7 apr 2017 -- 14:00
Aula Tonelli VI piano, Dipartimento di Matematica, Bologna
Abstract.
A tensor $T$ of rank $k$ is identifiable when it has a unique decomposition in terms of rank-$1$ tensors. There are cases in which the identifiability fails over $\mathbb{C}$, for general tensors of fixed rank. The failure, often, is due to the existence of an elliptic normal curve through general points of the corresponding variety of rank-$1$ tensors. After a brief introduction to the subject, we prove the existence of non-empty euclidean open subsets of some varieties of real $ k $-rank tensors, whose elements have $ 2 $ complex decompositions, but are identifiable over $\mathbb{R}$. Moreover we provide examples of non-trivial euclidean open subsets in certain spaces of symmetric tensors and of almost unbalanced tensors, whose elements have real rank equal to the complex rank and are identifiable over $\mathbb{R}$ but not over $\mathbb{C}$. On the contrary, there are examples of tensors of given real rank, for which identifiability over $ \mathbb{R} $ can't hold in non-trivial open subsets. These results have been obtained in collaboration with Cristiano Bocci and Luca Chiantini.