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Degeneracy in Candecomp/Parafac and Indscal Explained For Several Three-Sliced Arrays With A Two-Valued Typical Rank

Published online by Cambridge University Press:  01 January 2025

Alwin Stegeman
Alwin Stegeman*
Affiliation:
University of Groningen
*
Requests 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: [email protected]
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Abstract

The Candecomp/Parafac (CP) method decomposes a three-way array into a prespecified number R of rank-1 arrays, by minimizing the sum of squares of the residual array. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate sequences of solutions, in which several rank-1 arrays become highly correlated in all three modes and some elements of the rank-1 arrays become arbitrarily large. We consider the real-valued CP decomposition of all known three-sliced arrays, i.e., of size p× q × 3, with a two-valued typical rank. These are the 5 × 3 × 3 and 8 × 4 × 3 arrays, and the 3 × 3 × 4 and 3 × 3 × 5 arrays with symmetric 3 × 3 slices. In the latter two cases, CP is equivalent to the Indscal model. For a typical rank of {m, m+1}, we consider the CP decomposition with R=m of an array of rank m+1. We show that (in most cases) the CP objective function does not have a minimum but an infimum. Moreover, any sequence of feasible CP solutions in which the objective value approaches the infimum will become degenerate. We use the tools developed in Stegeman (2006), who considers p × p × 2 arrays, and present a framework of analysis which is of use to the future study of CP degeneracy related to a two-valued typical rank. Moreover, our examples show that CP uniqueness is not necessary for degenerate solutions to occur.

Keywords

CandecompParafacIndscalthree-way arraysdegenerate solutions
Type
Theory and Methods
Information
Psychometrika , Volume 72 , Issue 4 , December 2007 , pp. 601 - 619
Copyright
Copyright © 2007 The Psychometric Society

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Footnotes

The author is supported by the Dutch Organisation for Scientific Research (NWO), VENI grant 451-04-102.

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