Algebraic and Computational Aspects of Real Tensor Ranks

Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki (Autoren)

Buch | Softcover

108 Seiten

2016 | 1st ed. 2016
Springer Verlag, Japan
978-4-431-55458-5 (ISBN)

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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.

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.

Erscheint lt. Verlag	30.3.2016
Reihe/Serie	JSS Research Series in Statistics
Reihe/Serie	SpringerBriefs in Statistics
Zusatzinfo	5 Illustrations, black and white; VIII, 108 p. 5 illus.
Verlagsort	Tokyo
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Computerprogramme / Computeralgebra
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
	Naturwissenschaften
	Sozialwissenschaften ► Soziologie ► Empirische Sozialforschung
Schlagworte	Computational Aspects of Tensor Ranks • Full Rank Tensors • Maximal Rank of Real Tensors • Simultaneous Singular Value Decomposition • Typical Rank of Real Tensors
ISBN-10	4-431-55458-0 / 4431554580
ISBN-13	978-4-431-55458-5 / 9784431554585
Zustand	Neuware