Tensor rank problem and multivariate analysis of tensor normal distributions
| Project/Area Number |
24540131
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kyushu University |
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Principal Investigator |
Toshio Sakata 九州大学, 芸術工学研究科(研究院), 教授 (20117352)
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| Co-Investigator(Kenkyū-buntansha) |
SUMI Toshio 九州大学, 基幹教育研究院, 准教授 (50258513)
MIYAZAKI Mitsuhiro 京都教育大学, 教育学部, 准教授 (90219767)
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| Co-Investigator(Renkei-kenkyūsha) |
SASABUCHI Syoichi 九州大学, 芸術工学研究院 (20128028)
KURIKI Satoshi 統計数理研究所 (90195545)
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| Project Period (FY) |
2012-04-01 – 2016-03-31
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| Project Status |
Completed (Fiscal Year 2015)
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| Budget Amount *help |
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| Keywords | 3-tensors / typical rank / bilinear forms / determinantal ideal / matrix normal / similar test / テンソルデータの階数 / テンソルデータの典型階数 / 絶対列充足階数テンソル / 正則な双線形形式 / 行列・テンソル型正規分布の平均の片側検定 / テンソルの階数 / テンソルの典型階数 / テンソル正規分布 / 行列正規分布の片側検定 / テンソル正規分布の片側検定 / 制約ホロノミック勾配法 / 2x2x...x2テンソルの階数 / テンソル多変量解析 / Ｒによるグレブナ基定 / 計算代数統計 / 量子通信と階数 |
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| Outline of Final Research Achievements |
In statistics, a tensor means a multi-way array and it is an extenstion of matrix. To grasp the meanning of the datum, we decompose the tensor into a sum of rank one tensors, where rank one tensors are the most simple tensors. The rank of a tensor T is the mimimum length of such sum of rank one tensors to express T. A tensor rank is called a typical rank if the tensors with the rank has a positive measure. From a view of data analysis, typical rank is an important concept. In this study, we considered the typical ranks of (m,n,p) type 3-tensors. By using the concepts of absolutely nonsingular tensor, absolutely full column rank tensors and nonsingular bilinear form,determinantal ideal, we decided partially whether there is one typical rank or there are plural typical ranks over the real number field. Also we studied and constructed the one-sided similar test for the mean matrix of matrix type normal distributions.
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Report
(5 results)
Research Products
(41 results)
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[Presentation] About typical ranks of real tensors2015
Author(s)
Mitsuhiro Miyazaki, Toshio Sumi, Toshio Sakata
Organizer
SIAM Conference on Applied Algebraic Geometry
Place of Presentation
National Institute for Mathematical Sciences, 韓国、Daejeon（大田）
Year and Date
2015-08-03
Related Report
Int'l Joint Research
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