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Algebraic and Computational Aspects of Real Tensor Ranks

  • Book
  • © 2016

Overview

Authors:
  1. Toshio Sakata

    1. Faculty of Design, Kyushu University, Fukuoka, Japan

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  2. Toshio Sumi

    1. Faculty of Arts and Science, Kyushu University, Fukuoka, Japan

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    You can also search for this author in PubMed   Google Scholar

  3. Mitsuhiro Miyazaki

    1. Department of Mathematics, Kyoto University of Education, Kyoto, Japan

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  • Presents the first comprehensive treatment of maximal ranks and typical ranks over the real number file
  • Provides interesting ideas of determinant polynomials, determinantal ideals, absolutely nonsingular tensors and absolutely full column rank tensors
  • Includes numerical methods of determining ranks by simultaneous singular value decomposition through a theory of matrix star algebra

Part of the book series: SpringerBriefs in Statistics (BRIEFSSTATIST)

Part of the book sub series: JSS Research Series in Statistics (JSSRES)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-viii
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  2. Basics of Tensor Rank

    • Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 1-10

  3. 3-Tensors

    • Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 11-15

  4. Simple Evaluation Methods of Tensor Rank

    • Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 17-28

  5. Absolutely Nonsingular Tensors and Determinantal Polynomials

    • Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 29-37

  6. Maximal Ranks

    • Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 39-59

  7. Typical Ranks

    • Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 61-80

  8. Global Theory of Tensor Ranks

    • Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 81-91

  9. \(2\times 2\times \cdots \times 2\) Tensors

    • Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 93-101

  10. Back Matter

    Pages 103-108
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Keywords

About this book

This book provides comprehensive summaries of theoretical (algebraic) and computational aspects of tensor ranks, maximal ranks, and typical ranks, over the real number field. Although tensor ranks have been often argued in the complex number field, it should be emphasized that this book treats real tensor ranks, which have direct applications in statistics. The book provides several interesting ideas, including determinant polynomials, determinantal ideals, absolutely nonsingular tensors, absolutely full column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. In addition to reviews of methods to determine real tensor ranks in details, global theories such as the Jacobian method are also reviewed in details. The book includes as well an accessible and comprehensive introduction of mathematical backgrounds, with basics of positive polynomials and calculations by using the Groebner basis. Furthermore, this book provides insights into numerical methods of finding tensor ranks through simultaneous singular value decompositions.

Authors and Affiliations

  • Faculty of Design, Kyushu University, Fukuoka, Japan

    Toshio Sakata

  • Faculty of Arts and Science, Kyushu University, Fukuoka, Japan

    Toshio Sumi

  • Department of Mathematics, Kyoto University of Education, Kyoto, Japan

    Mitsuhiro Miyazaki

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