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Sufficient Conditions for Uniqueness in Candecomp/Parafac and Indscal with Random Component Matrices

Published online by Cambridge University Press:  01 January 2025

Alwin Stegeman ,
Jos M. F. ten Berge  and
Lieven De Lathauwer
Alwin Stegeman*
Affiliation:
University of Groningen
Jos M. F. ten Berge
Affiliation:
University of Groningen
Lieven De Lathauwer
Affiliation:
ETIS, UMR 8051, Cergy-Pontoise
*
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: a.w.stegeman@rug.nl
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Abstract

A key feature of the analysis of three-way arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled from a continuous distribution, and the third component matrix has full column rank. In this context, we obtain almost sure sufficient uniqueness conditions for the Candecomp/Parafac and Indscal models separately, involving only the order of the three-way array and the number of components in the decomposition. Both uniqueness conditions are closer to necessity than the classical uniqueness condition by Kruskal.

Keywords

CandecompParafacIndscalthree-way arraysuniqueness
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 2 , June 2006 , pp. 219 - 229
Copyright
Copyright © 2006 The Psychometric Society

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Footnotes

Part of this research was supported by (1) the Flemish Government: (a) Research Council K.U. Leuven: GOA-MEFISTO-666, GOA-Ambiorics, (b) F.W.O. project G.0240.99, (c) F.W.O. Research Communities ICCoS and ANMMM, (d) Tournesol project T2004.13; and (2) the Belgian Federal Science Policy Office: IUAP P5/22. Lieven De Lathauwer holds a permanent research position with the French Centre National de la Recherche Scientifique (C.N.R.S.). He also holds an honorary research position with the K.U. Leuven, Leuven, Belgium.

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