Algebraic and Computational Aspects of Real Tensor Ranks

Description

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 Front Matter....Pages i-viii Basics of Tensor Rank....Pages 1-10 3-Tensors....Pages 11-15 Simple Evaluation Methods of Tensor Rank....Pages 17-28 Absolutely Nonsingular Tensors and Determinantal Polynomials....Pages 29-37 Maximal Ranks....Pages 39-59 Typical Ranks....Pages 61-80 Global Theory of Tensor Ranks....Pages 81-91 \(2\times 2\times \cdots \times 2\) Tensors....Pages 93-101 Back Matter....Pages 103-108