Computing distances and means on manifolds with a metric-constrained Eikonal approach
Daniel Kelshaw, Luca Magri · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2025
Computing distances on Riemannian manifolds is a challenging problem with numerous applications, from physics, through statistics, to machine learning. In this article, we introduce the metric-constrained Eikonal solver to obtain continuous, differentiable representations of distance functions (geodesics) on manifolds. The differentiable nature of these representations allows for the direct computation of globally length-minimizing 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 Fréchet 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.