Fast low rank approximations of matrices and tensors

Electronic Journal of Linear Algebra

Volumn 22, Issue , 2011, Pages 1031-1048

  Friedland, S ^a   Mehrmann, V ^b   Miedlar, A ^b   Nkengla, M ^a  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^b TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

CUR decomposition; Least squares; Rank k approximation; Singular value decomposition; Tucker decomposition

Indexed keywords

EID: 80555145106     PISSN: None     EISSN: 10813810     Source Type: Journal    
DOI: 10.13001/1081-3810.1489     Document Type: Article

References (31)

1. O. Alter, P. Brown, and D. Botstein. Processing and modeling genome-wide expression data using singular value decomposition. Microarrays: Optical Technologies and Informatics, Proc. SPIE, 4266:171-186, 2001.
2. J. Demmel and W Kahan. Accurate singular values of bidiagonal matrices. SIAM J. Sci. Statist. Comput., 11(5):873-912, 1990.
3. A. Deshpande and S. Vempala. Adaptive sampling and fast low-rank matrix approximation. Electronic Colloquium on Computational Complexity (ECCC) TR06-042, March 2006.
4. P. Drineas, R. Kannan, and M.W. Mahoney. Fast Monte Carlo algorithms for matrices I- III: Computing a compressed approximate matrix decomposition. SIAM J. Comput., 36(1):132-206, 2006.
5. P. Drineas, M.W. Mahoney, and S. Muthukrishnan. Polynomial time algorithm for columnrow- based relative-error low-rank matrix approximation. Technical Report TR 2006-04, DIMACS, March 2006.
6. P. Drineas, M.W. Mahoney, and S. Muthukrishnan. Subspace sampling and relative-error matrix approximation: Column-row-based methods. Proceedings of the 14th Annual European Symposium on Algorithms (ESA), 304-314, 2006.
7. N. Eriksson. Tree construction using singular value decomposition, 347-358. L. Pachter and B. Sturmfels (editors), Algebraic Statistics for Computational Biology, Cambridge University Press, 2005.
8. M. Espig, L. Grasedyck, and W. Hackbusch. Black box low tensor-rank approximation using fiber-crosses. Constr. Approx., 30(3):557-597, 2009.
9. S. Friedland, M. Kaveh, A. Niknejad, and H. Zare. Fast Monte-Carlo low rank approximations for matrices. Proc. IEEE SoSE, 218-223, Los Angeles, 2006.
10. S. Friedland and A. Torokhti. Generalized rank-constrained matrix approximations. SIAM J. Matrix Anal. Appl., 29(2):656-659, 2007.
11. P.J. Gemperline, K.H. Miller, T.L. West, J.E. Weinstein, J.C. Hamilton, and J.T. Bray. Principal component analysis, trace elements, and blue crab shell disease. Analytical Chemistry, 64(9):523A-532A, 1992.
12. G.H. Golub and C.F. Van Loan. Matrix Computations, 3rd edition. John Hopkins Univ. Press, Baltimore, 1996.
13. S.A. Goreinov and E.E. Tyrtyshnikov. The maximum-volume concept in approximation by low-rank matrices. Contemporary Mathematics, 280:47-51, 2001.
14. S.A. Goreinov, E.E. Tyrtyshnikov, and N.L. Zamarashkin. Pseudo-skeleton approximations of matrices. Dokl. Akad. Nauk, 343(2):151-152, 1995.
15. S.A. Goreinov, E.E. Tyrtyshnikov, and N.L. Zamarashkin. A theory of pseudoskeleton approximations. Linear Algebra Appl., 261:1-21, 1997.
16. S. Har-Peled. Low rank matrix approximation in linear time. Manuscript, 2006, available at http://valis.cs.uiuc.edu/~sariel/papers/05/lrank/lrank.pdf.
17. J. Kamm and J.G. Nagy. Kronecker product and SVD approximations in image restoration. Linear Algebra Appl., 284:177-192, 1998.
18. P.M. Kroonenberg. Three-Mode Principal Component Analysis: Theory and Applications. DSWO Press, Leiden, 1983.
19. R.B. Lehoucq, D.C. Sorensen, and C. Yang. ARPACK User's Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods (Software, Environments, and Tools). SIAM Publications, 1998.
20. M. Lyons. Max Lyons Digital Image Gallery. Available at http://www.tawbaware.com/maxlyons/index.html.
21. M.W. Mahoney, M. Maggioni, and P. Drineas. Tensor-CUR decompositions for tensor-based data. Proceedings of the 12th Annual ACM SIGKDD Conference, 327-336, 2006.
22. MATLAB Image Processing Toolbox Version 5.4 (R2007a). The MathWorks, inc., 24 Prime Park Way, Natick, MA 01760-1500, USA, 2007.
23. MATLAB, Version 7.4.0.336 (R2007a). The MathWorks, inc., 24 Prime Park Way, Natick, MA 01760-1500, USA, 2007.
24. M. Nkengla. Low Rank Approximations of Matrice and Tensors. PhD Thesis, University of Illinois at Chicago, 2010.
25. P. Paschou, M. W. Mahoney, A. Javed, J.R. Kidd, A.J. Pakstis, S. Gu, K.K. Kidd, and P. Drineas. Intra- and interpopulation genotype reconstruction from tagging SNPs. Genome Res., 17:96-107, 2007.
26. M. Rudelson and R. Vershynin. Sampling from large matrices: An approach through geometric functional analysis. J. ACM, 54, 2007.
27. T. Sarlos. Improved approximation algorithms for large matrices via random projections. Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 143-152, 2006.
28. A. Stegeman, J.M.F. Ten Berge, and L. De Lathauwer. Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices. Psychometrika, 71:219-229, 2006.
29. G.W. Stewart. Updating a rank-revealing ULV decomposition. SIAM J. Matrix Anal. Appl., 14:494-499, 1993.
30. L.R. Tucker. Some mathematical notes on three-mode factor analysis. Psychometrika, 31(3):279-311, 1966.
31. M.A.O. Vasilescu and D. Terzopoulos. Multilinear image analysis for facial recognition. Proceedings of the International Conference on Pattern Recognition (ICPR'2002), 3:511-514, Quebec City, Canada, 2002.