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Algebraic and Computational Aspects of Real Tensor Ranks

  • Toshio Sakata
  • Toshio Sumi
  • Mitsuhiro Miyazaki

Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Also part of the JSS Research Series in Statistics book sub series (JSSRES)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 1-10
  3. Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 11-15
  4. Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 17-28
  5. Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 29-37
  6. Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 39-59
  7. Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 61-80
  8. Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 81-91
  9. Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
    Pages 93-101
  10. Back Matter
    Pages 103-108

About this book

Introduction

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.

Keywords

Computational Aspects of Tensor Ranks Full Rank Tensors Maximal Rank of Real Tensors Simultaneous Singular Value Decomposition Typical Rank of Real Tensors

Authors and affiliations

  • Toshio Sakata
    • 1
  • Toshio Sumi
    • 2
  • Mitsuhiro Miyazaki
    • 3
  1. 1.Faculty of DesignKyushu UniversityFukuokaJapan
  2. 2.Faculty of Arts and ScienceKyushu UniversityFukuokaJapan
  3. 3.Department of MathematicsKyoto University of EducationKyotoJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-4-431-55459-2
  • Copyright Information The Author(s) 2016
  • Publisher Name Springer, Tokyo
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-4-431-55458-5
  • Online ISBN 978-4-431-55459-2
  • Series Print ISSN 2191-544X
  • Series Online ISSN 2191-5458
  • Buy this book on publisher's site