On the non-existence of optimal solutions and the occurrence of "degeneracy" in the CANDECOMP/PARAFAC model

Psychometrika

Volumn 73, Issue 3, 2008, Pages 431-439

  Krijnen, Wim P ^a   Dijkstra, Theo K ^b   Stegeman, Alwin ^b  

^a HANZE UNIVERSITY OF APPLIED SCIENCES   (Netherlands)

^b UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

Bounded sequences; Candecomp; Factor analysis; Level sets; Parafac

Indexed keywords

EID: 52949104087     PISSN: 00333123     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11336-008-9056-1     Document Type: Article

References (45)

1. A.H. Andersen W.S. Rayens 2004 Structure-seeking multilinear methods for the analysis of fMRI data NeuroImage 22 728 739
2. C.F. Beckmann S.M. Smith 2005 Tensorial extensions of independent component analysis for multisubject FMRI analysis NeuroImage 25 294 311
3. R. Bro S. De Jong 1997 A Fast non-negativity-constrained least squares algorithm Journal of Chemometrics 11 393 401
4. J.D. Carroll J.J. Chang 1970 Analysis of individual differences in multidimensional scaling via an N-way generalization of 'Eckart-Young' decomposition Psychometrika 35 283 319
5. W.S. DeSarbo J.D. Carroll 1985 Three-way metric unfolding via alternating weighted least squares Psychometrika 50 275 300
6. De Silva, V., & Lim, L.-H. (2006). Tensor rank and the ill-posedness of the best low-rank approximation problem. (SCCM Technical Report, 06-06). Preprint.
7. R.A. Harshman 1970 Foundations of the PARAFAC procedure: models and conditions for an "exploratory" multi-modal factor analysis UCLA Working Papers in Phonetics 16 1 84
8. R.A. Harshman 1972 Determination and proof of minimum uniqueness conditions for PARAFAC1 UCLA Working Papers in Phonetics 22 111 117
9. Harshman, R.A., & DeSarbo, W.S. (1984). An application of PARAFAC to a small sample problem, demonstrating preprocessing, orthogonality constraints, and split-half diagnostic techniques. In H.G. Law, C.W. Snyder, J.A. Hattie, & R.P. McDonald (Eds.), Research methods for multi-mode data analysis (pp. 602-642). New York: Praeger.
10. Harshman, R.A., & Lundy, M.E. (1984). Data preprocessing and the Extended PARAFAC model. In H.G. Law, C.W. Snyder, J.A. Hattie, & R.P. McDonald (Eds.), Research methods for multi-mode data analysis (pp. 216-284). New York: Praeger.
11. R.A. Harshman P. Ladefoged L. Goldstein 1977 Factor analysis of tongue shapes Journal of the Acoustical Society of America 62 693 707
12. P.K. Hopke P. Paatero H. Jia R.T. Ross R.A. Harshman 1998 Three-way (Parafac) factor analysis: examination and comparison of alternative computational methods as applied to ill-conditioned data Chemometrics and Intelligent Laboratory Systems 43 25 42
13. H.A.L. Kiers 1989 A computational short-cut for INDSCAL with orthonormality constraints on positive semi-definite matrices of lower rank Computational Statistics Quarterly 2 119 135
14. H.A.L. Kiers 1989 An alternating least squares algorithm for fitting the two- and three-way DEDICOM model and the INDIOSCAL model Psychometrika 54 515 521
15. H.A.L. Kiers 1991 Simple structure in component analysis techniques for mixtures of qualitative and quantitative variables Psychometrika 56 197 212
16. W.P. Krijnen 2006 Convergence of the sequence of parameters generated by alternating least squares algorithms Computational Statistics and Data Analysis 51 481 489
17. W.P. Krijnen J.M.F. Ten Berge 1992 A constrained PARAFAC method for positive manifold data Applied Psychological Measurement 16 295 305
18. Krijnen, W.P., & Kiers, H.A.L. (1993). A confirmatory and an exploratory approach to simple structure in PARAFAC. In J.H.L. Oud & R.A.W. van Blokland-Vogelesang (Eds.), Advances in longitudinal and multivariate analysis in the behavioral sciences (pp. 165-177). Leiden: DSWO Press.
19. J.B. Kruskal 1976 More factors than subjects, tests and treatments: an indeterminacy theorem for canonical decomposition & individual differences scaling Psychometrika 41 281 293
