Canonical polyadic decomposition of third-order tensors: Reduction to generalized eigenvalue decomposition

SIAM Journal on Matrix Analysis and Applications

Volumn 35, Issue 2, 2014, Pages 636-660

  Domanov, Ignat ^a   De Lathauwer, Lieven ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

Candecomp Parafac decomposition; Canonical polyadic decomposition; Compound matrix; Khatri Rao product; Mixed discriminant; Permanent; Tensor

Indexed keywords

ALGEBRA; EIGENVALUES AND EIGENFUNCTIONS; MATRIX ALGEBRA;

CANDECOMP/PARAFAC DECOMPOSITIONS; CANONICAL POLYADIC DECOMPOSITIONS; COMPOUND MATRICES; KHATRI-RAO PRODUCTS; MIXED DISCRIMINANT; PERMANENT;

TENSORS;

EID: 84903935573     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/130916084     Document Type: Article

References (40)

1. A. D. Aleksandrov, Zur Theorie der gemischten Volumina von konvexen Körpern. IV. Die gemischten Diskriminanten und die gemischten Volumina, Mat. Sb., 3 (45), no. 2 (1938), pp. 227-251.
2. R. B. Bapat, Mixed discriminants of positive semidefinite matrices, Linear Algebra Appl., 126 (1989), pp. 107-124.
3. J. 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. A. Cichocki, D. Mandic, C. Caiafa, A.-H. Phan, G. Zhou, Q. Zhao, and L. De Lathauwer, Tensor decompositions for signal processing applications. From two-way to multiway component analysis, IEEE Signal Process. Mag., to appear.
5. P. Comon, X. Luciani, and A. L. F. de Almeida, Tensor decompositions, alternating least squares and other tales, J. Chemometrics, 23 (2009), pp. 393-405.
6. 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)
7. L. De Lathauwer, A short introduction to tensor-based methods for factor analysis and blind source separation, in ISPA 2011: Proceedings of the 7th International Symposium on Image and Signal Processing and Analysis, 2011, pp. 558-563.
8. I. Domanov and L. De Lathauwer, On the uniqueness of the canonical polyadic decomposition of third-order tensors - Part I: Basic results and uniqueness of one factor matrix, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 855-875.
9. I. Domanov and L. De Lathauwer, On the uniqueness of the canonical polyadic decomposition of third-order tensors - Part II: Uniqueness of the overall decomposition, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 876-903.
10. G. P. Egorychev, Proof of the van der Waerden conjecture for permanents, Siberian Math. J., 22 (1981), pp. 854-859.
11. R. A. Harshman, Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis, UCLA Working Papers in Phonetics, 16 (1970), pp. 1-84.
12. R. A. Harshman, Determination and proof of minimum uniqueness conditions for PARAFAC1, UCLA Working Papers in Phonetics, 22 (1972), pp. 111-117.
13. R. A. Harshman and M. E. Lundy, Parafac: Parallel factor analysis, Comput. Stat. Data Anal., 18 (1994), pp. 39-72.
14. F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of products, J. Math. Phys., 6 (1927), pp. 164-189.
15. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1990.
16. 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.
17. T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500.
18. P. M. Kroonenberg, Applied Multiway Data Analysis, Wiley, Hoboken, NJ, 2008.
19. J. B. Kruskal, Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics, Linear Algebra Appl., 18 (1977), pp. 95- 138.
20. J. M. Landsberg, Tensors: Geometry and Applications, AMS, Providence, RI, 2012.
21. S. E. Leurgans, R. T. Ross, and R. B. Abel, A decomposition for three-way arrays, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 1064-1083.
22. J. S. Lomont and M. S. Cheena, A multilinearity property of determinant functions, Linear Multilinear Algebra, 14 (1983), pp. 199-223.
23. M. Marcus, The Hadamard theorem for permanents, Proc. Amer. Math. Soc., 15 (1964), pp. 967-973.
24. M. Marcus, Finite Dimensional Multilinear Algebra: Part I, Pure Appl. Math. 23, Marcel Dekker, New York, 1973.
25. M. Marcus and H. Minc, On the relation between the determinant and the permanent, Illinois J. Math., 5 (1961), pp. 376-381.
26. H. Minc, Permanents, Encyclopedia Math. Appl. 9999, Addison-Wesley, Reading, MA, 1978.
27. J. Möcks, Topographic components model for event-related potentials and some biophysical considerations, IEEE Trans. Biomed. Eng., 35 (1988), pp. 482-484.
28. T. Muir, A Treatise on the Theory of Determinants: With Graduated Sets of Exercises for Use in Colleges and Schools, Macmillan and Co., London, 1882.
29. D. Nion and L. De Lathauwer, A Study of the Decomposition of a Third-Order Tensor in Rank-(L, L, 1) Terms, ESAT-STADIUS Internal Report 11-239, Department of Electrical Engineering (ESAT), KU Leuven, Leuven, Belgium, 2011.
30. L. Oeding and G. Ottaviani, Eigenvectors of tensors and algorithms for Waring decomposition, J. Symbolic Comput., 54 (2013), pp. 9-35.
31. E. Sanchez and B. Kowalski, Tensorial resolution: A direct trilinear decomposition, J. Chemometrics, 4 (1990), pp. 29-45.
32. R. Sands and F. Young, Component models for three-way data: An alternating least squares algorithm with optimal scaling features, Psychometrika, 45 (1980), pp. 39-67.
33. J. R. Schott, Kronecker product permutation matrices and their application to moment matrices of the normal distribution, J. Multivariate Anal., 87 (2003), pp. 177-190. (Pubitemid 37268355)
34. A. K. Smilde, R. Bro, and P. Geladi, Multi-way Analysis: Applications in the Chemical Sciences, Wiley, Chichester, UK, 2004.
35. L. Sorber, M. Van Barel, and L. De Lathauwer, Tensorlab v2.0, http://www.tensorlab. net/(2014).
36. L. Sorber, M. Van Barel, and L. De Lathauwer, Optimization-based algorithms for tensor decompositions: Canonical polyadic decomposition, decomposition in rank-(Lr,Lr,1) terms, and a new generalization, SIAM J. Optim., 23 (2013), pp. 695-720.
37. M. Sørensen and L. De Lathauwer, Blind signal separation via tensor decomposition with Vandermonde factor: Canonical polyadic decomposition, IEEE Trans. Signal Process., 61 (2013), pp. 5507-5519.
38. M. Sørensen and L. De Lathauwer, New Uniqueness Conditions for the Canonical Polyadic Decomposition of Third-Order Tensors, ESAT-STADIUS Internal Report 13-05, Department of Electrical Engineering (ESAT), KU Leuven, Leuven, Belgium, 2013.
39. M. Sørensen, L. De Lathauwer, P. Comon, S. Icart, and L. Deneire, Canonical polyadic decomposition with a columnwise orthonormal factor matrix, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 1190-1213.
40. J. Ten Berge and J. Tendeiro, The link between sufficient conditions by Harshman and by Kruskal for uniqueness in Candecomp/Parafac, J. Chemometrics, 23 (2009), pp. 321-323.