On Generic Identifiability of 3-Tensors of Small Rank

Abstract

We introduce an inductive method for the study of the uniqueness of\ndecompositions of tensors, by means of tensors of rank 1. The method is based\non the geometric notion of weak defectivity. For three-dimensional tensors of\ntype (a, b, c), a\\le b\\le c, our method proves that the decomposition is unique\n(i.e. k-identifiability holds) for general tensors of rank k, as soon as k\\le\n(a+1)(b+1)/16. This improves considerably the known range for identifiability.\nThe method applies also to tensor of higher dimension. For tensors of small\nsize, we give a complete list of situations where identifiability does not\nhold. Among them, there are 4\\times4\\times4 tensors of rank 6, an interesting\ncase because of its connection with the study of DNA strings.\n