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

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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.