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This paper advances nonparametric regression on Riemannian manifolds by developing rigorous simultaneous inference procedures for tangent vector fields, addressing the geometric complexities where responses reside in distinct tangent spaces. By deriving explicit bias and covariance structures that account for manifold geometry, the method enables robust uncertainty quantification for spatially varying data, such as global wind patterns, without relying on standard Euclidean assumptions. The work represents a significant theoretical contribution to statistical learning on curved domains, offering improved tools for analyzing complex physical systems modeled via differential geometry.
Riemannian Simultaneous Inference for Tangent Vector Field Regression
arXiv:2609.21910v1 Announce Type: cross Abstract: We consider nonparametric tangent vector field regression on a Riemannian manifold without boundary. Because responses at different points lie in different tangent spaces, the proposed kernel estimator first parallel transports nearby responses to the target tangent space and then forms a volume-corrected local average. We first derive its uniform second-order bias, finite-bandwidth covariance, and stochastic rate. For simultaneous inference, the tangent norm is written as a supremum over the unit tangent bundle. Exact covariance whitening gives a unit-variance Gaussian field whose correlation length is of order $h$ along the base manifold and of order one along the fibre. Its local covariance geometry leads to a Gumbel limit with an explicit intrinsic constant. Combining this limit with Gaussian approximation and cross-fitted covariance estimation yields a feasible simultaneous confidence tube for the regression field. We further discuss improved finite-sample inference with bandwidth selection and high-order bias corrections. Simulations on various manifolds support the proposed inference procedure. A randomized reconstruction of global wind data illustrates how the tube's cross-sections describe spatially varying uncertainty.
Abstract:We consider nonparametric tangent vector field regression on a Riemannian manifold without boundary. Because responses at different points lie in different tangent spaces, the proposed kernel estimator first parallel transports nearby responses to the target tangent space and then forms a volume-corrected local average. We first derive its uniform second-order bias, finite-bandwidth covariance, and stochastic rate. For simultaneous inference, the tangent norm is written as a supremum over the unit tangent bundle. Exact covariance whitening gives a unit-variance Gaussian field whose correlation length is of order $h$ along the base manifold and of order one along the fibre. Its local covariance geometry leads to a Gumbel limit with an explicit intrinsic constant. Combining this limit with Gaussian approximation and cross-fitted covariance estimation yields a feasible simultaneous confidence tube for the regression field. We further discuss improved finite-sample inference with bandwidth selection and high-order bias corrections. Simulations on various manifolds support the proposed inference procedure. A randomized reconstruction of global wind data illustrates how the tube's cross-sections describe spatially varying uncertainty.
Machine Learning (stat.ML); Machine Learning (cs.LG); Methodology (stat.ME)
arXiv:2609.21910 [stat.ML]
(or arXiv:2609.21910v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2609.21910 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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