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More Factors than Subjects, Tests and Treatments: An Indeterminacy Theorem for Canonical Decomposition and Individual differences Scaling

Published online by Cambridge University Press:  01 January 2025

Joseph B. Kruskal
Joseph B. Kruskal*
Affiliation:
Bell Laboratories
*
Requests for reprints should be sent to Joseph B. Kruskal, Bell Laboratories, Murray Hill, New Jersey 07974.
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Abstract

Some methods that analyze three-way arrays of data (including INDSCAL and CANDECOMP/PARAFAC) provide solutions that are not subject to arbitrary rotation. This property is studied in this paper by means of the “triple product” [A, B, C] of three matrices. The question is how well the triple product determines the three factors. The answer: up to permutation of columns and multiplication of columns by scalars—under certain conditions. In this paper we greatly expand the conditions under which the result is known to hold. A surprising fact is that the nonrotatability characteristic can hold even when the number of factors extracted is greater than every dimension of the three-way array, namely, the number of subjects, the number of tests, and the number of treatments.

Keywords

three-way arraytrilinear decompositionrotation problemarray rankfactor analysismultidimensional scaling
Type
Original Paper
Information
Psychometrika , Volume 41 , Issue 3 , September 1976 , pp. 281 - 293
Copyright
Copyright © 1976 The Psychometric Society

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Footnotes

This paper is being published in place of Dr. Kruskal's presidential address to the Psychometric Society, April, 1975. Further results like those in this paper, as well as a surprising connection with an area of mathematics called arithmetic complexity theory, will be found in a more recent paper [Kruskal, in press].

References

Reference Notes

Brockett, R. W. & Dobkin, D. On the number of multiplications required for matrix multiplications. Unpublished manuscript.Google Scholar
Brockett, R. W. & Dobkin, D. On the optimal evaluation of a set of bilinear forms. Unpublished manuscript.Google Scholar
Carroll, J. D. Personal communication.Google Scholar
Harshman, R. A. Determination and proof of minimum uniqueness conditions for PARAFAC1. Working Papers in Phonetics No. 22, University of California at Los Angeles, 1972.Google Scholar
Harshman, R. A. Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multimodel factor analysis. Working Papers in Phonetics No. 16, University of California at Los Angeles, 1970.Google Scholar

References

Carroll, J. D. & Chang, J. J. Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart-Young decomposition. Psychometrika, 1970, 35, 283–319.CrossRefGoogle Scholar
Fiduccia, C. M. On obtaining upper bounds on the complexity of matrix multiplication. In Miller, Raymond E. & Thatcher, James W.(Eds.), Complexity of computer computations. New York: Plenum Press. 1972, 31–40.CrossRefGoogle Scholar
Kruskal, Joseph B. Trilinear decomposition of three-way arrays: Rank and uniqueness in arithmetic complexity and in statistical models. Linear algebra and its applications, in press.Google Scholar
Strassen, V. Gaussian elimination is not optimal. Numerische Mathematik, 1968, 13, 354–356.CrossRefGoogle Scholar
Tucker, L. R. The extension of factor analysis to three dimensional matrices. In Frederiksen, C. H. & Gulliksen, H.(Eds.), Contributions to mathematical psychology, 1964, New York: Holt, Rinehart & Winston.Google Scholar