An uniqueness condition for the 4-way CANDECOMP/PARAFAC model with collinear loadings in three modes

In this paper we investigate the uniqueness of the 4-way CANDECOMP/PARAFAC (CP) model in the case where file only possible linear dependencies between the columns of the loading matrices take die form of collinear loadings. For this special configuration we state a necessary and sufficient condition for having full column rank of the Khatri-Rao product of two loading matrices. This allows to derive a sufficient condition for uniqueness of the 4-way CP model with collinear loadings in at most three modes. The result is illustrated by analyzing 4-way fluorescence data.