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The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Author

Listed:
  • Alessandra Bernardi

    (Dipartimento di Matematica, Università di Trento, 38123 Trento, Italy)

  • Enrico Carlini

    (Dipartimento di Scienze Matematiche, Politecnico di Torino, 10129 Turin, Italy)

  • Maria Virginia Catalisano

    (Dipartimento di Ingegneria Meccanica, Energetica, Gestionale e dei Trasporti, Università degli studi di Genova, 16145 Genoa, Italy)

  • Alessandro Gimigliano

    (Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy)

  • Alessandro Oneto

    (Barcelona Graduate School of Mathematics, and Universitat Politècnica de Catalunya, 08034 Barcelona, Spain)

Abstract

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X . The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

Suggested Citation

  • Alessandra Bernardi & Enrico Carlini & Maria Virginia Catalisano & Alessandro Gimigliano & Alessandro Oneto, 2018. "The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition," Mathematics, MDPI, vol. 6(12), pages 1-86, December.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:12:p:314-:d:189041
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    Citations

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    Cited by:

    1. Wenjing Jing & Linzhang Lu & Qilong Liu, 2022. "Discriminative Nonnegative Tucker Decomposition for Tensor Data Representation," Mathematics, MDPI, vol. 10(24), pages 1-16, December.

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