Singular value decompositions and low rank approximations of tensors

IEEE Transactions on Signal Processing

Volumn 58, Issue 3 PART 1, 2010, Pages 1171-1182

  Weiland, Siep ^a   Van Belzen, Femke ^a  

^a EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Low rank approximations; Multidimensional signal processing; Multilinear algebra; Tensors

Indexed keywords

APPROXIMATION PROBLEMS; LOW RANK APPROXIMATIONS; MODEL REDUCTION; MULTIDIMENSIONAL SIGNAL PROCESSING; MULTILINEAR ALGEBRA; MULTILINEAR FUNCTIONS; SINGULAR VALUES;

ALGEBRA; DATA COMPRESSION; NUMERICAL ANALYSIS; SIGNAL PROCESSING; TENSORS;

SINGULAR VALUE DECOMPOSITION;

EID: 80052325478     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2009.2034308     Document Type: Article

References (27)

1. R. P. Argalwal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications. Cambridge, U.K.: Cambridge Univ. Press, 2001.
2. P. Astrid, S.Weiland, K.Willcox, and T. Backx, "Missing point estimation in models described by proper orthogonal decomposition," IEEE Trans. Autom. Control, vol. 53, no. 10, pp. 2237-2251, Oct. 2008.
3. B. W. Bader and T. G.Kolda, "Algorithm 862:MATLAB tensor classes for fast algorithm prototyping," ACM Trans. Math. Softw., vol. 32, no. 4, Dec. 2006.
4. D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. New York: Academic, 1982.
5. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2005.
6. J. D. Carroll and J. J. Chang, "Analysis of individual differences in multidimensional scaling via an n-way generalization of 'Eckart-Young' decomposition," Psychometrika, vol. 35, pp. 287-319, 1970.
7. J. Chen and Y. Saad, "On the tensor SVD and the optimal low rank orthogonal approximation of tensors," SIAM J. Matrix Anal. Applicat., vol. 30, no. 4, pp. 1709-1734, 2009.
8. P. Comon, J. G. McWhirter and I. K. Proudler, Eds., "Tensor Decompositions: State of the Art and Applications," in Mathematics in Signal Processing V. London, U.K.: Oxford Univ. Press, 2001.
9. R. Constantini, L. Sbaiz, and S. Süsstrunk, "Higher order SVD analysis for dynamic texture synthesis," IEEE Trans. Image Process., vol. 17, no. 1, Jan. 2008.
10. V. de Silva and L.-H. Lim, "Tensor rank and the ill-posedness of the best low-rank approximation problem," SIAM J. Matrix Anal. Applicat., vol. 30, no. 3, pp. 1084-1127, 2008.
11. C. Eckart and G. Young, "The approximation of one matrix by another of lower rank," Psychometrika, vol. 1, pp. 211-218, 1936.
12. R. Fletcher, Practical Methods of Optimization, Vol. 2 Constrained Optimization. New York: Wiley, 1981.
13. G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. Baltimore, MD: John Hopkins Univ. Press, 1996.
14. R. A. Harshman, "Foundations of the PARAFAC procedure: Model and conditions for an 'explanatory' multi-mode factor analysis," UCLA Working Papers in Phonetics, vol. 16, pp. 1-84, 1970.
15. T. G. Kolda, "Orthogonal tensor decompositions," SIAM J. Matrix Anal. Applicat., vol. 23, no. 1, pp. 243-255, 2001.
16. T. G. Kolda, "A counterexample to the possibility of an extension of the Eckart-Young low-rank approximation theorem for the orthogonal rank tensor decomposition," SIAM J. Matrix Anal. Applicat., vol. 24, no. 3, pp. 762-767, 2003. (Pubitemid 36913885)
17. D. Leibovici and R. Sabatier, "A singular value decomposition of a κ-way array for a principal component analysis of multiway data, PTA-k," Linear Algebra Applicat., vol. 269, pp. 307-329, 1998. (Pubitemid 128387565)
18. L. De Lathauwer, "Signal processing based on multilinear algebra," Ph.D. thesis, K.U. Leuven, Leuven, Belgium, 1997.
19. L. De Lathauwer, B. De Moor, and J. Vandewalle, "A multilinear singular value decomposition," SIAM J. Matrix Anal. Applicat., vol. 21, no. 4, pp. 1253-1278, 2000. (Pubitemid 33217068)
20. n) approximation of higher order tensors," SIAM J. Matrix Anal. Applicat., vol. 21, no. 4, pp. 1324-1342, 2000. (Pubitemid 33217072)
21. L.-H. Lim, "What's possible and what's not possible in tensor decompositions-A freshman's view," presented at the Amer. Inst. Mathematics Workshop Tensor Decompositions, Jul. 2004.
22. C. D. Meyer, Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: SIAM, 2000.
23. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic, 1970.
24. L. R. Tucker, "Some mathematical notes on three-mode factor analysis," Psychometrika, vol. 31, no. 3, pp. 279-311, 1966.
25. M. A. O. Vasilecsu and D. Terzopoulus, "Multilinear subspace analysis of image ensembles," presented at the 2003 IEEE Computer Society Conf. Computer Vision and Pattern Recognition (CVPR'03), 2003.
26. M. A. O. Vasilecsu and D. Terzopoulus, "Multilinear independent component analysis," presented at the 2005 IEEE Computer Society Conf. Computer Vision and Pattern Recognition (CVPR'05), 2005.
27. T. Zhang and G. H. Golub, "Rank-one approximation to high order tensors," SIAM J. Matrix Anal. Applicat., vol. 23, no. 2, pp. 534-550, 2001.