Relating two proposed methods for speedup of algorithms for fitting two- and three-way principal component and related multilinear models

Multilinear analysis methods such as component (and three-way component) analysis of very large data sets can become very computationally demanding and even infeasible unless some method is used to compress the data and/or speed up the algorithms. We discuss two previously proposed speedup methods. (a) Alsberg and Kvalheim have proposed use of data simplification along with some new analysis algorithms. We show that their procedures solve the same problem as (b) the more general approach proposed (in a different context) by Carroll, Pruzansky, and Kruskal. In the latter approach, a speed improvement is attained by applying any (three-mode) PCA algorithm to a small (three-way) array derived from the original data. Hence, it can employ the new algorithms by Alsberg and Kvalheim, but, as is shown in the present paper, it is easier and often more efficient to apply standard (three-mode) PCA algorithms to the small array. Finally, it is shown how the latter approach for speed improvement can also be used for other three-way models and analysis methods (e.g., PARAFAC/CANDECOMP and constrained three-mode PCA).