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Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices

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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.

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Authors and Affiliations

  1. University of Groningen, Groningen

    Alwin Stegeman & Jos M. F. Ten Berge

  2. ETIS, UMR 8051, Cergy-Pontoise

    Lieven De Lathauwer

  3. Heijmans Institute of Psychological Research, University of Groningen, Grote Kruisstraat 2/1, 9712 TS, Groningen, The Netherlands

    Alwin Stegeman

Authors
  1. Alwin Stegeman
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  2. Jos M. F. Ten Berge
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  3. Lieven De Lathauwer
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Correspondence to Alwin Stegeman.

Additional information

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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Stegeman, A., Berge, J.M.F.T. & Lathauwer, L.D. Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices. Psychometrika 71, 219–229 (2006). https://doi.org/10.1007/s11336-006-1278-2

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  • DOI: https://doi.org/10.1007/s11336-006-1278-2

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