Limited memory block krylov subspace optimization for computing dominant singular value decompositions

SIAM Journal on Scientific Computing

Volumn 35, Issue 3, 2013, Pages

  Liu, Xin ^a   Wen, Zaiwen ^b   Zhang, Yin ^c  

^a CHINESE ACADEMY OF SCIENCES   (China)

^b SHANGHAI JIAO TONG UNIVERSITY   (China)

^c Rice University   (United States)

Author keywords

Dominant singular value decomposition; Eigenvalue decomposition; Krylov subspace; Subspace optimization

Indexed keywords

COMPUTATIONAL EFFICIENCY; DATA MINING; EIGENVALUES AND EIGENFUNCTIONS; ITERATIVE METHODS; LINEAR TRANSFORMATIONS; MATLAB; PRINCIPAL COMPONENT ANALYSIS; UBIQUITOUS COMPUTING;

DATA-INTENSIVE APPLICATION; DENSE MATRIX; DOMINANT SINGULAR VALUE DECOMPOSITION; EIGEN-VALUE; EIGENVALUE DECOMPOSITION; KRYLOV SUBSPACE; LIMITED MEMORY; MEMORY BLOCKS; PRINCIPAL-COMPONENT ANALYSIS; SUBSPACE OPTIMIZATION;

SINGULAR VALUE DECOMPOSITION;

EID: 84884952807     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/120871328     Document Type: Article

References (32)

1. E. Anderson, Z. Bai, J. Dongarra, A. Greenbaum, A. McKenney, J. Du Croz, S. Hammerling, J. Demmel, C. Bischof, and D. Sorensen, LAPACK: A portable linear algebra library for high-performance computers, in Proceedings of the 1990 ACM/IEEE Conference on Supercomputing (Supercomputing '90), 1990, pp. 2-11.
2. Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000.
3. M. Bollḧofer and Y. Notay, JADAMILU: A software code for computing selected eigenvalues of large sparse symmetric matrices, Comput. Phys. Comm., 177 (2007), pp. 951-964.
4. E. J. Cand̀es, X. Li, Y. Ma, and J. Wright, Robust principal component analysis?, J. ACM, 58 (2011), pp. 1-37.
5. Y.-H. Dai, Fast algorithms for projection on an ellipsoid, SIAM J. Optim., 16 (2006), pp. 986-1006.
6. E. D. Dolan and J. J. Moŕe, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), pp. 201-213.
7. P. Drineas, R. Kannan, and M. W. Mahoney, Fast Monte Carlo algorithms for matrices II: Computing a low-rank approximation to a matrix, SIAM J. Comput., 36 (2006), pp. 158-183.
8. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins Stud. Math. Sci., The Johns Hopkins University Press, Baltimore, MD, 1996.
9. N. I. M. Gould, D. Orban, and P. L. Toint, Numerical methods for large-scale nonlinear optimization, Acta Numer., 14 (2005), pp. 299-361.
10. N. Halko, P. G. Martinsson, and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), pp. 217-288.
11. V. Herńandez, J. E. Roḿan, A. Toḿas, and V. Vidal, A Survey of Software for Sparse Eigenvalue Problems, Technical report STR-6, Scalable Library for Eigenvalue Problem Computations, Universidad Polit́ecnica de Valencia, Valencia, Spain, 2009.
12. A. V. Knyazev, A preconditioned conjugate gradient method for eigenvalue problems and its implementation in a subspace, in Numerical Treatment of Eigenvalue Problems, Vol. 5, Internat. Ser. Numer. Math. 96, Birkḧauser, Basel, 1991, pp. 143-154.
13. A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23 (2001), pp. 517-541.
14. R. M. Larsen, Lanczos Bidiagonalization with Partial Reorthogonalization, Technical report DAIMI PB-357, Aarhus University, Aarhus, Denmark, 1998.
15. R. B. Lehoucq, Implicitly restarted Arnoldi methods and subspace iteration, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 551-562.
16. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, Software Environ. Tools 6, SIAM, Philadelphia, 1998.
17. Z. Lin, M. Chen, L. Wu, and Y. Ma, The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices, Technical report UILU-ENG-09-2215, University of Illinois at Urbana-Champaign, Urbana, IL, 2009, Math. Program., submitted.
18. H. Rutishauser, Computational aspects of F. L. Bauer's simultaneous iteration method, Numer. Math., 13 (1969), pp. 4-13.
19. H. Rutishauser, Simultaneous iteration method for symmetric matrices, Numer. Math., 16 (1970), pp. 205-223.
20. Y. Saad, Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems, Math. Comp., 42 (1984), pp. 567-588.
21. Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester; Halsted Press, New York, 1992.
22. Y. Saad, J. R. Chelikowsky, and S. M. Shontz, Numerical methods for electronic structure calculations of materials, SIAM Rev., 52 (2010), pp. 3-54.
23. D. C. Sorensen, Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations, in Parallel Numerical Algorithms (Hampton, VA, 1994), ICASE/LaRC Interdiscip. Ser. Sci. Eng. 4, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997, pp. 119-165.
24. D. C. Sorensen, Numerical methods for large eigenvalue problems, Acta Numer., 11 (2002), pp. 519-584.
25. A. Stathopoulos and C. F. Fischer, A Davidson program for finding a few selected extreme eigenpairs of a large, sparse, real, symmetric matrix, Comput. Phys. Comm., 79 (1994), pp. 268-290.
26. G. W. Stewart, Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices, Numer. Math., 25 (1975/76), pp. 123-136.
27. G. W. Stewart, Matrix Algorithms Volume II: Eigensystems, SIAM, Philadelphia, 2001.
28. W. J. Stewart and A. Jennings, A simultaneous iteration algorithm for real matrices, ACM Trans. Math. Software, 7 (1981), pp. 184-198.
29. M. Turk and A. Pentland, Eigenfaces for recognition, J. Cognit. Neurosci., 3 (1991), pp. 71-86.
30. C. Yang, J. C. Meza, and L.-W. Wang, A trust region direct constrained minimization algorithm for the Kohn-Sham equation, SIAM J. Sci. Comput., 29 (2007), pp. 1854-1875.
31. X. Yuan and J. Yang, Sparse and low-rank matrix decomposition via alternating direction method, Pac. J. Optim., 9 (2013), pp. 167-180.
32. Y. Yuan, Subspace techniques for nonlinear optimization, in Some Topics in Industrial and Applied Mathematics, Ser. Contemp. Appl. Math. CAM 8, Higher Education Press, Beijing, 2007, pp. 206-218.