Algebraic and Computational Aspects of Real Tensor Ranks

Nonfiction, Science & Nature, Mathematics, Statistics, Computers, Application Software
Cover of the book Algebraic and Computational Aspects of Real Tensor Ranks by Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki, Springer Japan
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Author: Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki ISBN: 9784431554592
Publisher: Springer Japan Publication: March 18, 2016
Imprint: Springer Language: English
Author: Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
ISBN: 9784431554592
Publisher: Springer Japan
Publication: March 18, 2016
Imprint: Springer
Language: English

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.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

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