Determinantal point fields and machine learning

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K20884
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: Kyushu University
• Principal Investigator: Shirai Tomoyuki 九州大学, マス・フォア・インダストリ研究所, 教授 (70302932)
• Co-Investigator(Kenkyū-buntansha): 河原 吉伸 大阪大学, 大学院情報科学研究科, 教授 (00514796)
• Project Period (FY): 2020-07-30 – 2024-03-31
• Keywords: 行列式確率場 / 立方格子複体 / パーシステントホモロジー / 全域木・連結成分

Outline of Final Research Achievements

To deepen understanding of determinantal probability in machine learning, the following research was conducted: enumeration of connected components when the number of cycles on a complete bipartite graph is fixed, determination of eigenvalues/eigenvectors of Laplacians on cubical complexes and their application in counting weighted spanning acycles and the system of partial differential equations satisfied by a family of exponential generating functions, comparison of representability of statistical models when modeling determinantal probability using L-matrices encompassing positive definite matrices and skew-symmetric matrices, approximation of Ginibre random point fields via perturbed lattices using distances based on persistence diagram and histograms of nearest-neighbors in point configurations. These studies have been published in international academic journals and are currently under submission.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

行列式確率場は本来純粋数学的興味から研究が始まった研究対象であるが、機械学習への応用が提案されて以来，応用分野でも関心をもたれてきた．本研究は行列式確率場を具体的な問題に応用するという観点から実施し，特にハッシング，セルラーネットワーク，機械学習などの具体例に現れる行列式確率場に関わる数学的問題をとりあげた．これらの研究成果は，数学としての確率論および組合せ論の一分野に貢献するとともに，実世界の問題において応用される可能性も期待される．