Fixed point and Bregman iterative methods for matrix rank minimization

Mathematical Programming

Volumn 128, Issue 1-2, 2011, Pages 321-353

  Ma, Shiqian ^a   Goldfarb, Donald ^a   Chen, Lifeng ^a  

^a Columbia University ^*  (United States)

Author keywords

Bregman distances; Fixed point iterative method; Matrix completion problem; Matrix rank minimization; Nuclear norm minimization; Singular value decomposition

Indexed keywords

BREGMAN DISTANCES; FIXED POINTS; MATRIX COMPLETION PROBLEM; MATRIX RANK MINIMIZATION; NUCLEAR NORM MINIMIZATION; SINGULAR VALUES;

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; ECONOMIC ANALYSIS; MATHEMATICAL PROGRAMMING; RELAXATION PROCESSES; SIGNAL PROCESSING; SINGULAR VALUE DECOMPOSITION;

ITERATIVE METHODS;

EID: 79957957723     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-009-0306-5     Document Type: Article

References (45)

1. F.R. Bach 2008 Consistency of trace norm minimization J. Mach. Learn. Res. 9 Jun 1019 1048 2417263
2. Bertalmío, M.; Sapiro, G.; Caselles, V.; Ballester, C.: Image inpainting. In: Proceedings of SIGGRAPH 2000, New Orleans, USA (2000)
3. Borwein J.M.; Lewis A.S.: Convex Analysis and Nonlinear Optimization. Springer, New York (2003)
4. L. Bregman 1967 The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming USSR Comput. Math. Math. Phys. 7 200 217 10.1016/0041-5553(67)90040- 7
5. S. Burer R.D.C. Monteiro 2003 A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization Math. Program. (Ser. B) 95 329 357 1976484 1030.90077 10.1007/s10107-002-0352-8 (Pubitemid 44757858)
6. S. Burer R.D.C. Monteiro 2005 Local mimima and convergence in low-rank semidefinite programming Math. Program. 103 3 427 444 2166543 1099.90040 10.1007/s10107-004-0564-1 (Pubitemid 40925236)
7. Cai, J.; Candès, E.J.; Shen, Z.: A singular value thresholding algorithm for matrix completion. Preprint available at http://arxiv.org/abs/ 0810.3286 (2008)
8. Candès, E.J.; Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. (2009)
9. 1-MAGIC: recovery of sparse signals via convex programming. Technical Report, Caltech (2005)
10. E.J. Candès J. Romberg T. Tao 2006 Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information IEEE Trans. Inf. Theory 52 489 509 10.1109/TIT.2005.862083 (Pubitemid 43174093)
11. Candès, E.J.; Tao, T.: The power of convex relaxation: near-optimal matrix completion. Preprint available at http://arxiv.org/abs/0903. 1476 (2009)
12. Dai, W.; Milenkovic, O.: Subspace pursuit for compressive sensing: closing the gap between performance and complexity. Preprint available at arXiv: 0803.0811 (2008)
13. D. Donoho 2006 Compressed sensing IEEE Trans. Inf. Theory 52 1289 1306 2241189 10.1109/TIT.2006.871582
14. 1-norm minimization problems when the solution may be sparse. Technical Report, Department of Statistics, Stanford University (2006)
15. Donoho, D.; Tsaig, Y.; Drori, I.; Starck, J.C.: Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit. IEEE Trans. Inf. Theory (2006) (submitted)
16. P. Drineas R. Kannan M.W. Mahoney 2006 Fast Monte Carlo algorithms for matrices ii: computing low-rank approximations to a matrix SIAM J. Comput. 36 158 183 2231644 1111.68148 10.1137/S0097539704442696 (Pubitemid 46374022)
17. Fazel, M.: Matrix rank minimization with applications. Ph.D. thesis, Stanford University (2002)
18. Fazel, M.; Hindi, H.; Boyd, S.: A rank minimization heuristic with application to minimum order system approximation. In: Proceedings of the American Control Conference, vol. 6, pp. 4734-4739 (2001) (Pubitemid 32837075)
19. Figueiredo, M.A.T.; Nowak, R.D.; Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1(4) (2007)
20. Ghaoui, L.E.; Gahinet, P.: Rank minimization under LMI constraints: a framework for output feedback problems. In: Proceedings of the European Control Conference (1993)
21. K. Goldberg T. Roeder D. Gupta C. Perkins 2001 Eigentaste: a constant time collaborative filtering algorithm Inf. Retr. 4 2 133 151 0989.68052 10.1023/A:1011419012209 (Pubitemid 33636930)
