Construction of a Mathematical Foundation for a Probabilistic Advancement of Principal Flow Analysis on Riemannian Manifolds toward Applications in Industrial Data Analysis

Reference No.	2026a031
Type/Category	Grant for Young Researchers and Students-Short-term Joint Research
Title of Research Project	Construction of a Mathematical Foundation for a Probabilistic Advancement of Principal Flow Analysis on Riemannian Manifolds toward Applications in Industrial Data Analysis
Principal Investigator	Hiroki Tanaka（Department of Statistics and Data Science, National University of Singapore; RIKEN Center for Advanced Intelligence Project (AIP) (planned)・Graduate student）
Research Period
Keyword(s) of Research Fields	Geometric Statistical Analysis, Stochastic Differential Equations, Nonlinear Principal Component Analysis, Geometric Data Analysis, Statistical Machine Learning, Dimensionality Reduction, Stochastic Analysis on Manifolds, Riemannian Manifolds
Abstract for Research Report	The aim of this study is to probabilistically refine, using stochastic differential equations (SDEs), the statistical framework of the Principal Flow/Fixed Boundary Flows on Riemannian manifolds proposed by Yao et al. (2014 and 2023, JASA), and thereby to construct a new mathematical framework with potential applications to industrial data analysis. Conventional Principal Flow has been formulated as a deterministic ordinary differential equation, making it difficult to sufficiently reflect the uncertainty and noise structure contained in data. In this study, by probabilistically analyzing Principal Flow through diffusion processes on manifolds, we establish a new concept of probabilistic confidence bands that reflect geometric structure. This is expected to provide an analytical method for nonlinear data arising in industrial fields—such as shape data, directional data, and network-structured data—that simultaneously enables dimensionality reduction, uncertainty quantification, and incorporation of geometric structure. The results of this study will not only deepen the fundamental theory but also create mathematical models that can serve as seeds for industry–academia collaborative research with applications in image analysis, medical data analysis, materials science, and related areas, thereby contributing to the realization of IMI’s philosophy of “Mathematics for Industry.” This study is also related to international collaborative research involving NUS, Keio University, and the RIKEN AIP. The results will, together with the deepening of fundamental theory, create new seeds for industry–academia collaborative research and contribute to the realization of IMI’s philosophy of “Mathematics for Industry.”
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Hiroki Tanaka（Department of Statistics and Data Science, National University of Singapore; RIKEN Center for Advanced Intelligence Project (AIP) (planned)・Graduate Student） Kei Kobayashi（Keio University; RIKEN Center for Advanced Intelligence Project (AIP)・Professor）