## Arkiv för Matematik

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

### On the dimension of contact loci and the identifiability of tensors

#### 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.

#### Article information

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

Dates
Revised: 1 December 2017
First available in Project Euclid: 19 June 2019

https://projecteuclid.org/euclid.afm/1560968133

Digital Object Identifier
doi:10.4310/ARKIV.2018.v56.n2.a4

Mathematical Reviews number (MathSciNet)
MR3893774

Zentralblatt MATH identifier
07021438

#### Citation

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. https://projecteuclid.org/euclid.afm/1560968133