Best low multilinear rank approximation of higher-order tensors, based on the riemannian trust-region scheme

SIAM Journal on Matrix Analysis and Applications

Volumn 32, Issue 1, 2011, Pages 115-135

  Ishteva, Mariya ^a,b   Absil, P A ^a   Van Huffel, Sabine ^c   De Lathauwer, Lieven ^c,d  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

^b Georgia Institute of Technology   (United States)

^c UNIVERSITY OF LEUVEN   (Belgium)

^d Group Science Engineering and Technology   (Belgium)

Author keywords

Grassmann manifold; Higher order tensor; Multilinear algebra; Rank reduction; Riemannian manifold; Singular value decomposition; Trust region scheme

Indexed keywords

GRASSMANN MANIFOLD; HIGHER-ORDER TENSOR; MULTILINEAR ALGEBRA; RANK REDUCTION; RIEMANNIAN MANIFOLD; SINGULAR VALUES; TRUST-REGION SCHEME; MULTI-LINEAR ALGEBRAS; TRUST REGION;

ALGEBRA; ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; COST FUNCTIONS; ITERATIVE METHODS; SIGNAL PROCESSING; SINGULAR VALUE DECOMPOSITION; CONJUGATE GRADIENT METHOD; HIGHER ORDER STATISTICS;

TENSORS;

EID: 79952430949     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/090764827     Document Type: Article

References (48)

