Unified construction of geometric theory for information space and morphological information science

• Project/Area Number: 16KT0132
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 特設分野
• Research Field: Mathematical Sciences in Search of New Cooperation
• Research Institution: Nagoya Institute of Technology
• Principal Investigator: Matsuzoe Hiroshi 名古屋工業大学, 工学(系)研究科(研究院), 教授 (90315177)
• Co-Investigator(Kenkyū-buntansha): 高津 飛鳥 首都大学東京, 理学研究科, 准教授 (90623554)
• Project Period (FY): 2016-07-19 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2018: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 情報幾何学 / マルコフ埋め込み / Wasserstein幾何学 / 幾何学 / 微分幾何学 / 幾何学的統計理論 / 形態情報科学

Outline of Final Research Achievements

In information geometry, the invariance of geometric structures for statistical models under Markov embedding is very important. However, our research group found that the construction of Markov embedding is not unique, and the embedding can be modified. As a result, it is shown that different invariants corresponding to the new Markov embedding are obtained.
The result of this study, which was not expected at the beginning of this research project, is an important development on the basic properties of information geometry.

Academic Significance and Societal Importance of the Research Achievements

情報幾何学における不変性の要請は，情報空間の幾何学構成において非常に重要な基本原則である．しかしながら，その不変性に任意性があり，従来とは異なる自然な埋め込みを構成することで，新しい幾何学構造の不変性が得られることを示した．この研究成果は情報空間の微分幾何学的基礎理論として，非常に重要であると考えている．

Research Products

• [Int'l Joint Research] Istituto dei Sistemi Complessi (ISC-CNR)/Politecnico di Torino(イタリア)
• [Int'l Joint Research] Istituto dei Sistemi Complessi (ISC-CNR)/Politecnico di Torino/Universita di Firenze(イタリア)
• [Int'l Joint Research] Istituto dei Sistemi Complessi (ISC-CNR)/Politecnico di Torino(イタリア)
• [Journal Article] Normalization Problems for Deformed Exponential Families (2019)
  • Author(s): Hiroshi Matsuzoe, Antonio M. Scarfone, Tatsuaki Wada
  • Journal Title: Lecture Notes in Computer Science Volume: 11712 Pages: 279-287
  • DOI: 10.1007/978-3-030-26980-7_29
  • ISBN: 9783030269791, 9783030269807
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Convergence of combinatorial Ricci flows to degenerate circle patterns (2019)
  • Author(s): Asuka Takatsu
  • Journal Title: Transactions of the American Mathematical Society Volume: 印刷中 Issue: 11 Pages: 7597-7617
  • DOI: 10.1090/tran/7778
  • Peer Reviewed / Open Access
• [Journal Article] To logconcavity and beyond (2019)
  • Author(s): Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
  • Journal Title: Communications in Contemporary Mathematics Volume: 印刷中 Issue: 02 Pages: 1950009-1950009
  • DOI: 10.1142/s0219199719500093
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A Sequential Structure of Statistical Manifolds on Deformed Exponential Family (2017)
  • Author(s): Matsuzoe Hiroshi、Scarfone Antonio M.、Wada Tatsuaki
  • Journal Title: Lecture Notes in Computer Science Volume: 10589 Pages: 223-230
  • DOI: 10.1007/978-3-319-68445-1_26
  • ISBN: 9783319684444, 9783319684451
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Normalization problems for deformed exponential families (2019)
  • Author(s): Hiroshi Matsuzoe, Antonio M. Scarfone, Tatsuaki Wada
  • Organizer: 4th conference on Geometric Science of Information
  • Int'l Joint Research
• [Presentation] Cencov の定理再訪 (2019)
  • Author(s): 高津 飛鳥, 松添 博
  • Organizer: 日本数学会2019年度秋季総合分科会
• [Presentation] Geometry of generalized entropy (2019)
  • Author(s): Hiroshi Matsuzoe
  • Organizer: Information Geometry and Affine Differential Geometry III
  • Int'l Joint Research / Invited
• [Presentation] Sequential Structures of Statistical Manifolds (2018)
  • Author(s): Hiroshi Matsuzoe
  • Organizer: Entropy
  • Int'l Joint Research
• [Presentation] Dually flat structures for q-Gaussian families (2018)
  • Author(s): Hiroshi Matsuzoe
  • Organizer: IMS Annual Meeting on Probability and Statistics
  • Int'l Joint Research
• [Presentation] Sequential structures of statistical manifolds on deformed exponential families (2018)
  • Author(s): Hiroshi Matsuzoe
  • Organizer: 5th International Conference on Recent Advances in Pure and Applied Mathematics
  • Int'l Joint Research
• [Presentation] Sequential structures of statistical manifolds (2018)
  • Author(s): Hiroshi Matsuzoe
  • Organizer: International Colloquium on Differential Geometry and its Related Fields
  • Int'l Joint Research
• [Presentation] 変形指数型分布族の幾何学 (2018)
  • Author(s): 松添博
  • Organizer: ミニワークショップ統計多様体の幾何学とその周辺(10)
  • Invited
• [Presentation] 退化した Circle Pattern に収束するトーラス上の組合せリッチ流 (2018)
  • Author(s): 高津飛鳥
  • Organizer: 2018年度日本数学会秋季総合分科会
• [Presentation] To log concavity and beyond (2018)
  • Author(s): Asuka Takatsu
  • Organizer: Workshop on barycenters, convexity on metric spaces and positive operators
  • Int'l Joint Research / Invited
• [Presentation] Hessian structures on deformed exponential families II (2018)
  • Author(s): Hiroshi Matsuzoe
  • Organizer: The Ninth International Conference on Guided Self-Organisation
• [Presentation] 統計モデルの幾何学 (2017)
  • Author(s): 松添博
  • Organizer: 多様体上の微分方程式
  • Invited
• [Presentation] A sequential structure of statistical manifolds on deformed exponential family (2017)
  • Author(s): Hiroshi Matsuzoe
  • Organizer: 3rd conference on Geometric Science of Information
• [Presentation] Wasserstein幾何と情報幾何 (2017)
  • Author(s): 高津飛鳥
  • Organizer: 東北大学理学研究科数学専攻談話会
  • Invited
• [Presentation] Riemannian Wasserstein geometry on the space of Gaussian measures over the Wiener space (2016)
  • Author(s): Asuka Takatsu
  • Organizer: 第１回日本-台湾微分幾何研究集会
  • Place of Presentation: 早稲田大学
  • Year and Date: 2016-12-13
  • Int'l Joint Research / Invited
• [Presentation] Construction of model selection criterion for q-exponential family (2016)
  • Author(s): Hiroshi Matsuzoe
  • Organizer: International Colloquium on Differential Geometry and its Related Fields
  • Place of Presentation: St. Cyril and St. Methodius University of Veliko Tarnovo
  • Year and Date: 2016-09-06
  • Int'l Joint Research / Invited
• [Presentation] Curvature Dimension condition from the viewpoint of Information geometry (2016)
  • Author(s): Asuka Takatsu
  • Organizer: Follow-up Workshop to JTP "Optimal Transportation"
  • Place of Presentation: Hausdorff Research Institute for Mathematics
  • Year and Date: 2016-08-29
  • Int'l Joint Research / Invited