New method estimates and explains vector fields on curved spaces
Riemannian Simultaneous Inference for Tangent Vector Field Regression
Machine Learning
Summary
Understanding how vector quantities like wind or magnetic fields change over curved surfaces is hard because measurements live in different local directions. The authors created a way to combine these local measurements into a smooth estimate by carefully moving them to a common viewpoint and averaging. They also developed tools to confidently say how uncertain their estimates are at every location at once. Their method works well on simulated tests and on real global wind data, showing where estimates are more or less reliable.
What this means in practice
- •For meteorology teams: Provide spatially consistent estimates and uncertainty bounds for wind vector fields on Earth’s curved surface.
- •For geospatial engineers: Improve smoothing and uncertainty quantification of vector measurements that live on curved geographic surfaces.
Authors
Xiaotian Chang, Yangdi Jiang, Qirui Hu
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.