Matrix completion via an alternating direction method

IMA Journal of Numerical Analysis

Volumn 32, Issue 1, 2012, Pages 227-245

  Chen, Caihua ^a   He, Bingsheng ^a   Yuan, Xiaoming ^b  

^a NANJING UNIVERSITY   (China)

^b HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

Author keywords

alternating direction method; convex programming; low rank; matrix completion; noise; nuclear norm

Indexed keywords

CONVEX OPTIMIZATION; RELAXATION PROCESSES;

ALTERNATING DIRECTION METHODS; LOW RANK; MATRIX COMPLETION; NOISE; NUCLEAR NORM;

NUMERICAL METHODS;

EID: 84863068818     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/drq039     Document Type: Article

References (56)

1. ABERNETHY, F., EVGENIOU, T. & VERT, J. P. (2006) Low-rank matrix factorization with attributes. Technical Report N24/06/MM. Paris, France: Ecole des Mines de Paris.
2. ABERNETHY, F., EVGENIOU, T. & VERT, J. P. (2009) A new approach to collaborative filtering: Operator estimation with spectral regularization. J. Mach. Learn. Res., 10, 803-826.
3. AMIT, Y., FINK, M., SREBRO, N. & ULLMAN, S. (2007) Uncovering shared structures in multiclass classification. Proceedings of the International Conference on Machine Learning (Z. Ghahramani ed.). New York: ACM, pp. 17-24.
4. ARGYRIOU, A., EVGENIOU, T. & PONTIL, M. (2008) Convex multi-task feature learning. Mach. Learn., 73, 243-272.
5. BERTSEKAS, D. P. (1982) Constrained Optimization and Lagrange Multiplier Methods. New York: Academic Press.
6. BERTSEKAS, D. P. & TSITSIKLIS, J. N. (1989) Parallel and Distributed Computation: Numerical Methods. Englewood Cliffs, NJ: Prentice Hall.
7. CAI, J. F., CANDÈS, E. J. & SHEN, Z. W. (2010) A singular value thresholding algorithm for matrix completion. SIAM J. Optim., 20, 1956-1982.
8. CANDÈS, E. J. & PLAN, Y. (2009) Matrix completion with noise. Proc. IEEE, 98, 925-936.
9. CANDÈS, E. J. & RECHT, B. (2008) Exact matrix completion via convex optimization. Found. Comput. Math., 9, 717-772.
10. CANDÈS, E. J. & TAO, T. (2009) The power of convex relaxation: Near-optimal matrix completion. IEEE Trans. Inf. Theory, 56, 2053-2080.
11. CHEN, G. & TEBOULLE, M. (1994) A proximal-based decomposition method for convex minimization problems. Math. Program., 64, 81-101.
12. CHEN, P. & SUTER, D. (2004) Recovering the missing components in a large noisy low-rank matrix: Application to SFM source. IEEE Trans. Pattern Anal. Mach. Intell., 26, 1051-1063.
13. EAVES, B. C. (1971) On the basic theorem of complementarity. Math. Program., 1, 68-75.
14. ECKSTEIN, J. (1994) Some saddle-function splitting methods for convex programming. Optim. Methods Softw., 4, 75-83.
15. ECKSTEIN, J. & FUKUSHIMA, M. (1994) Some reformulations and applications of the alternating direction method of multipliers: Large Scale Optimization: State of the Art (W. W. Hager, D. W. Hearn & P. M. Pardalos eds). Dordrecht, The Netherlands: Kluwer Academic Publishers, pp. 115-134.
16. ESSER, E. (2009) Applications of Lagrangian-based alternating direction methods and connections to split Bregman. Manuscript. Available at http://www.math.ucla.edu/applied/cam/.
17. FAZEL, M. (2002) Matrix rank minimization with applications. Ph.D. Thesis, Stanford University, Menlo Park, CA, USA.
18. FAZEL, M., HINDI, H. & BOYD, S. (2001) A rank minimization heuristic with application to minimum order system approximation. Proceedings of the American Control Conference, pp. 4734-4739.
19. FUKUSHIMA, M. (1992) Application of the alternating direction method of multipliers to separable convex programming problems. Comput. Optim. Appl., 1, 93-111.
20. GABAY, D. (1983) Application of the method of multipliers to variational inequalities. Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problem (M. Fortin & R. Glowinski eds). Amsterdam, The Netherlands: North-Holland, pp. 299-331.
21. GABAY, D. & MERCIER, B. (1976) A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl., 2, 17-40.
22. GLOWINSKI, R. (1984) Numerical Methods for Nonlinear Variational Problems. New York: Springer.
23. GLOWINSKI, R. & LE TALLEC, P. (1989) Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics. SIAM Studies in Applied Mathematics, vol. 9. Philadelphia, PA: SIAM.
24. GOLDBERG, K., ROEDER, T., GUPTA, D. & PERKINS, C. (2001) Eigentaste: A constant time collaborative filtering algorithm. Inf. Retr., 4, 133-151.
