Real identifiability vs. complex identifiability

Abstract

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