A Method to Avoid Diverging Components in the Candecomp/Parafac Model for Generic $I\timesJ\times2$ Arrays

Abstract

Computing the Candecomp/Parafac (CP) solution of R components (i.e., the best rank-R approximation) for a generic $I\times J\times 2$ array may result in diverging components, also known as “degeneracy.” In such a case, several components are highly correlated in all three modes, and their component weights become arbitrarily large. Evidence exists that this is caused by the nonexistence of an optimal CP solution. Instead of using CP, we propose to compute the best approximation by means of a generalized Schur decomposition (GSD), which always exists. The obtained GSD solution is the limit point of the sequence of CP updates (whether it features diverging components or not) and can be separated into a nondiverging CP part and a sparse Tucker3 part or into a nondiverging CP part and a smaller GSD part. We show how to obtain both representations and illustrate our results with numerical experiments.