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Development of a new method for data science via the fusion of dynamical systems and computational topology

Research Project

Project/Area Number 19KK0068
Research Category

Fund for the Promotion of Joint International Research (Fostering Joint International Research (B))

Allocation TypeMulti-year Fund
Review Section Medium-sized Section 12:Analysis, applied mathematics, and related fields
Research InstitutionTokyo Institute of Technology (2023)
Chubu University (2019-2022)

Principal Investigator

荒井 迅  東京工業大学, 情報理工学院, 教授 (80362432)

Co-Investigator(Kenkyū-buntansha) 平岡 裕章  京都大学, 高等研究院, 教授 (10432709)
大林 一平  岡山大学, サイバーフィジカル情報応用研究コア, 教授 (30583455)
竹内 博志  滋賀大学, データサイエンス教育研究センター, 助教 (40851457)
Project Period (FY) 2019-10-07 – 2025-03-31
Project Status Granted (Fiscal Year 2023)
Budget Amount *help
¥18,330,000 (Direct Cost: ¥14,100,000、Indirect Cost: ¥4,230,000)
Fiscal Year 2024: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2023: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2022: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2021: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2020: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2019: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Keywords位相的データ解析 / 計算トポロジー / 力学系
Outline of Research at the Start

本研究では,力学系と計算トポロジーを融合させることで,双方の分野の困難を突破する新しい手法を提案する事を目指す.まず,力学系で開発された Connection Matrix の理論とパーシステントホモロジー理論を融合させることで,時系列に適用可能な位相的データ解析法を構築する.また逆に,計算トポロジーの理論に基き,位相的な構造の変化に着目した,応用可能性の高い新しい力学系の安定性概念を定義することを目指す.
これらの目的を達成するため,力学系の研究と位相的データ解析の双方で指導的な役割を果たしてきた,米国ラトガース大学のミシャイコフ教授のグループとの連携を進める.

Outline of Annual Research Achievements

本研究は,力学系と計算トポロジーを融合することにより,新しいデータ解析の手法を開発すること,また開発した手法を用いて様々な分野の具体的な応用問題に貢献することを目指す.さらに,具体的な問題への応用という観点から力学系の安定性理論を見直し,新しい力学系の安定性概念を提案することも目的とする.主な道具は,力学系理論からはコンレイ指数理論や分岐理論,一様双曲性証明アルゴリズムであり,計算トポロジー理論からはパーシステントホモロジー理論やその逆問題解析法などである.この目的のため,力学系の研究と位相的データ解析の双方の分野において世界的に指導的な役割を果たしてきた,米国ラトガース大学のミシャイコフ教授のグループと連携を進める.アルゴリズムの実装から具体的な応用まで幅広く相互に技術を交換し,データ解析の新しい枠組みを展開することを目指す.本年度は,パーシステントホモロジーの逆問題に関する解析手法の発展や,その応用について進展があった.また,昨年度に引き続き,動的なデータ解析を目指す上で重要な,サンプル写像に対するパーシステントホモロジー理論の基礎づけや,計算アルゴリズムについての研究も進めた.応用面では,位相幾何学的な手法で得られる解析結果を,統計的因果推論や大域的な力学系解析手法などの結果と比較し,また相互に組み合わせることで信頼性を高める手法について研究を進めている.そのような手法を神経生理学などの異分野から提供される様々な実データに対して適用し,生物学的な知見と位相的データ解析の結果を比較する研究を進めている.

Current Status of Research Progress
Current Status of Research Progress

3: Progress in research has been slightly delayed.

Reason

昨年度までの新型コロナ感染による影響もあり,海外へ渡航しての研究活動は予定より遅れている.サンプル写像に対するパーシステントホモロジー理論の基礎づけや,実データへの応用可能性についての検討,さらに具体的なデータ科学者との連携は比較的順調に進んでいるが,研究のまとめに向かう年度としては進捗はやや遅れぎみと判断した.

Strategy for Future Research Activity

サンプル写像のパーシステントホモロジー理論を基礎に,データ駆動型のコンレイ指数理論を構築することを引き続き目指す.さらに,その計算アルゴリズムの高速化と並列化のため,様々な近似アルゴリズムやデータ構造などを取り入れる研究を行なう.また,神経整理学に由来するデータの解析への応用について検討を始め,どのような幾何構造がデータの表現に最適であるかの検討や,それに即したアルゴリズムの開発を進める.

Report

(5 results)
  • 2023 Research-status Report
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (29 results)

All 2024 2023 2022 2021 2020 2019 Other

All Journal Article (5 results) (of which Peer Reviewed: 5 results,  Open Access: 3 results) Presentation (22 results) (of which Int'l Joint Research: 12 results,  Invited: 15 results) Remarks (1 results) Funded Workshop (1 results)

  • [Journal Article] Stable Volumes for Persistent Homology2023

    • Author(s)
      Ippei Obayashi
    • Journal Title

      Journal of Applied and Computational Topology

      Volume: 7 Issue: 4 Pages: 671-706

    • DOI

      10.1007/s41468-023-00119-8

    • Related Report
      2023 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Algebraic stability theorem for derived categories of zigzag persistence modules2022

    • Author(s)
      Hiraoka Yasuaki、Ike Yuichi、Yoshiwaki Michio
    • Journal Title

      Journal of Topology and Analysis

      Volume: - Pages: 1-45

    • DOI

      10.1142/s1793525322500091

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The persistent homology of a sampled map: from a viewpoint of quiver representations2021

