., MICCHELLI, C. A., PONTIL, M. and YING, Y. (2008). A spectral regularization framework for multi-task structure learning. In Advances in Neural Information Processing Systems (J. C. Platt, D. Koller, Y. Singer and S. Roweis, eds.) 20 25-32. MIT Press, Cambridge, MA.
5. AUBIN, J.-P. and EKELAND, I. (1984). Applied Nonlinear Analysis. Wiley, New York. MR0749753
6. BACH, F. R. (2008). Consistency of trace norm minimization. J. Mach. Learn. Res. 9 1019-1048. MR2417263
7. BICKEL, P. J., RITOV, Y. and TSYBAKOV, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705-1732. MR2533469
8. BUNEA, F., SHE, Y. and WEGKAMP, M. H. (2011). Optimal selection of reduced rank estimators of high-dimensional matrices. Ann. Statist. 39 1282-1309.
9. CANDÈS, E. J. and PLAN, Y. (2010). Matrix completion with noise. Proc. IEEE 98 925-936.
10. CANDÈS, E. J. and PLAN, Y. (2010). Tight oracle bounds for low-rank matrix recovery from a mininal number of noisy random measurements. Available at arXiv:1001.0339.
11. CANDÈS, E. J. and RECHT, B. (2009). Exact matrix completion via convex optimization. Found. Comput. Math. 9 717-772. MR2565240
12. CANDÈS, E. J. and TAO, T. (2010). The power of convex relaxation: Near-optimal matrix completion. IEEE Trans. Inform. Theory 56 2053-2080. MR2723472
13. GAIFFAS, S. and LECUÉ, G. (2010). Sharp oracle inequalities for the prediction of a highdimensional matrix. Available at arXiv:1008.4886.
14. GIRAUD, C. (2011). Low rank multivariate regression. Electron. J. Stat. 5 775-799.
15. GROSS, D. (2011). Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory 57 1548-1566.
16. KESHAVAN, R. H., MONTANARI, A. and OH, S. (2010). Matrix completion from noisy entries. J. Mach. Learn. Res. 11 2057-2078. MR2678022
17. KOLTCHINSKII, V. (2009). The Dantzig selector and sparsity oracle inequalities. Bernoulli 15 799-828. MR2555200
18. KOLTCHINSKII, V. (2010). von Neumann entropy penalization and low rank matrix approximation. Available at arXiv:1009.2439.
19. NEGAHBAN, S., RAVIKUMAR, P.,WAINWRIGHT, M. J. and YU, B. (2010). A unified framework for high-dimensional analysis of M-estimators with decomposable regularizers. Available at arXiv:1010.2731.
20. NEGAHBAN, S. andWAINWRIGHT, M. J. (2011). Estimation of (near) low rank matrices with noise and high-dimensional scaling. Ann. Statist. 39 1069-1097.
21. NEGAHBAN, S. and WAINWRIGHT, M. J. (2010). Restricted strong convexity and weighted matrix completion: Optimal bounds with noise. Available at arXiv:1009.2118.
22. RECHT, B. (2009). A simpler approach to matrix completion. Available at arXiv:0910.0651.
23. RECHT, B., FAZEL, M. and PARRILO, P. A. (2010). Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52 471-501. MR2680543
24. ROHDE, A. and TSYBAKOV, A. (2011). Estimation of high-dimensional low rank matrices. Ann. Statist. 39 887-930.
25. STEWART, G. W. and SUN, J. G. (1990). Matrix Perturbation Theory. Academic Press, Boston, MA. MR1061154
26. TROPP, J. A. (2010). User-friendly tail bounds for sums of random matrices. Available at arXiv:1004.4389.
27. TSYBAKOV, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York. MR2724359
28. WATSON, G. A. (1992). Characterization of the subdifferential of some matrix norms. Linear Algebra Appl. 170 33-45. MR1160950