On uniqueness of the canonical tensor decomposition with some form of symmetry

SIAM Journal on Matrix Analysis and Applications

Volumn 32, Issue 2, 2011, Pages 561-583

  Stegeman, Alwin ^a  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

Candecomp; Indscal; Parafac; Symmetry; Tensor decomposition; Uniqueness

Indexed keywords

CANDECOMP; INDSCAL; PARAFAC; TENSOR DECOMPOSITION; UNIQUENESS;

TENSORS;

EID: 80051528849     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/100814615     Document Type: Article

References (48)

1. E. ACAR AND B. YENER, Unsupervised multiway data analysis: A literature survey, IEEE Trans. Knowl. Data Eng., 21(2009), pp. 1-15.
2. A. BELOUCHRANI, K. ABED-MERAIM, J.-F. CARDOSO, AND E. MOULINES, A blind source separation technique using second order statistics, IEEE Trans. Signal Process., 45(1997), pp. 434-444. (Pubitemid 127765906)
3. J. D. CARROLL AND J. J. CHANG, Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart-Young decomposition, Psychometrika, 35(1970), pp. 283-319.
4. P. COMON, Independent component analysis, a new concept?, Signal Process., 36(1994), pp. 287-314.
5. P. COMON, J. M. F. TEN BERGE, L. DE LATHAUWER, AND J. CASTAING, Generic and typical ranks of multiway arrays, Linear Algebra Appl., 430(2009), pp. 2997-3007.
6. L. DE LATHAUWER, B. DE MOOR, AND J. VANDEWALLE, An introduction to independent component analysis, J. Chemom., 14(2000), pp. 123-149.
7. L. DE LATHAUWER, A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization, SIAM J. Matrix Anal. Appl., 28(2006), pp. 642-666. (Pubitemid 47109355)
8. L. DE LATHAUWER AND J. CASTAING, Tensor-based techniques for the blind separation of DS-CDMA signals, Signal Process., 87(2007), pp. 322-336. (Pubitemid 44602255)
9. L. DE LATHAUWER, J. CASTAING, AND J.-F. CARDOSO, Fourth-order cumulant-based blind identification of underdetermined mixtures, IEEE Trans. Signal Process., 55(2007), pp. 2965-2973. (Pubitemid 46845686)
10. L. DE LATHAUWER AND J. CASTAING, Blind identification of underdetermined mixtures by simultaneous matrix diagonalization, IEEE Trans. Signal Process., 56(2008), pp. 1096-1105. (Pubitemid 351386270)
11. V. DE SILVA AND L.-H. LIM, Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl., 30(2008), pp. 1084-1127.
12. M. DE VOS, D. NION, S. VAN HUFFEL, AND L. DE LATHAUWER, A Combination of Parallel Factor and Independent Component Analysis, Technical report 09-04, Katholieke Universiteit Leuven, Leuven, Belgium, 2009.
13. A. FERRÉOL, L. ALBERA, AND P. CHEVALIER, Fourth-order blind identification of underdetermined mixtures of sources (FOBIUM), IEEE Trans. Signal Process., 53(2005), pp. 1640-1653. (Pubitemid 40679139)
14. R. A. HARSHMAN, Foundations of the Parafac procedure: Models and conditions for an "explanatory" multimodal factor analysis, UCLA Working Papers in Phonetics, 16(1970), pp. 1-84.
15. F. L. HITCHCOCK, The expression of a tensor or apolyadic as a sum of products, J. Math. Phys., 6(1927), pp. 164-189.
16. F. L. HITCHCOCK, Multiple invariants and generalized rank of ap-way matrix or tensor, J. Math. Phys., 7(1927), pp. 39-79.
17. A. HYVÄRINEN, J. KARHUNEN, AND E. OJA, Independent Component Analysis, John Wiley, New York, 2001.
18. T. JIANG AND N. D. SIDIROPOULOS, Kruskal's permutation lemma and the identification of Candecomp/Parafac and bilinear models with constant modulus constraints, IEEE Trans. Signal Process., 52(2004), pp. 2625-2636.
19. H. A. L. KIERS AND I. VAN MECHELEN, Three-way component analysis: Principles and illustrative application, Psychological Methods, 6(2001), pp. 84-110. (Pubitemid 33610297)
20. T. G. KOLDA AND B. W. BADER, Tensor decompositions and applications, SIAM Rev., 51(2009), pp. 455-500.
21. W. P. KRIJNEN, T. K. DIJKSTRA, AND A. STEGEMAN, On the non-existence of optimal solutions and the occurrence of "degeneracy" in the Candecomp/Parafac model, Psychometrika, 73(2008), pp. 431-439.
22. P. M. KROONENBERG, Applied Multiway Data Analysis, Wiley Ser. Probab. Stat., John Wiley, New York, 2008.
