Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds.

Daniel Kelshaw, Luca Magri

Submission Track: Extended Abstract

Keywords: Riemannian manifolds, geodesics, distance, geodesic flow

TL;DR: We propose a method for obtaining a continuous, differentiable representation of the distance function for arbitrary 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.

Submission Number: 48