Arnoldi-Enhanced Multivariate Hermite Interpolation of Manifold-Valued Data
Abstract
This paper integrates the multivariate Arnoldi process into tangent space Hermite interpolation for manifold-valued data. A G-Arnoldi recurrence constructs a discrete orthogonal polynomial basis, improving numerical conditioning for high-degree approximation. The analysis uses a Markov inequality with a geometry-dependent exponent r; for the convex parameter domains considered here, r = 2 is available. Under suitable regularity and sampling assumptions, we establish C^0 and C^1 error bounds and analyze their propagation through the exponential map. When only function values are sampled, these bounds are O(sqrt(M) n^{-m}) and O(sqrt(M) n^{2-m}), respectively. When first derivatives are also sampled, the C^1 bound improves to O(sqrt(M(d+1)) n^{1-m}). Here n is the polynomial degree, d the parameter dimension, m the target regularity, and M the number of sampling points. For sampling sequences satisfying the fill distance condition with M \asymp n^{2d}, the corresponding bounds become O(n^{d-m}), O(n^{d+2-m}), and O(n^{d+1-m}), yielding superalgebraic convergence for targets admitting smooth extensions. Numerical experiments on SO(3) and S^2 compare the accuracy and computational cost of the Arnoldi-enhanced method with THI+Kriging, with supplementary experiments demonstrating approximation without derivative samples.
Related
- A Direct Approach to Hermite Interpolation
- ROOT ISOLATION FOR ANALYTIC FUNCTIONS USING CUBIC HERMITE INTERPOLATION.
- Conjectures of Bernstein and Erdős for weighted Lagrange interpolation with Hermite weight
- The conditional expectation of the product of the first n−1 Hermite polynomials in a multivariate normal distribution with respect to the n-th variable. A fresh perspective on the Kibble-Slepian formula
- Triple Products of Correlated Multivariate Hermite Polynomials
BibTeX
@misc{li2026arnoldienhanced,
title = {{Arnoldi-Enhanced Multivariate Hermite Interpolation of Manifold-Valued Data}},
author = {Yuxuan Li and Qiang Niu and Wubin Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.25176},
eprint = {2605.25176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.25176},
}
RIS
TY - UNPB
TI - Arnoldi-Enhanced Multivariate Hermite Interpolation of Manifold-Valued Data
AU - Yuxuan Li
AU - Qiang Niu
AU - Wubin Zhou
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2605.25176
UR - https://arxiv.org/abs/2605.25176
AB - This paper integrates the multivariate Arnoldi process into tangent space Hermite interpolation for manifold-valued data. A G-Arnoldi recurrence constructs a discrete orthogonal polynomial basis, improving numerical conditioning for high-degree approximation. The analysis uses a Markov inequality with a geometry-dependent exponent r; for the convex parameter domains considered here, r = 2 is available. Under suitable regularity and sampling assumptions, we establish C^0 and C^1 error bounds and analyze their propagation through the exponential map. When only function values are sampled, these bounds are O(sqrt(M) n^{-m}) and O(sqrt(M) n^{2-m}), respectively. When first derivatives are also sampled, the C^1 bound improves to O(sqrt(M(d+1)) n^{1-m}). Here n is the polynomial degree, d the parameter dimension, m the target regularity, and M the number of sampling points. For sampling sequences satisfying the fill distance condition with M \asymp n^{2d}, the corresponding bounds become O(n^{d-m}), O(n^{d+2-m}), and O(n^{d+1-m}), yielding superalgebraic convergence for targets admitting smooth extensions. Numerical experiments on SO(3) and S^2 compare the accuracy and computational cost of the Arnoldi-enhanced method with THI+Kriging, with supplementary experiments demonstrating approximation without derivative samples.
ER -