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.

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Cite

Text

Kelshaw and Magri. "Manifold-Augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds.." NeurIPS 2023 Workshops: NeurReps, 2023.

Markdown

[Kelshaw and Magri. "Manifold-Augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds.." NeurIPS 2023 Workshops: NeurReps, 2023.](https://mlanthology.org/neuripsw/2023/kelshaw2023neuripsw-manifoldaugmented/)

BibTeX

@inproceedings{kelshaw2023neuripsw-manifoldaugmented,
  title     = {{Manifold-Augmented Eikonal Equations: Geodesic Distances and Flows on Differentiable Manifolds.}},
  author    = {Kelshaw, Daniel and Magri, Luca},
  booktitle = {NeurIPS 2023 Workshops: NeurReps},
  year      = {2023},
  url       = {https://mlanthology.org/neuripsw/2023/kelshaw2023neuripsw-manifoldaugmented/}
}