@inproceedings{
kelshaw2023manifoldaugmented,
title={Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds.},
author={Daniel Kelshaw and Luca Magri},
booktitle={NeurIPS 2023 Workshop on Symmetry and Geometry in Neural Representations},
year={2023},
url={https://openreview.net/forum?id=LQoejMxeiv}
}
Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order modelling, statistical inference, and interpolation. In this work, we propose a model-based parameterisation for distance fields and geodesic flows on manifolds, exploiting solutions of a manifold-augmented Eikonal equation. We demonstrate how the geometry of the manifold impacts the distance field, and exploit the geodesic flow to obtain globally length-minimising curves directly. This work opens opportunities for statistics and reduced-order modelling on differentiable manifolds.
Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds.
Daniel Kelshaw Luca Magri