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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
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.

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