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.

We propose a new sufficient condition for verifying whether general rank-$r$ complex tensors of arbitrary order admit a unique decomposition as a linear combination of rank-$1$ tensors. A practical algorithm is proposed for verifying this condition, with which it was established that in all spaces of dimension less than 15000, with a few known exceptions, listed in the paper, generic identifiability holds for ranks up to one less than the generic rank of the space. This is the largest possible rank value for which generic identifiability can hold, except for spaces with a perfect shape. The algorithm can also verify the identifiability of a given specific rank-$r$ decomposition, provided that it can be shown to correspond to a nonsingular point of the $r$th order secant variety. For sufficiently small rank, which nevertheless improves upon the known bounds for specific identifiability, some local equations of this variety are known, allowing us to verify this property. As a particular example of our appro...

We prove that the general symmetric tensor in $S^d\mathbb {C}^{n+1}$ of rank $r$ is identifiable, provided that $r$ is smaller than the generic rank. That is, its Waring decomposition as a sum of $r$ powers of linear forms is unique. Only three exceptional cases arise, all of which were known in the literature. Our original contribution regards the case of cubics ( $d=3$ ), while for $d\ge 4$ we rely on known results on weak defectivity by Ballico, Ciliberto, Chiantini, and Mella.

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several...