Arkiv för Matematik

  • Ark. Mat.
  • Volume 56, Number 2 (2018), 265-283.

On the dimension of contact loci and the identifiability of tensors

Edoardo Ballico, Alessandra Bernardi, and Luca Chiantini

Full-text: Open access


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.

Article information

Ark. Mat., Volume 56, Number 2 (2018), 265-283.

Received: 10 July 2017
Revised: 1 December 2017
First available in Project Euclid: 19 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Ballico, Edoardo; Bernardi, Alessandra; Chiantini, Luca. On the dimension of contact loci and the identifiability of tensors. Ark. Mat. 56 (2018), no. 2, 265--283. doi:10.4310/ARKIV.2018.v56.n2.a4.

Export citation