On uniqueness of the canonical tensor decomposition with some form of symmetry

We study the 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 was obtained by Kruskal in 1977 [Linear Algebra Appl., 18 (1977), pp. 95-138]. More recently, Kruskal's uniqueness condition has been generalized to n >= 3, 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. We consider the decomposition with some form of symmetry, and prove necessary, sufficient, and necessary and sufficient uniqueness conditions analogous to the asymmetric case. For n = 3, 4, 5, we also prove generic uniqueness bounds on R. Most of these conditions are easy to check. Throughout, we emphasize the analogies and striking differences between the symmetric and asymmetric cases