Deformation and flatness of information geometric structure and their applications

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04997
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: University of Fukui
• Principal Investigator: Ohara Atsumi 福井大学, 学術研究院工学系部門, 教授 (90221168)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 非指数型確率分布 / 情報幾何学 / 統計多様体 / 共形平坦化 / 対称錐 / 非線形拡散現象

Outline of Final Research Achievements

A1. Mathematical analysis by conformal flattening technique in information geometry: conformal flattening technique which transform non-flat statistical manifolds to dually flat ones is derived and its properties are investigated. The non-flat manifolds can be regarded as a generalized exponential family.
A2. Application of information geometry on probability simplex and a cone of positive definite symmetric matrices corresponding to generalized exponential family: Computational algorithms dealing with generalized exponential densities are performed on their parameter spaces. Using dually flat geometry defined on these spaces, we developed the algorithms such as robust parameter estimation, mathematical optimization, and pattern classification.

Free Research Field

情報幾何

Academic Significance and Societal Importance of the Research Achievements

本研究の意義は，計画に沿った成果をあげることで関連する多くの学問分野に新しい知見と刺激を与える波及効果を及ぼすことに加えて，前述したような高い社会的要請にも対処できるデータ解析手法を提示するという実用面への貢献もあげることができる．