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Degeneracy in Candecomp/Parafac Explained for p × p × 2 Arrays of Rank p + 1 or Higher

Published online by Cambridge University Press:  01 January 2025

Alwin Stegeman
Alwin Stegeman*
Affiliation:
University of Groningen
*
Request for reprints should be sent to Alwin Stegeman, Heijmans Institute of Psychological Research, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands. E-mail: a.w.stegeman@rug.nl
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Abstract

The Candecomp/Parafac (CP) model decomposes a three-way array into a prespecified number R of rank-1 arrays and a residual array, in which the sum of squares of the residual array is minimized. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate solutions, in which some components are highly correlated in all three modes and the elements of these components become arbitrarily large. We consider the real-valued CP model in which p × p × 2 arrays of rank p + 1 or higher are decomposed into p rank-1 arrays and a residual array. It is shown that the CP objective function does not have a minimum in these cases, but an infimum. Moreover, any sequence of CP approximations, of which the objective value approaches the infimum, will become degenerate. This result extends Ten Berge, Kiers, & De Leeuw (1988), who consider a particular 2 × 2 × 2 array of rank 3.

Keywords

CandecompParafacthree-way arraysdegenerate solutions

Information

Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 3 , September 2006 , pp. 483 - 501
Copyright
Copyright © 2006 The Psychometric Society

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Footnotes

The author is obliged to Jos ten Berge and Henk Kiers for helpful comments. Also, the author would like to thank the Associate Editor and the anonymous reviewers for many suggestions on how to improve the contents and the presentation of the paper.

References

Apostol, T.M. (1969). Calculus, Vol. II, (2nd ed.). Dordrecht: Wiley.Google Scholar
Carroll, J.D., Chang, J.J. (1970). Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart–Young decomposition. Psychometrika, 35, 283–319.CrossRefGoogle Scholar
Harshman, R.A. (1970). Foundations of the Parafac procedure: Models and conditions for an “explanatory” multimodal factor analysis. UCLA Working Papers in Phonetics, 16, 1–84.Google Scholar
Harshman, R.A. (2004). The problem and nature of degenerate solutions or decompositions of 3-way arrays. Talk at the Tensor Decompositions Workshop, July 19–23. Palo Alto, CA: AIM.Google Scholar
Harshman, R.A., Lundy, M.E. (1984). Data preprocessing and the extended Parafac model. In Law, H.G., Snyder, C.W. Jr., Hattie, J.A., McDonald, R.P. (Eds.), Research methods for multimode data analysis (pp. 216–284). Dordrecht: Praeger.Google Scholar
Ja’ Ja’, J. (1979). Optimal evaluation of pairs of bilinear forms. SIAM Journal on Computing, 8, 443–462.Google Scholar
Kroonenberg, P.M. (1983). Three-mode principal component analysis, Leiden: DSWO.Google Scholar
Kruskal, J.B. (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.CrossRefGoogle Scholar
Kruskal, J.B. (1989). Rank, decomposition, and uniqueness for 3-way and N-way arrays. In Coppi, R., Bolasco, S. (Eds.), Multiway data analysis (pp. 7–18). Amsterdam: North-Holland.Google Scholar
Kruskal, J.B., Harshman, R.A., Lundy, M.E. (1989). How 3-MFA data can cause degenerate Parafac solutions, among other relationships. In Coppi, R., Bolasco, S. (Eds.), Multiway data analysis (pp. 115–121). Amsterdam: North-Holland.Google Scholar
Lim, L.-H. (2005). Optimal solutions to non-negative Parafac/multilinear NMF always exist. Talk at the Workshop on Tensor Decompositions and Applications, August 29—September 2. Marseille: CIRM, Luminy.Google Scholar
Mitchell, B.C., Burdick, D.S. (1994). Slowly converging Parafac sequences: Swamps and two-factor degeneracies. Journal of Chemometrics, 8, 155–168.CrossRefGoogle Scholar
Paatero, P. (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.Google Scholar
Paatero, P. (2000). Construction and analysis of degenerate Parafac models. Journal of Chemometrics, 14, 285–299.3.0.CO;2-1>CrossRefGoogle Scholar
Rayens, W.S., Mitchell, B.C. (1997). Two-factor degeneracies and a stabilization of Parafac. Chemometrics and Intelligent Laboratory Systems, 38, 173–181.CrossRefGoogle Scholar
Sidiropoulos, N.D. (2004). Low-rank decomposition of multi-way arrays: A signal processing perspective. In Proceedings of IEEE Sensor Array and Multichannel (SAM) Signal Processing Workshop, July 18–21, Sitges, Barcelona, Spain.CrossRefGoogle Scholar
Smilde, A., Bro, R., Geladi, P. (2004). Multi-way analysis: Applications in the chemical sciences, Dordrecht: Wiley.CrossRefGoogle Scholar
Stegeman, A. (2005a). Degeneracy in Candecomp/Parafac explained for random p × q × 2 arrays. Technical Report. Department of Psychology, University of Groningen, Groningen, The Netherlands. Submitted.Google Scholar
Stegeman, A. (2005b). Degeneracy in Candecomp/Parafac explained for 5 × 3 × 3 arrays of rank 6 or higher. Technical Report. Department of Psychology, University of Groningen, Groningen, The Netherlands.Google Scholar
Ten Berge, J.M.F. (1991). Kruskal’s polynomial for 2 × 2 × 2 arrays and a generalization to 2 × n × n arrays. Psychometrika, 56, 631–636.CrossRefGoogle Scholar
Ten Berge, J.M.F. (2004). Partial uniqueness in Candecomp/Parafac. Journal of Chemometrics, 18, 12–16.CrossRefGoogle Scholar
Ten Berge, J.M.F., Kiers, H.A.L. (1999). Simplicity of core arrays in three-way principal component analysis and the typical rank of p × q × 2 arrays. Linear Algebra and its Applications, 294, 169–179.CrossRefGoogle Scholar
Ten Berge, J.M.F., Kiers, H.A.L., De Leeuw, J. (1988). Explicit Candecomp/Parafac solutions for a contrived 2 × 2 × 2 array of rank three. Psychometrika, 53, 579–584.CrossRefGoogle Scholar
Tucker, L.R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.CrossRefGoogle ScholarPubMed
Zijlstra, B.J.H., Kiers, H.A.L. (2002). Degenerate solutions obtained from several variants of factor analysis. Journal of Chemometrics, 16, 596–605.CrossRefGoogle Scholar