23. J. B. KRUSKAL, Three-way arrays: Rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics, Linear Algebra Appl., 18(1977), pp. 95-138.
24. J. B. KRUSKAL, R. A. HARSHMAN, AND M. E. LUNDY, How 3-MFA data can cause degenerate Parafac solutions, among other relationships, in Multiway Data Analysis, R. Coppi and S. Bolasco, eds., North-Holland, Amsterdam, 1989, pp. 115-121.
25. X. LIU AND N. D. SIDIROPOULOS, Cramér-Rao lower bounds for low-rank decomposition of multidimensional arrays, IEEE Trans. Signal Process., 49(2001), pp. 2074-2086. (Pubitemid 32866214)
26. D.-T. PHAM AND J.-F. CARDOSO, Blind separation of instantaneous mixtures of nonstationary sources, IEEE Trans. Signal Process., 49(2001), pp. 1837-1848. (Pubitemid 32866193)
27. J. A. RHODES, A concise proof of Kruskal's theorem on tensor decomposition, Linear Algebra Appl., 432(2010), pp. 1818-1824.
28. R. ROCCI AND P. GIORDANI, A weak degeneracy revealing decomposition for the Candecomp/Parafac model, J. Chemom., 24(2010), pp. 57-66.
29. N. D. SIDIROPOULOS AND R. BRO, On the uniqueness of multilinear decomposition of N-way arrays, J. Chemom., 14(2000), pp. 229-239.
30. N. D. SIDIROPOULOS, G. GIANNAKIS, AND R. BRO, Blind Parafac receivers for DS-CDMA systems, IEEE Trans. Signal Process., 48(2000), pp. 810-823. (Pubitemid 30576249)
31. N. D. SIDIROPOULOS, R. BRO, AND G. GIANNAKIS, Parallel factor analysis in sensor array processing, IEEE Trans. Signal Process., 48(2000), pp. 2377-2388.
32. A. SMILDE, R. BRO, AND P. GELADI, Multi-way Analysis: Applications in the Chemical Sciences, John Wiley, New York, 2004.
33. A. STEGEMAN, Degeneracy in Candecomp/Parafac explained for p × p × 2 arrays ofrankp + 1 orhigher, Psychometrika, 71(2006), pp. 483-501. (Pubitemid 44788544)
34. A. STEGEMAN, Degeneracy in Candecomp/Parafac explained for several three-sliced arrays with a twovalued typical rank, Psychometrika, 72(2007), pp. 601-619.
35. A. STEGEMAN, Low-rank approximation of generic p × q × 2 arrays and diverging components in the Candecomp/Parafac model, SIAM J. Matrix Anal. Appl., 30(2008), pp. 988-1007.
36. A. STEGEMAN, Using the simultaneous generalized Schur decomposition as a Candecomp/Parafac algorithm for ill-conditioned data, J. Chemom., 23(2009), pp. 385-392.
37. A. STEGEMAN, On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode, Linear Algebra Appl., 431(2009), pp. 211-227.
38. A. STEGEMAN, On uniqueness of the nth order tensor decomposition into rank-1 terms with linear independence in one mode, SIAM J. Matrix Anal. Appl., 31(2010), pp. 2498-2516.
39. A. STEGEMAN, Candecomp/Parafac-From Diverging Components to a Decomposition in Block Terms, Technical report, University of Groningen, The Netherlands, 2011.
40. 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(2006), pp. 219-229. (Pubitemid 44116380)
41. A. STEGEMAN AND N. D. SIDIROPOULOS, On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition, Linear Algebra Appl., 420(2007), pp. 540-552. (Pubitemid 44751783)
42. A. STEGEMAN AND L. DE LATHAUWER, A method to avoid diverging components in the Candecomp/Parafac model for generic I × J × 2 arrays, SIAM J. Matrix Anal. Appl., 30(2009), pp. 1614-1638.
43. J. M. F. TEN BERGE AND N. D. SIDIROPOULOS, On uniqueness in Candecomp/Parafac, Psychometrika, 67(2002), pp. 399-409.
44. J. M. F. TEN BERGE, N. D. SIDIROPOULOS, AND R. ROCCI, Typical rank and Indscal dimensionality for symmetric three-way arrays of order I × 2 × 2 or I × 3 × 3, Linear Algebra Appl., 388(2004), pp. 363-377.
45. J. M. F. TEN BERGE, A. STEGEMAN, AND M. BENNANI DOSSE, The Carroll and Chang conjecture of equal Indscal components when Candecomp/Parafac gives perfect fit, Linear Algebra Appl., 430(2009), pp. 818-829.
46. G. TOMASI AND R. BRO, A comparison of algorithms for fitting the Parafac model, Comput. Statist. Data Anal., 50(2006), pp. 1700-1734. (Pubitemid 43052351)
47. A. YEREDOR, Blind source separation via the second characteristic function, Signal Process., 80(2000), pp. 897-902.
48. A. YEREDOR, Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation, IEEE Trans. Signal Process., 50(2002), pp. 1545-1553.