Tensor rank problem and multivariate analysis of tensor normal distributions

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540131
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kyushu University
• Principal Investigator: Toshio Sakata 九州大学, 芸術工学研究科(研究院), 教授 (20117352)
• 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)
• 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)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 3-tensors / typical rank / bilinear forms / determinantal ideal / matrix normal / similar test

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.

Free Research Field

数理統計学