Abstract
We study the uniqueness of the decomposition of an th order tensor (also called -way array) into a sum of rank-1 terms (where each term is the outer product of vectors). This decomposition is also known as Parafac or Candecomp, and a general uniqueness condition for was obtained by Kruskal in 1977 [Linear Algebra Appl., 18 (1977), pp. 95–138]. More recently, Kruskal’s uniqueness condition has been generalized to , 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 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 , 4, 5, we also prove generic uniqueness bounds on . Most of these conditions are easy to check. Throughout, we emphasize the analogies and striking differences between the symmetric and asymmetric cases.
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