1. P.-A. Absil, C. G. Baker, and K. A. Gallivan, Trust-region methods on Riemannian manifolds, Found. Comput. Math., 7 (2007), pp. 303-330.
2. P.-A. Absil, M. Ishteva, L. De Lathauwer, and S.Van Huffel, A geometric Newton method for Oja's vector field, Neural Comput., 21 (2009), pp. 1415-1433.
3. P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008.
4. E. Acar, C. A. Bingol, H. Bingol, R. Bro, and B. Yener, Multiway analysis of epilepsy tensors, Bioinformatics, 23 (2007), pp. i10-i18. (Pubitemid 47244381)
5. R. Badeau and R. Boyer, Fast multilinear singular value decomposition for structured tensors, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1008-1021.
6. G. Beylkin and M. J. Mohlenkamp, Numerical operator calculus in higher dimensions, Proc. Natl. Acad. Sci. USA, 99 (2002), pp. 10246-10251.
7. G. Beylkin and M. J. Mohlenkamp, Algorithms for numerical analysis in high dimensions, SIAM J. Sci. Comput., 26 (2005), pp. 2133-2159. (Pubitemid 41641365)
8. P. Comon, Independent component analysis, a new concept?, Signal Process., 36 (1994), pp. 287-314.
9. A. R. Conn, N. I. M. Gould, and Ph. L. Toint, Trust-Region Methods, MPS-SIAM Series on Optimization, SIAM, Philadelphia, PA, 2000.
10. L. De Lathauwer and J. Castaing, Tensor-based techniques for the blind separation of DSCDMA signals, Signal Process., 87 (2007), pp. 322-336. (Pubitemid 44602255)
11. L. De Lathauwer, B. De Moor, and J. Vandewalle, An introduction to independent component analysis, J. Chemometrics, 14 (2000), pp. 123-149.
12. L. De Lathauwer, B. De Moor, and J. Vandewalle, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253-1278. (Pubitemid 33217068)
13. N) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1324-1342. (Pubitemid 33217072)
14. N) reduction in multilinear algebra, Linear Algebra Appl., 391 (2004), pp. 31-55.
15. M. De Vos, L. De Lathauwer, B. Vanrumste, S. Van Huffel, and W. Van Paesschen, Canonical decomposition of ictal scalp EEG and accurate source localization: Principles and simulation study, J. Comput. Intell. Neurosci., 2007 (2007), pp. 1-10.
16. M. De Vos, A. Vergult, L. De Lathauwer, W. De Clercq, S. Van Huffel, P. Dupont, A. Palmini, and W. Van Paesschen, Canonical decomposition of ictal scalp EEG reliably detects the seizure onset zone, NeuroImage, 37 (2007), pp. 844-854. (Pubitemid 47248306)
17. 3) approximation of a tensor, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 248-271.
18. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
19. M. Haardt, F. Roemer, and G. Del Galdo, Higher-order SVD-based subspace estimation to improve the parameter estimation accuracy in multidimensional harmonic retrieval problems, IEEE Trans. Signal Process., 56 (2008), pp. 3198-3213.
20. W. Hackbusch and B. Khoromskij, Tensor-product approximation to multidimensional integral operators and Green's functions, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1233-1253.
21. F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of products, J. Math. Phys., 6 (1927), pp. 164-189.
22. F. L. Hitchcock, Multiple invariants and generalized rank of a p-way matrix or tensor, J. Math. Phys., 7 (1927), pp. 39-79.
23. M. Ishteva, P.-A. Absil, S. Van Huffel, and L. De Lathauwer, Tucker compression and local optima, Chemometrics Intell. Lab. Syst., (2010), DOI 10.1016/j.chemolab.2010.06.006.
24. M. Ishteva, L. De Lathauwer, P.-A. Absil, and S. Van Huffel, Dimensionality reduction for higher-order tensors: Algorithms and applications, Int. J. Pure Appl. Math., 42 (2008), pp. 337-343.
25. 3) approximation of tensors, Numer. Algorithms, 51 (2009), pp. 179-194.
26. T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500.
27. P. M. Kroonenberg, Applied Multiway Data Analysis, Wiley, New York, 2008.
28. P. M. Kroonenberg and J. de Leeuw, Principal component analysis of three-mode data by means of alternating least squares algorithms, Psychometrika, 45 (1980), pp. 69-97.
29. P. McCullagh, Tensor Methods in Statistics, Chapman and Hall, London, 1987.
30. J. J. Moré and D. C. Sorensen, Computing a trust region step, SIAM J. Sci. Statist. Comput., 4 (1983), pp. 553-572.
31. M. Mørup, L. K. Hansen, C. S. Herrmann, J. Parnas, and S. M. Arnfred, Parallel factor analysis as an exploratory tool for wavelet transformed event-related EEG, NeuroImage, 29 (2006), pp. 938-947. (Pubitemid 43121218)
32. C. L. Nikias and J. M. Mendel, Signal processing with higher-order spectra, IEEE Signal Process. Mag., 10 (1993), pp. 10-37.
33. C. L. Nikias and A.P. Petropulu, Higher-Order Spectra Analysis. A Nonlinear Signal Processing Framework, Prentice Hall, Englewood Cliffs, NJ, 1993.
34. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed., Springer Ser. Oper. Res., Springer, New York, 2006.
35. I. V. Oseledets, D. V. Savostianov, and E. E. Tyrtyshnikov, Tucker dimensionality reduction of three-dimensional arrays in linear time, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 939-956.
36. J.-M. Papy, L. De Lathauwer, and S. Van Huffel, Exponential data fitting using multilinear algebra: The single-channel and the multichannel case, Numer. Linear Algebra Appl., 12 (2005), pp. 809-826. (Pubitemid 41441205)
37. J.-M. Papy, L. De Lathauwer, and S. Van Huffel, Exponential data fitting using multilinear algebra: The decimative case, J. Chemometrics, 23 (2009), pp. 341-351.
38. B. Savas and L. Eldén, Krylov Subspace Methods for Tensor Computations, Technical report LITH-MAT-R-2009-02-SE, Department of Mathematics, Linköping University, Linköping, Sweden, 2009.
39. B. Savas and L.-H. Lim, Quasi-Newton methods on Grassmannians and multilinear approximations of tensors, SIAM J. Sci. Comput., 32 (2010), pp. 3352-3393.
40. N. Sidiropoulos, R. Bro, and G. Giannakis, Parallel factor analysis in sensor array processing, IEEE Trans. Signal Process., 48 (2000), pp. 2377-2388.
41. N. Sidiropoulos, G. Giannakis, and R. Bro, Blind PARAFAC receivers for DS-CDMA systems, IEEE Trans. Signal Process., 48 (2000), pp. 810-823. (Pubitemid 30576249)
42. A. Smilde, R. Bro, and P. Geladi, Multi-way Analysis. Applications in the Chemical Sciences, Wiley, Chichester, U.K., 2004.
43. T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20 (1983), pp. 626-637.
44. A. Swami and G. Giannakis, Bibliography on HOS, Signal Process., 60 (1997), pp. 65-126.
45. Ph. L. Toint, Towards an efficient sparsity exploiting Newton method for minimization, in Sparse Matrices and Their Uses, I. S. Duff, ed., Academic Press, London, 1981, pp. 57-88.
46. L. R. Tucker, The extension of factor analysis to three-dimensional matrices, in Contributions to Mathematical Psychology, H. Gulliksen and N. Frederiksen, eds., Holt, Rinehart and Winston, New York, 1964, pp. 109-127.
47. L. R. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, 31 (1966), pp. 279-311.
48. M. A. O. Vasilescu and D. Terzopoulos, Multilinear subspace analysis for image ensembles, in Proceedings of the 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Madison, WI, 2003, pp. 93-99.