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The Typical Rank of Tall Three-Way Arrays

Published online by Cambridge University Press:  01 January 2025

Jos M. F. Ten Berge
Jos M. F. Ten Berge*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Jos ten Berge, Heijmans 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

The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank of I × J × K arrays with K = 2 to a particular class of arrays with K ≥ 2. It is shown that the typical rank is I when the array is tall in the sense that JK − J < I < JK. In addition, typical rank results are given for the case where I equals JK − J.

Keywords

three-way ranktensorial rankCANDECOMPPARAFACthree-way component analysis
Type
Original Paper
Information
Psychometrika , Volume 65 , Issue 4 , December 2000 , pp. 525 - 532
Copyright
Copyright © 2000 The Psychometric Society

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Footnotes

The author is obliged to Henk Kiers, Tom Snijders, and Philip Thijsse for helpful comments.

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