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

In the Candecomp/Parafac (CP) model, a three-way array 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 2R+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 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.