The image by multilinear map to the real projective space and an application to the rank of tensors

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05151
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyushu University
• Principal Investigator: Sumi Toshio 九州大学, 基幹教育院, 教授 (50258513)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: テンソル / ランク / 多重線形写像 / 群作用

Outline of Final Research Achievements

A 3-tensor is a 3-way array. The determination of the rank, a measure of complexity of computation, of a tensor is difficult (NP-hard). In this study, I tried to determine the ranks of certain tensors. I found a condition for tensors with size (m,n,p) such that the set of tensors with size (m,n,p) has p+1 as the minimal typical rank, which is the minimal number occurring as a rank with positive probability, to have p+1 as the rank. Further I study representative spaces, and in particular, the dimensions of the fixed point sets of a representative space by cyclic subgroups. I showed there exists a finite simple group which does not have the expectative property.

Free Research Field

トポロジー

Academic Significance and Societal Importance of the Research Achievements

テンソルの階数１テンソルの和への分解は，行列においては，特異値分解に対応し，その拡張である．テンソルとは高次元配列のことである．テンソルの階数は，従来，計算の複雑度の尺度として用いられていたが，近年，テンソルの階数１テンソルの和への分解（の近似）が，シグナルプロセッシング，データマイニング，コンピュータビジョン，グラフ解析など様々な分野で応用が見られるようになった．実験データとして得られることが多い実数体上のテンソルの階数の研究はまだ非常に少ないため，それらの研究を行った．