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On the dimension of contact loci and the identifiability of tensors

Edoardo Ballico Università di Trento Alessandra Bernardi Università di Trento Luca Chiantini Università di Siena

Algebraic Geometry mathscidoc:1912.43015

Arkiv for Matematik, 56, (2), 265 – 283, 2018
Let X⊂Pr be an integral and non-degenerate variety. Set n:=dim(X). We prove that if the (k+n−1)-secant variety of X has (the expected) dimension (k+n−1)(n+1)−1<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⊂X with ♯(S)=k. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures G1,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<r, this is a known result in many cases), beating several bounds on the identifiability of tensors.
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@inproceedings{edoardo2018on,
  title={On the dimension of contact loci and the identifiability of tensors},
  author={Edoardo Ballico, Alessandra Bernardi, and Luca Chiantini},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204101510210265571},
  booktitle={Arkiv for Matematik},
  volume={56},
  number={2},
  pages={265 – 283},
  year={2018},
}
Edoardo Ballico, Alessandra Bernardi, and Luca Chiantini. On the dimension of contact loci and the identifiability of tensors. 2018. Vol. 56. In Arkiv for Matematik. pp.265 – 283. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204101510210265571.
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