On Uniqueness of thenth Order Tensor Decomposition into Rank-1 Terms with Linear Independence in One Mode

Abstract

We study uniqueness of the decomposition of an nth order tensor (also called n-way array) into a sum of R rank-1 terms (where each term is the outer product of n vectors). This decomposition is also known as Parafac or Candecomp, and a general uniqueness condition for $n=3$ has been obtained by Kruskal in 1977 [Linear Algebra Appl., 18 (1977), pp. 95–138]. More recently, Kruskal's uniqueness condition has been generalized to $n\geq3$, and less restrictive uniqueness conditions have been obtained for the case where the vectors of the rank-1 terms are linearly independent in (at least) one of the n modes. For this case, only $n=3$ and $n=4$ have been studied. We generalize these results by providing a framework of analysis for arbitrary $n\geq3$. Our results include necessary, sufficient, necessary and sufficient, and generic uniqueness conditions. For the sufficient uniqueness conditions, the rank of a matrix needs to be checked. The generic uniqueness conditions have the form of a bound on R in terms of the dimensions of the tensor to be decomposed.