Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds
Abstract
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.
Cite
@article{arxiv.2310.06157,
title = {Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds},
author = {Daniel Kelshaw and Luca Magri},
journal= {arXiv preprint arXiv:2310.06157},
year = {2023}
}
Comments
Accepted to NeurIPS 2023: Symmetry and Geometry in Neural Representations Workshop