Sampling Design and Confidence Tubes for Integral Curves on Riemannian Manifolds
Abstract
We estimate the integral curve of an unknown vector field from noisy tangent vector field on a known compact Riemannian manifold. We consider two sampling settings: independent observations at random locations that have already been collected, and a fixed response budget that allows subsequent locations to be chosen using pilot information. Under suitable regularity assumptions, the passive kernel estimator has curve error of order n−2/(d+3) in probability, with sample size n and manifold dimension d. Its functional central limit theorem has a limit characterized by a linear stochastic differential equation. For active sampling, an independent pilot directs one new batch of measurements to neighborhoods of the target curve. Under localization, smoothness and design conditions, this concentration yields expected maximum curve error of order N−1/2, including the pilot in the total response budget N. Both methods provide simultaneous geodesic confidence tubes with asymptotically exact coverage under mild conditions. Simulations show narrower active tubes in most reported settings, with improved small-sample coverage in some three-dimensional examples. Ocean-drifter observations illustrate mean-flow reconstruction on the sphere, and a canopy-flow LDA application uses retrospective active sampling to quantify lateral and vertical departures of local mean streamlines.
Related
- CURS: an exact method for sampling on Riemannian manifolds
- Spline interpolation on compact riemannian manifolds
- Kernel Density Regression Estimation with Covariates on Riemannian Manifolds
- Smoothness estimation for Whittle–Matérn processes on closed Riemannian manifolds
- The Parabolic Harnack Inequality on Weighted Riemannian Manifolds
BibTeX
@misc{chang2026sampling,
title = {{Sampling Design and Confidence Tubes for Integral Curves on Riemannian Manifolds}},
author = {Xiaotian Chang and Qirui Hu and Cyrus Mostajeran},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00771},
eprint = {2611.00771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00771},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sampling Design and Confidence Tubes for Integral Curves on Riemannian Manifolds
AU - Xiaotian Chang
AU - Qirui Hu
AU - Cyrus Mostajeran
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00771
UR - https://arxiv.org/abs/2611.00771
AB - We estimate the integral curve of an unknown vector field from noisy tangent vector field on a known compact Riemannian manifold. We consider two sampling settings: independent observations at random locations that have already been collected, and a fixed response budget that allows subsequent locations to be chosen using pilot information. Under suitable regularity assumptions, the passive kernel estimator has curve error of order n^−2/(d+3) in probability, with sample size n and manifold dimension d. Its functional central limit theorem has a limit characterized by a linear stochastic differential equation. For active sampling, an independent pilot directs one new batch of measurements to neighborhoods of the target curve. Under localization, smoothness and design conditions, this concentration yields expected maximum curve error of order N^−1/2, including the pilot in the total response budget N. Both methods provide simultaneous geodesic confidence tubes with asymptotically exact coverage under mild conditions. Simulations show narrower active tubes in most reported settings, with improved small-sample coverage in some three-dimensional examples. Ocean-drifter observations illustrate mean-flow reconstruction on the sphere, and a canopy-flow LDA application uses retrospective active sampling to quantify lateral and vertical departures of local mean streamlines.
ER -