25. GOLUB, G. H. & VAN LOAN, C. F. (1996) Matrix Computation, 3rd edn. Baltimore, MD: The Johns Hopkins University Press.
26. HE, B., LIAO, L.-Z., HAN, D. & YANG, H. (2002) A new inexact alternating directions method for monotone variational inequalities. Math. Program., 92, 103-118.
27. HE, B., XU, M. H. & YUAN, X. M. (2011) Solving large-scale least squares covariance matrix problems by alternating direction methods. SIAM J. Matrix Anal. Appl., 32, 136-152.
28. HE, B. & YANG, H. (1998) Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. Oper. Res. Lett., 23, 151-161.
29. HIGHAM, N. J. (1988) Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Appl., 103, 103-118.
30. JAMES, B. & STAN, L. (2007) The netflix prize. Proceedings of KDD Cup and Workshop. Available at http://wwwcs.uic.edu/-liub/KDD-cup-2007/proceedings. html.
31. KESHAVAN, R. H., MONTANARI, A. & OH, S. (2010) Matrix completion from noisy entries. J. Mach. Learn., 11, 2057-2078.
32. KONTOGIORGIS, S. & MEYER, R. R. (1998) A variable-penalty alternating directions method for convex optimization. Math. Program., 83, 29-53.
33. LARSEN, R. M. (2005) PROPACK-Software for Large and Sparse SVD Calculations. Available at http://sun.stanford.edu/-rmunk/PROPACK/.
34. LIN, Z. C., CHEN, M. M., & MA, Y. (2009) The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices. Manuscript.
35. LIU, Y. J., SUN, D. F. & TOH, K. C. (2009) An implementable proximal point algorithmic framework for nuclear norm minimization. Manuscript.
36. LIU, Z. & VANDENBERGHE, L. (2009) Interior-point method for nuclear norm approximation with application to system identification. SIAM J. Matrix Anal. Appl., 31, 1235-1256.
37. MA, S. Q., GOLDFARB, D. & CHEN, L. F. (2010) Fixed point and Bregman iterative methods for matrix rank minimization. Math. Program. (to appear).
38. MARTINET, B. (1970) Régularisation d'inéquations variationnelles par approximations successives. Rev. Française Informat. Recherche Opérationnelle, 4, 154-158.
39. NETFLIX (2006) Netflix Prize. Available at http://www.netflixprize.com/.
40. NG, M., WEISS, P. A. & YUAN, X. M. (2010) Solving constrained total-variation problems via alternating direction methods. SIAM J. Sci. Comput., 32, 2710-2736.
41. NOCEDAL, J. & WRIGHT, S. J. (2006) Numerical Optimization. New York: Springer.
42. OSHER, S., MAO, Y., DONG, B. & YIN, W. (2008) Fast linearized Bregman iteration for compressive sensing and sparse denoising. Commun. Math. Sci., 8, 93-111.
43. PONG, T. K., TSENG, P., JI, S. & YE, J. (2010) Trace norm regularization: Reformulations, algorithms, and multitask learning. SIAM J. Optim., 20, 3465-3489.
44. RECHT, B., FAZEL, M. & PARRILO, P. A. (2010) Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev., 52, 471-501.
45. RECHT, B., XU, W. & HASSIBI, B. (2009) Null Space Conditions and Thresholds for Rank Minimization. Available at http://pages.cs.wisc.edu/-brecht/ publications.html.
46. ROCKAFELLAR, R. T. (1976) Monotone operators and the proximal point algorithm. SIAM J. Control Optim., 14, 877-898.
47. SCHMIDT, R. O. (1986) Multiple emitter location and signal parameter estimation. IEEE Trans. Ant. Propag., 34, 276-280.
48. TOH, K. C. & YUN, S. W. (2010) An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. Pacific J. Optim., 6, 615-640.
49. TOMASI, C. & KANADE, T. (1992) Shape and motion from image streams under orthography: A factorization method. Int. J. Comput. Vis., 9, 137-154.
50. TSENG, P. (1997) Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J. Optim., 7, 951-965.
51. WEN, Z. W., GOLDFARB, D. & YIN, W. (2010) Alternating direction augmented Lagrangian methods for semidefinite programming. Math. Program. Comput., 3-4, 203-230.
52. YANG, J. & ZHANG, Y. (2011) Alternating direction algorithms for l1-problems in compressive sensing. SIAM J. Sci. Comput., 33, 250-278.
53. YE, C. H. & YUAN, X. M. (2007) A descent method for structured monotone variational inequalities. Optim. Methods Softw., 22, 329-338.
54. YUAN, X. M. (2009) Alternating direction methods for sparse covariance selection (submitted).
55. YUAN, X. M. & YANG, J. (2009) Sparse and low-rank matrix decomposition via alternating direction methods. Pac. j. Optim.
56. ZHU, Z. S., SO, A. M. C. & YE, Y. Y. (2009) Fast and near-optimal matrix completion via randomized basis pursuit. Manuscript.