Algebraic and Computational Aspects of Real Tensor Ranks

1. Auflage 2016
ISBN: 978-4-431-55458-5
Verlag: Springer Japan

Buch, Englisch, 108 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 1883 g

Reihe: JSS Research Series in Statistics

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.

Zielgruppe

Research

Autoren/Hrsg.

Sakata, Toshio

Miyazaki, Mitsuhiro

Sumi, Toshio

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Vektoranalysis, Physikalische Felder
• Mathematik | Informatik Mathematik Topologie Algebraische Topologie

Inhaltsverzeichnis

Basics of Tensor Rank.- 3-Tensors.- Simple Evaluation
Methods of Tensor Rank.- Absolutely Nonsingular Tensors and Determinantal
Polynomials.- Maximal Ranks.- Typical Ranks.- Global Theory of Tensor Ranks.- 2
× 2 × · · · × 2 Tensors.