Kruskal's condition for uniqueness in Candecomp/Parafac when ranks and k-ranks coincide

Authors:
Alwin Stegeman;Jos M. F. Ten Berge
Affiliations:
Heijmans Institute of Psychological Research, University of Groningen, Grote Kruisstraat 2/1, Groningen, 9712 TS, The Netherlands;Heijmans Institute of Psychological Research, University of Groningen, Grote Kruisstraat 2/1, Groningen, 9712 TS, The Netherlands
Venue:
Computational Statistics & Data Analysis
Year:
2006

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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. Kruskal has previously shown that the three component matrices involved are essentially unique when the sum of their k-ranks is at least twice the rank of the decomposition plus 2. It was proved that Kruskal's sufficient condition is also necessary when the rank of the decomposition is 2 or 3. If the rank is 4 or higher, the condition is not necessary for uniqueness. However, when the k-ranks of the component matrices equal their ranks, necessity of Kruskal's condition still holds in the rank-4 case. Ten Berge and Sidiropoulos conjectured that Kruskal's condition is necessary for all cases of rank 4 and higher where ranks and k-ranks coincide. In the present paper we show that this conjecture is false.