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Algebraic and Computational Aspects of Real Tensor Ranks

Part of the SpringerBriefs in Statistics Book Series

DOI: 10.1007/978-4-431-55459-2

ISBN: 978-4-431-55458-5

Publisher: Springer Japan - Springer Journals

Mar 19, 2016

Available

Basics of Tensor Rank

Sakata, Toshio; Sumi, Toshio; Miyazaki, Mitsuhiro

Pages: 1 –10

[In statistical data analysis, a tensor is a multi-array datum. Just as the complexity of a matrix datum is described by its matrix rank, the complexity of a tensor datum is described by its tensor rank. In this chapter, we review several concepts of tensor rank such as rank-1 tensors and tensor...
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3-Tensors

Sakata, Toshio; Sumi, Toshio; Miyazaki, Mitsuhiro

Pages: 11 –15

[3-mode real tensors are the main focus of this book. Several concepts are explained in this chapter by specialization for 3-tensors.]
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Simple Evaluation Methods of Tensor Rank

Sakata, Toshio; Sumi, Toshio; Miyazaki, Mitsuhiro

Pages: 17 –28

[In this chapter, we illustrate simple evaluations of the rank of 3-mode tensors, which might facilitate readers’ understanding of tensor rank.]
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Absolutely Nonsingular Tensors and Determinantal Polynomials

Sakata, Toshio; Sumi, Toshio; Miyazaki, Mitsuhiro

Pages: 29 –37

[In this chapter, we define absolute nonsingularity for 3-tensors over R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
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Maximal Ranks

Sakata, Toshio; Sumi, Toshio; Miyazaki, Mitsuhiro

Pages: 39 –59

[In this chapter, we consider the maximal rank of tensors with format (m, n, 2) or (m, n, 3). The rank of a tensor with format (m, n, 2) is computable because of the list of representatives of GL(m,F)×GL(n,F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}...
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Typical Ranks

Sakata, Toshio; Sumi, Toshio; Miyazaki, Mitsuhiro

Pages: 61 –80

[Let m and n be positive integers with m≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...
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Global Theory of Tensor Ranks

Sakata, Toshio; Sumi, Toshio; Miyazaki, Mitsuhiro

Pages: 81 –91

[The generic rank is considered under the complex number field, and it corresponds with the dimension of the secant variety. The dimension is studied in the area of algebraic geometry. In this chapter, we introduce known results and discuss the typical rank from the point of view of the Jacobian...
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2×2×⋯×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2\times \cdots \times 2$$\end{document} Tensors

Sakata, Toshio; Sumi, Toshio; Miyazaki, Mitsuhiro

Pages: 93 –101

[We study an upper bound of the ranks of n-tensors with format (2,2,…,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...

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