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Simplicity and Typical Rank Results for 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 M.F. ten Berge, University of Groningen, Groningen, The Netherlands. E-mail: [email protected]
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Abstract

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Matrices can be diagonalized by singular vectors or, when they are symmetric, by eigenvectors. Pairs of square matrices often admit simultaneous diagonalization, and always admit block wise simultaneous diagonalization. Generalizing these possibilities to more than two (non-square) matrices leads to methods of simplifying three-way arrays by nonsingular transformations. Such transformations have direct applications in Tucker PCA for three-way arrays, where transforming the core array to simplicity is allowed without loss of fit. Simplifying arrays also facilitates the study of array rank. The typical rank of a three-way array is the smallest number of rank-one arrays that have the array as their sum, when the array is generated by random sampling from a continuous distribution. In some applications, the core array of Tucker PCA is constrained to have a vast majority of zero elements. Both simplicity and typical rank results can be applied to distinguish constrained Tucker PCA models from tautologies. An update of typical rank results over the real number field is given in the form of two tables.

Keywords

tensor decompositiontensor ranktypical ranksparse arraysCandecompParafacTucker component analysis
Type
Editorial Notes
Information
Psychometrika , Volume 76 , Issue 1 , January 2011 , pp. 3 - 12
Copyright
Copyright © 2010 The Psychometric Society

Footnotes

This research was done jointly with Henk Kiers, Roberto Rocci, Alwin Stegeman, and Jorge Tendeiro. The author is obliged to Henk Kiers and Mohammed Bennani Dosse for helpful comments.

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