Algebraic and Computational Aspects of Real Tensor Ranks

Authors: Sakata, Toshio, Sumi, Toshio, Miyazaki, Mitsuhiro
Presents the first comprehensive treatment of maximal ranks and typical ranks over the real number file
Provides interesting ideas of determinant polynomials, determinantal ideals, absolutely nonsingular tensors and absolutely full column rank tensors
Includes numerical methods of determining ranks by simultaneous singular value decomposition through a theory of matrix star algebra

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.

Number of Illustrations and Tables
5 b/w illustrations
Topics
Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences
Statistics and Computing / Statistics Programs
Statistics for Social Science, Behavorial Science, Education, Public Policy, and Law

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