    • Author(s)
      Takeuchi Hiroshi
    • Journal Title

      Journal of Applied and Computational Topology

      Volume: 2021 Issue: 2 Pages: 179-213

    • DOI

      10.1007/s41468-021-00065-3

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Percolation on Homology Generators in Codimension One2020

    • Author(s)
      Hiraoka Yasuaki、Mikami Tatsuya
    • Journal Title

      Topological Data Analysis and Beyond. Workshop at NeurIPS 2020

      Volume: 15 Pages: 307-342

    • DOI

      10.1007/978-3-030-43408-3_12

    • ISBN
      9783030434076, 9783030434083
    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Zeta functions of periodic cubical lattices and cyclotomic-like polynomials2020

    • Author(s)
      Y. Hiraoka, H. Ochiai, T. Shirai
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 84

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Presentation] かたちのデータマイニング:パーシステンス図編2024

    • Author(s)
      竹内博志
    • Organizer
      数学と現象 in 山中湖
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] 良い「かたち」を探す2024

    • Author(s)
      竹内博志
    • Organizer
      北陸応用数理研究会
    • Related Report
      2023 Research-status Report
    • Invited
  • [Presentation] The Conley index and the topological dimension of hyperbolic Julia set2023

    • Author(s)
      Zin Arai
    • Organizer
      Dynamical Systems Seminar, Academia Sinica
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Moduli of stability on the first bifurcation curve of the Henon map2023

    • Author(s)
      Zin Arai
    • Organizer
      2023 NCTS Workshop on Dynamical Systems
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] An introduction to set-oriented computations for dynamics and their applications2023

    • Author(s)
      Zin Arai
    • Organizer
      Hokkaido Summer Institute at Research Center of Mathematics for Social Creativity
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A data analysis framework with linear models and persistent homology2023

    • Author(s)
      Ippei Obayashi
    • Organizer
      TDA Week 2023
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Topological descriptor of thermal conductivity in amorphous Si2023

    • Author(s)
      Ippei Obayashi, Emi Minamitani, Takuma Shiga, and Makoto Kashiwagi
    • Organizer
      ICIAM2023
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the disconnectedness of the Julia set of the Henon map2023

    • Author(s)
      Zin Arai
    • Organizer
      Atelier franco-japonais sur Real and Complex Dynamics of Henon’s map
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The Conley index and the topological dimension of hyperbolic Julia sets2023

    • Author(s)
      Zin Arai
    • Organizer
      Ergodic Theory, Dynamical Systems, and Climate Sciences
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] かたちのデータマイニング2023

    • Author(s)
      竹内博志
    • Organizer
      数学と現象 in 長瀞
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] パーシステントホモロジーの材料科学への応用2023

    • Author(s)
      大林一平
    • Organizer
      データサイエンスと幾何学の物性への応用
    • Related Report
      2022 Research-status Report
  • [Presentation] On the disconnectedness of the Julia set for the Henon map2022

    • Author(s)
      Zin Arai
    • Organizer
      複素力学系と関連分野
    • Related Report
      2022 Research-status Report
  • [Presentation] On the disconnectedness of the Julia set of the Henon map2022

    • Author(s)
      Zin Arai
    • Organizer
      ランダム力学系・非自励力学系研究の展望:理論と応用
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] The Conley index and the topological dimension of hyperbolic Julia sets2022

    • Author(s)
      Zin Arai
    • Organizer
      力学系の理論と諸分野への応用
    • Related Report
      2022 Research-status Report
  • [Presentation] Moduli of stability on the first bifurcation curve of the Henon map2021

    • Author(s)
      Zin Arai
    • Organizer
      RIMS Symposium: Recent Developments in Dynamical Systems and their Application
    • Related Report
      2021 Research-status Report
  • [Presentation] On characterizing rare events in persistent homology2021

    • Author(s)
      Yasuaki Hiraoka
    • Organizer
      Beyond TDA -- Persistent topology and its applications in data sciences
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On characterizing rare events in persistent homology2021

    • Author(s)
      Yasuaki Hiraoka
    • Organizer
      Thematic Einstein Semester on Geometric and Topological Structure of Materials
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The persistent homology of a sampled map: on failed reconstructions2021

    • Author(s)
      Hiroshi Takeuchi
    • Organizer
      Asia Pacific Seminar on Applied Topology and Geometry
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Characterizing rare events in persistent homology2021

    • Author(s)
      Yasuaki Hiraoka
    • Organizer
      Joint Mathematics Meetings, AMS Special Session on Applied Topology
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Characterizing rare events in persistent homology2020

    • Author(s)
      Yasuaki Hiraoka
    • Organizer
      Topological Data Analysis and Beyond
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Period doubling bifurcations from complex and algebraic point of view2019

    • Author(s)
      Zin Arai
    • Organizer
      Applied and Computational Math Seminar
    • Related Report
      2019 Research-status Report
  • [Presentation] Application of persistent homology to granular materials and sampled dynamical systems2019

    • Author(s)
      Hiroshi Takeuchi
    • Organizer
      Applied and Computational Math Seminar
    • Related Report
      2019 Research-status Report
  • [Remarks] Web pape of Zin Arai

    • URL

      http://www.isc.chubu.ac.jp/zin_arai/

    • Related Report
      2020 Research-status Report
  • [Funded Workshop] TDA week 20232023

    • Related Report
      2023 Research-status Report

URL: 

Published: 2019-10-10   Modified: 2024-12-25  

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