Algebraic and Computational Aspects of Real Tensor Ranks

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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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.

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

• 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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