Functional Estimation of Manifold-Valued Diffusion Processes

2026-03-21

AI Summary

This work addresses the challenge of analyzing nonstationary, high-dimensional time series that often lie on low-dimensional manifolds. The authors propose a nonparametric method based on Nadaraya–Watson kernel regression to jointly estimate, from a single trajectory, the drift vector field, diffusion matrix, and invariant density of a manifold-valued Itô diffusion process. They establish, for the first time, asymptotic consistency and asymptotic normality of these estimators on boundaryless, smooth, complete manifolds, and introduce a tangent space estimator tailored for dependent data. The theoretical framework integrates stochastic differential equations, manifold ergodicity, and nonparametric statistics. Numerical experiments across diverse manifold structures demonstrate the method’s effectiveness and robustness.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: State Estimation
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Nonstationary high-dimensional time series are increasingly encountered in biomedical research as measurement technologies advance. Owing to the homeostatic nature of physiological systems, such datasets are often located on, or can be well approximated by, a low-dimensional manifold. Modeling such datasets by manifold-valued Itô diffusion processes has been shown to provide valuable insights and to guide the design of algorithms for clinical applications. In this paper, we propose Nadaraya-Watson type nonparametric estimators for the drift vector field and diffusion matrix of the process from one trajectory. Assuming a time-homogeneous stochastic differential equation on a smooth complete manifold without boundary, we show that as the sampling interval and kernel bandwidth vanish with increasing trajectory length, recurrence of the process yields asymptotic consistency and normality of the drift and diffusion estimators, as well as the associated occupation density. Analysis of the diffusion estimator further produces a tangent space estimator for dependent data, which has its own interest and is essential for drift estimation. Numerical experiments across a range of manifold configurations support the theoretical results.

Problem

• manifold-valued diffusion processes
• drift estimation
• diffusion matrix estimation
• nonstationary time series
• functional estimation

Innovation

• manifold-valued diffusion processes
• nonparametric estimation
• Nadaraya-Watson estimator
• asymptotic consistency
• tangent space estimation