Abstract
Let $X \subset \mathbb{P}^r$ be an integral and non-degenerate variety. Set $n := \mathrm{dim} \: (X)$. We prove that if the $(k+n-1)$-secant variety of $X$ has (the expected) dimension $(k+n-1) (n+1)-1 \lt r$ and $X$ is not uniruled by lines, then $X$ is not $k$-weakly defective and hence the $k$-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique $S \subset X$ with $\sharp (S) = k$. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures $\mathcal{G}_{1,d}$. If $X$ is the Segre embedding of a multiprojective space we prove identifiability for the $k$-secant variety (assuming that the $(k+n-1)$-secant variety has dimension $(k+n-1) (n+1)-1 \lt r$, this is a known result in many cases), beating several bounds on the identifiability of tensors.
Citation
Edoardo Ballico. Alessandra Bernardi. Luca Chiantini. "On the dimension of contact loci and the identifiability of tensors." Ark. Mat. 56 (2) 265 - 283, October 2018. https://doi.org/10.4310/ARKIV.2018.v56.n2.a4