Simplicity transformations for three-way arrays with symmetric slices

Abstract

Tucker three-way PCA and Candecomp/Parafac are two well-known methods of generalizing principal component analysis to three way data. Candecomp/Parafac yields component matrices that are typically unique up to jointly permuting and rescaling columns. Tucker-3 analysis, on the other hand, has full transformational freedom. That is, the fit does not change when the component matrices are postmultiplied by nonsingular transformation matrices, provided that the inverse transformations are applied to the so-called core array. This freedom of transformation can be used to create a simple structure in the component matrices, and/or in the core array. We address the question of how a core array, or, in fact, any three-way array can be transformed to have a maximum number of zero elements. Direct applications are in Tucker-3 analysis (to facilitate the interpretation of a Tucker-3 solution), in constrained Tucker-3 analysis, and as a mathematical tool to examine rank and generic or typical rank of three-way arrays. Only arrays with symmetric frontal slices are considered.