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Interpolation in path-connected metric spaces using splines on embedded graphs
Abstract
We study a local graph-based interpolation framework for data defined on graph vertices embedded in a metric space. The resulting interpolants are constructed from variational graph splines defined on small neighborhoods, yielding continuous-domain extensions of the graph data. We analyze their behavior through energy estimates, Lipschitz continuity, and stability properties, and compare them with k-Nearest Neighbor (kNN) interpolation as a natural baseline due to its simplicity and widespread use. Numerical experiments on the manifold of symmetric positive definite matrices illustrate the relative performance of the proposed local interpolation schemes and kNN interpolation across varying levels of smoothness and noise.
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BibTeX
@misc{fuselier2026interpolation,
title = {{Interpolation in path-connected metric spaces using splines on embedded graphs}},
author = {E. Fuselier and J. P. Ward and J. Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15172},
eprint = {2610.15172},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15172},
}
RIS
TY - UNPB
TI - Interpolation in path-connected metric spaces using splines on embedded graphs
AU - E. Fuselier
AU - J. P. Ward
AU - J. Zhao
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15172
UR - https://arxiv.org/abs/2610.15172
AB - We study a local graph-based interpolation framework for data defined on graph vertices embedded in a metric space. The resulting interpolants are constructed from variational graph splines defined on small neighborhoods, yielding continuous-domain extensions of the graph data. We analyze their behavior through energy estimates, Lipschitz continuity, and stability properties, and compare them with k-Nearest Neighbor (kNN) interpolation as a natural baseline due to its simplicity and widespread use. Numerical experiments on the manifold of symmetric positive definite matrices illustrate the relative performance of the proposed local interpolation schemes and kNN interpolation across varying levels of smoothness and noise.
ER -