黎曼流形上切向量场回归的联合推断
Riemannian Simultaneous Inference for Tangent Vector Field Regression
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- Nanyang Technological University（南洋理工大学）
- Shanghai University of Finance and Economics（上海财经大学）
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中文总结 AI 辅助
针对黎曼流形上的切向量场回归，提出基于平行移动和体积校正的核估计器，推导其渐近性质并构建联合置信管，模拟与风数据验证了方法的有效性。
中文摘要 AI 辅助
我们考虑在无边界黎曼流形上的非参数切向量场回归。由于不同点处的响应位于不同的切空间中，所提出的核估计器首先将邻近响应平行移动到目标切空间，然后形成体积校正的局部平均。我们首先推导其均匀二阶偏差、有限带宽协方差和随机速率。对于联合推断，切范数被写成单位切丛上的上确界。精确协方差白化给出单位方差高斯场，其相关长度沿基流形为$h$量级，沿纤维为一阶量级。其局部协方差几何导致具有显式内在常数的Gumbel极限。将该极限与高斯近似和交叉拟合协方差估计相结合，为回归场产生可行的联合置信管。我们进一步讨论通过带宽选择和高阶偏差校正改进有限样本推断。在各种流形上的模拟支持所提出的推断程序。全球风数据的随机重建说明了管的横截面如何描述空间变化的的不确定性。
英文摘要
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.
