Riemannian Simultaneous Inference for Tangent Vector Field Regression
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.
Related
- Simultaneous inference for additive regression
- Riemannian Density-Driven Optimal Control: Tangent-Space LQR for Second-Order Multi-Agent Systems on Curved Manifolds
- Tangent bundles of Quot schemes of vector bundles on curves
- Kernel Density Regression Estimation with Covariates on Riemannian Manifolds
- The Yang-Mills flow in vector bundles over higher dimensional Riemannian manifolds
BibTeX
@misc{chang2026riemannian,
title = {{Riemannian Simultaneous Inference for Tangent Vector Field Regression}},
author = {Xiaotian Chang and Yangdi Jiang and Qirui Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21910},
eprint = {2609.21910},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21910},
}
RIS
TY - UNPB
TI - Riemannian Simultaneous Inference for Tangent Vector Field Regression
AU - Xiaotian Chang
AU - Yangdi Jiang
AU - Qirui Hu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.21910
UR - https://arxiv.org/abs/2609.21910
AB - 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.
ER -