Simplicity transformations for three-way arrays with symmetric slices, and applications to Tucker-3 models with sparse core arrays

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 A (e.g., for subjects or objects), B (e.g., for variables) and C (e.g., for occasions) that are typically unique up to jointly permuting and rescaling columns. Tucker-3 analysis, on the other hand, has full transforma- tional freedom. That is, the fit does not change when A, B, and C are postmultiplied by nonsingular transformation matrices, provided that the inverse transformations are applied to the so-called core array G. This freedom of transformation can be used to create a simple structure in A, B, C, and/or in G. This paper deals with the latter possibility exclusively. It revolves around 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, where simplicity of the core may facilitate the interpretation of a Tucker-3 solution, and in constrained Tucker-3 analysis, where hypotheses involving sparse cores are taken into account. In the latter cases, it is important to know what degree of sparseness can be attained as a tautology, by using the transforma- tional freedom. In addition, simplicity transformations have proven useful as a mathematical tool to examine rank and generic or typical rank of three-way arrays. So far, a number of simplicity results have been attained, pertaining to arrays sampled randomly from con- tinuous distributions. These results do not apply to three-way arrays with symmetric slices in one direction. The present paper offers a number of simplicity results for arrays with symmetric slices of order $2 × 2$, $3 × 3$ and $4 × 4$. Some generalizations to higher orders are also discussed. As a mathematical application, the problem of determining the typical rank of $4 × 3 × 3$ and $5 × 3 × 3$ arrays with symmetric slices will be revisited, using a sparse form with only eight out of 36 elements nonzero for the former case and 10 out of 45 elements nonzero for the latter one, that can be attained almost surely for such arrays. The issue of maximal simplicity of the targets to be presented will be addressed, either by formal proofs or by relying on simulation results.

Publication
Linear Algebra and its Applications, 430