20. J.B. Kruskal 1977 Three-way arrays: rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics Linear Algebra and Its Applications 18 95 138
21. Kruskal, J.B. (1989). Rank, decomposition, and uniqueness for 3-way and n-way arrays. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 7-18). Amsterdam: Elsevier.
22. Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1983). Some relationships between Tucker's three-mode factor analysis and PARAFAC/CANDECOMP. Paper presented at the annual meeting of the Psychometric Society, Los Angeles.
23. Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1985). Several mathematical relationships between PARAFAC/CANDECOMP and three-mode factor analysis. Paper presented at the annual meeting of the classification society, St. John's, Newfoundland.
24. Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1989). How 3-MFA data can cause degenerate PARAFAC solutions, among other relationships. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 115-122). Amsterdam: Elsevier.
25. S. Leurgans R.T. Ross 1992 Multilinear models: applications in spectroscopy Statistical Science 7 289 319
26. Lim, L.-H. (2005). Optimal solutions to non-negative Parafac/multilinear NMF always exist. Talk at the Workshop on tensor decompositions and applications, CIRM, Luminy, Marseille, France. Available online at http://www.etis.ensea.fr/~wtda.
27. Luenberger, D.G. (1969). Optimization by vector space methods. New York: Wiley.
28. J.R. Magnus H. Neudecker 1979 The commutation matrix: some properties for applications The Annals of Statistics 7 381 394
29. J.P. Meyer 1980 Causal attribution for success and failure. A multivariate investigation of dimensionally, formation, and consequences Journal of Personality and Social Psychology 38 704 718
30. B.C. Mitchell D.S. Burdick 1994 Slowly converging PARAFAC sequences: swamps and two-factor degeneracies Journal of Chemometrics 8 155 168
31. Ortega, J.M., & Rheinboldt, W.C. (1970). Iterative solution of nonlinear equations in several variables. San Diego: Academic Press.
32. P. Paatero 1999 The multilinear engine-a table-driven least squares program for solving multilinear problems, including the n-way parallel factor analysis model Journal of Computational and Graphical Statistics 8 854 888
33. P. Paatero 2000 Construction and analysis of degenerate PARAFAC models Journal of Chemometrics 14 285 299
34. Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). New York: McGraw-Hill.
35. J. Schur 1911 Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen Journal für die Reine und Angewante Mathematiek 140 1 28
36. A. Smilde 1992 Three-way analysis: Problems and prospects Chemometrics and Intelligent Laboratory Systems 15 143 157
37. A. Stegeman 2006 Degeneracy in Candecomp/Parafac explained for p×p×2 arrays of rank p+1 or higher Psychometrika 71 483 501
38. Stegeman, A. (2008a). Low-rank approximation of generic p×q×2 arrays and diverging components in the Candecomp/Parafac model. SIAM Journal on Matrix Analysis and Applications, to appear.
39. Stegeman, A. (2008b). Degeneracy in Candecomp/Parafac and Indscal explained for several three-sliced arrays with a two-valued typical rank. Psychometrika, to appear.
40. A. Stegeman N.D. Sidiropoulos 2007 On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition Linear Algebra and Its Applications 420 540 552
41. G.P.H. Styan 1973 Hadamard products and multivariate statistical analysis Linear Algebra and Its Applications 6 217 240
42. J.M.F. Ten Berge H.A.L. Kiers J. De Leeuw 1988 Explicit CANDECOMP-PARAFAC solutions for a contrived 2×2×2 array of rank three Psychometrika 53 579 584
43. J.M.F. Ten Berge H.A.L. Kiers W.P. Krijnen 1993 The problem of negative saliences and asymmetry in INDSCAL Journal of Classification 10 115 124
44. G. Tomasi R. Bro 2006 A comparison of algorithms for fitting the Parafac model Computational Statistics and Data Analysis 50 1700 1734
45. B.J.H. Zijlstra H.A.L. Kiers 2002 Degenerate solutions obtained from several variants of factor analysis Journal of Chemometrics 16 596 605