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On Uniqueness in Candecomp/Parafac

Published online by Cambridge University Press:  01 January 2025

Jos M. F. ten Berge  and
Nikolaos D. Sidiropoulos
Jos M. F. ten Berge*
Affiliation:
University of Groningen
Nikolaos D. Sidiropoulos
Affiliation:
University of Minnesota
*
Requests for reprints should be sent to Jos M.E ten Berge, Heÿmans 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

One of the basic issues in the analysis of three-way arrays by CANDECOMP/PARAFAC (CP) has been the question of uniqueness of the decomposition. Kruskal (1977) has proved that uniqueness is guaranteed when the sum of thek-ranks of the three component matrices involved is at least twice the rank of the solution plus 2. Since then, little has been achieved that might further qualify Kruskal's sufficient condition. Attempts to prove that it is also necessary for uniqueness (except for rank 1 or 2) have failed, but counterexamples to necessity have not been detected. The present paper gives a method for generating the class of all solutions (or at least a subset of that class), given a CP solution that satisfies certain conditions. This offers the possibility to examine uniqueness for a great variety of specific CP solutions. It will be shown that Kruskal's condition is necessary and sufficient when the rank of the solution is three, but that uniqueness may hold even if the condition is not satisfied, when the rank is four or higher.

Keywords

CandecompParafacuniquenessthree-way arrays
Type
Articles
Information
Psychometrika , Volume 67 , Issue 3 , September 2002 , pp. 399 - 409
Copyright
Copyright © 2002 The Psychometric Society

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Footnotes

The authors are obliged to Henk Kiers for commenting on a previous draft, and to Tom Snijders for suggesting a proof mentioned in the appendix.

References

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