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

• URL: http://arxiv.org/abs/2310.06157v2
• Date: Thu, 2 Nov 2023 17:18:22 GMT
• Title: Manifold-augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds
• Authors: Daniel Kelshaw, Luca Magri
• Abstract summary: We show how the geometry of a 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 manifold.
• Score: 5.0401589279256065
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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