On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode

Abstract

In the Candecomp/Parafac (CP) model, a three-way array X ̲ is written as the sum of R outer vector product arrays and a residual array. The former comprise the columns of the component matrices A , B and C . For fixed residuals, ( A , B , C ) is unique up to trivial ambiguities, if 2 R + 2 is less than or equal to the sum of the k -ranks of A , B and C . This classical result was shown by Kruskal in 1977. In this paper, we consider the case where one of A , B , C has full column rank, and show that in this case Kruskal’s uniqueness condition implies a recently obtained uniqueness condition. Moreover, we obtain Kruskal-type uniqueness conditions that are weaker than Kruskal’s condition itself. Also, for ( A , B , C ) with rank ( A ) = R - 1 and C full column rank, we obtain easy-to-check necessary and sufficient uniqueness conditions. We extend our results to the Indscal decomposition in which the array X ̲ has symmetric slices and A = B is imposed. We consider the real-valued CP and Indscal decompositions, but our results are also valid for their complex-valued counterparts.