On Generic Identifiability of 3-Tensors of Small Rank

Abstract

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