# Computing distances and means on manifolds with a metric-constrained Eikonal approach

arXiv: https://arxiv.org/abs/2404.08754 · HTML: https://mlsanity.com/paper/2404.08754/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 37 references that are in the map.
- First posted on arXiv 12 Apr 2024.

## Builds on (most cited first, 6 of 6)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Retrieval-Augmented Generation for Large Language Models: A Survey](https://mlsanity.com/paper/2312.10997/) (2023)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Geodesic Distance Function Learning via Heat Flow on Vector Fields](https://mlsanity.com/paper/1405.0133/) (2014)

## Built on by (most cited first, 3 of 3)

- [Equivariant Eikonal Neural Networks: Grid-Free, Scalable Travel-Time Prediction on Homogeneous Spaces](https://mlsanity.com/paper/2505.16035/) (2025)
- [Proper Latent Decomposition](https://mlsanity.com/paper/2412.00785/) (2024)
- [Data-augmented Learning of Geodesic Distances in Irregular Domains through Soner Boundary Conditions](https://mlsanity.com/paper/2503.04579/) (2025)

## Abstract (arXiv)

Computing distances on Riemannian manifolds is a challenging problem with numerous applications, from physics, through statistics, to machine learning. In this paper, we introduce the metric-constrained Eikonal solver to obtain continuous, differentiable representations of distance functions on manifolds. The differentiable nature of these representations allows for the direct computation of globally length-minimising paths on the manifold. We showcase the use of metric-constrained Eikonal solvers for a range of manifolds and demonstrate the applications. First, we demonstrate that metric-constrained Eikonal solvers can be used to obtain the Fr'echet mean on a manifold, employing the definition of a Gaussian mixture model, which has an analytical solution to verify the numerical results. Second, we demonstrate how the obtained distance function can be used to conduct unsupervised clustering on the manifold -- a task for which existing approaches are computationally prohibitive. This work opens opportunities for distance computations on manifolds.