22. Goldfarb, D.; Ma, S.: Convergence of fixed point continuation algorithms for matrix rank minimization. Technical Report, Department of IEOR, Columbia University (2009)
23. 1-regularized minimization with applications to compressed sensing. Technical Report, CAAM TR07-07 (2007)
24. Hiriart-Urruty J.B.; Lemaréchal C.: Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods. Springer, New York (1993)
25. Horn R.A.; Johnson C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
26. Keshavan, R.H.; Montanari, A.; Oh, S.: Matrix completion from a few entries. Preprint available at http://arxiv.org/abs/0901.3150 (2009)
27. 1-regularized least-squares IEEE J. Sel. Top. Signal Process. 4 1 606 617 10.1109/JSTSP.2007.910971 (Pubitemid 351266626)
28. N. Linial E. London Y. Rabinovich 1995 The geometry of graphs and some of its algorithmic applications Combinatorica 15 215 245 1337355 0827.05021 10.1007/BF01200757
29. Liu, Z.; Vandenberghe, L.: Interior-point method for nuclear norm approximation with application to system identification. Preprint available at http://www.ee.ucla.edu/~vandenbe/publications/nucnrm.pdf (2008)
30. B.K. Natarajan 1995 Sparse approximation solutions to linear systems SIAM J. Comput. 24 2 227 234 1320206 0827.68054 10.1137/S0097539792240406
31. S. Osher M. Burger D. Goldfarb J. Xu W. Yin 2005 An iterative regularization method for total varitaion-based image restoration SIAM MMS 4 2 460 489 2162864 1090.94003 (Pubitemid 43885248)
32. Recht, B.; Fazel, M.; Parrilo, P.: Guaranteed minimum rank solutions of matrix equations via nuclear norm minimization. Preprint available at http://arxiv.org/abs/0706.4138 (2007)
33. Rennie, J.D.M.; Srebro, N.: Fast maximum margin matrix factorization for collaborative prediction. In: Proceedings of the International Conference of Machine Learning (2005)
34. L. Rudin S. Osher E. Fatemi 1992 Nonlinear total variation based noise removal algorithms Physica D 60 259 268 0780.49028 10.1016/0167-2789(92)90242-F
35. P.T. Spellman G. Sherlock M.Q. Zhang V.R. Iyer K. Anders M.B. Eisen P.O. Brown D. Botstein B. Futcher 1998 Comprehensive identification of cell cycle-regulated genes of the yeast saccharomyces cerevisiae by microarray hybridization Mol. Biol. Cell 9 3273 3297 (Pubitemid 28551928)
36. Srebro, N.: Learning with matrix factorizations. Ph.D. thesis, Massachusetts Institute of Technology (2004)
37. Srebro, N.; Jaakkola, T.: Weighted low-rank approximations. In: Proceedings of the Twentieth International Conference on Machine Learning (ICML-2003) (2003)
38. J.F. Sturm 1999 Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones Opt. Methods Softw. 11 12 625 653 1778433 10.1080/ 10556789908805766
39. R. Tibshirani 1996 Regression shrinkage and selection via the lasso J. R. Stat. Soc. B 58 267 288 1379242 0850.62538
40. J. Tropp 2006 Just relax: convex programming methods for identifying sparse signals IEEE Trans. Inf. Theory 51 1030 1051 2238069 10.1109/TIT.2005. 864420 (Pubitemid 46444890)
41. O. Troyanskaya M. Cantor G. Sherlock P. Brown T. Hastie R. Tibshirani D. Botstein R.B. Altman 2001 Missing value estimation methods for DNA microarrays Bioinformatics 17 6 520 525 10.1093/bioinformatics/17.6.520 (Pubitemid 32600498)
42. R.H. Tütüncü K.C. Toh M.J. Todd 2003 Solving semidefinite-quadratic-linear programs using SDPT3 Math. Program. Ser. B 95 189 217 1030.90082 10.1007/s10107-002-0347-5 (Pubitemid 44757853)
43. E. van den Berg M.P. Friedlander 2008 Probing the Pareto frontier for basis pursuit solutions SIAM J. Sci. Comput. 31 2 890 912 2466141 1193.49033 10.1137/080714488
44. Wen, Z.; Yin, W.; Goldfarb, D.; Zhang, Y.: A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation. Technical Report, Department of IEOR, Columbia University (2009)
45. 1-minimization with applications to compressed sensing SIAM J. Imaging Sci. 1 1 143 168 2475828 1203.90153 10.1137/070703983