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On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model

Published online by Cambridge University Press:  01 January 2025

Wim P. Krijnen ,
Theo K. Dijkstra  and
Alwin Stegeman
Wim P. Krijnen
Affiliation:
Hanze University Groningen
Theo K. Dijkstra
Affiliation:
University of Groningen
Alwin Stegeman*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Alwin Stegeman, Heymans Institute for 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

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The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal.

Keywords

CandecompParafaclevel setsbounded sequencesfactor analysis
Type
Theory and Methods
Information
Psychometrika , Volume 73 , Issue 3 , September 2008 , pp. 431 - 439
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NC
This article distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Copyright
Copyright © 2008 The Author(s)

Footnotes

A. Stegeman is supported by the Dutch Organisation for Scientific Research (NWO), Veni grant 451-04-102.

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