Real identifiability vs complex identifiability

TL;DR

This paper investigates the differences between real and complex tensor identifiability, showing that some tensors are uniquely decomposable over the reals despite having multiple complex decompositions, with implications for tensor analysis.

Contribution

It demonstrates the existence of open sets of tensors with unique real decompositions but multiple complex decompositions, highlighting a nuanced distinction between real and complex identifiability.

Findings

Existence of open subsets with real but not complex identifiability.

Examples of symmetric tensors with equal real and complex ranks that are real identifiable.

Instances where real identifiability fails in non-trivial open subsets.

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.