Algebraic and Computational Aspects of Real Tensor Ranks

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Springer, Mar 18, 2016 - Mathematics - 108 pages
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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Contents

1 Basics of Tensor Rank
1
2 3Tensors
11
3 Simple Evaluation Methods of Tensor Rank
17
4 Absolutely Nonsingular Tensors and Determinantal Polynomials
29
5 Maximal Ranks
38
6 Typical Ranks
61
7 Global Theory of Tensor Ranks
81
8 2times2times汥瑀瑯步渠times2 Tensors
92
References
103
Index
107
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