Real identifiability vs complex identifiability

Abstract

Let $T$ be a real tensor of (real) rank $r$. $T$ is 'identifiable' when it has a unique decomposition in terms of rank $1$ tensors. There are cases in which the identifiability fails over the complex field, for general tensors of rank $r$. This behavior is quite peculiar when the rank $r$ is submaximal. Often, the failure is due to the existence of an elliptic normal curve through general points of the corresponding Segre, Veronese or Grassmann variety. We prove the existence of nonempty euclidean open subsets of some variety of tensors of rank $r$, whose elements have several decompositions over $\mathbb C$, but only one of them is formed by real summands. Thus, in the open sets, tensors are not identifiable over $\mathbb C$, but are identifiable over $\mathbb R$. We also provide examples of non trivial euclidean open subsets in a whole space of symmetric tensors (of degree $7$ and $8$ in three variables) and of almost unbalanced tensors Segre Product ($\mathbb P^2\times \mathbb P^4\times \mathbb P^9$) whose elements have typical real rank equal to the complex rank, and are identifiable over $\mathbb R$, but not over $\mathbb C$. On the contrary, we provide examples of tensors of given real rank, for which real identifiability cannot hold in non-trivial open subsets.