[Paper Review] Functional Estimation of Manifold-Valued Diffusion Processes

Jacob McErlean, Hau Tieng Wu|arXiv (Cornell University)|Mar 21, 2026

Functional Brain Connectivity Studies0 citations

TL;DR

The paper develops Nadaraya-Watson type nonparametric estimators to recover drift and diffusion for manifold-valued Itô diffusion from a single trajectory, with tangent-space-based drift estimation and asymptotic guarantees under Harris recurrence.

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.

Motivation & Objective

• Motivate modeling high-dimensional biomedical time series on low-dimensional manifolds with diffusion dynamics.
• Develop kernel-based estimators for drift and diffusion on manifolds from observed data.
• Address curvature-induced bias and enable tangent-space based drift estimation.
• Establish asymptotic consistency and normality under Harris recurrence and appropriate sampling.
• Provide algorithmic steps and theoretical results validated by simulations.

Proposed method

• Adopt a manifold-valued diffusion model with Itô/SDE on a smooth complete manifold embedded in Euclidean space.
• Use Nadaraya-Watson type kernel estimators to estimate occupation density and the observable drift and diffusion from a single trajectory.
• Introduce a distance-like function based on embedding to define a kernel weighting and estimate the diffusion matrix.
• Estimate the tangent space via the eigenstructure of the estimated diffusion matrix and project the drift onto the tangent space.
• Compute the drift estimator by projecting a Euclidean-style estimator onto the estimated tangent space to correct curvature bias.
• Leverage Harris recurrence and a generalized Nummelin splitting framework to derive asymptotic results and a central limit theorem.

Experimental results

Research questions

• RQ1 How can drift and diffusion for manifold-valued diffusion processes be nonparametrically estimated from a single trajectory?
• RQ2 What are the effects of manifold curvature on drift estimation and how can tangent-space projections mitigate bias?
• RQ3 Under Harris recurrence, what are the asymptotic properties (consistency and normality) of the proposed estimators?
• RQ4 How should bandwidth and sampling rates be chosen to ensure reliable estimation on manifolds?

Key findings

• Proposed manifold-adaptive Nadaraya-Watson estimators for drift, diffusion, and occupation density that account for curvature and ambient embedding.
• A tangent-space based drift estimator is necessary to correct curvature-induced bias in manifold settings.
• Diffusion estimation remains robust to curvature and provides a usable tangent space for drift recovery.
• Asymptotic consistency and normality of the estimators are established under Harris recurrence with appropriate bandwidth and sampling schemes.
• A generalized ratio-limit approach and Darling-Kac framework underpin the asymptotic analysis and Gaussian mixture characterizations.
• Numerical experiments across manifold configurations corroborate the theoretical results.