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

TL;DR

The paper introduces a metric-constrained Eikonal solver that learns a differentiable distance function $\varphi(q; p)$ on a manifold $M$ by parameterising it with a neural network and enforcing the Eikonal constraint $\|\nabla \varphi\|_g = 1$. This framework yields globally length-minimising geodesics via the gradient flow and supports direct computation of the log* map, enabling efficient Fréchet-mean estimation and manifold clustering. The authors validate the approach on Euclidean space, the unit hypersphere, and the non-convex Peaks manifold, and demonstrate a Fréchet mean computation on a Gaussian-mixture manifold with competitive accuracy. Their results show accurate distance fields, compatible geodesic flows, and practical speedups, with an open-source implementation to facilitate adoption in geometry-aware learning and statistics on manifolds.

Abstract

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 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.

Figures (8)

• Figure 1: Overview of the manifolds used throughout the paper, visualised in $\mathbb{R}^3$. Panel (\ref{['fig:manifolds:sphere']}) depicts a patch of the two-sphere, panel (\ref{['fig:manifolds:gmm']}) shows a model defined by the probability density function of a Gaussian mixture model, and panel (\ref{['fig:manifolds:peaks']}) shows a multi-modal manifold.
• Figure 2: Results for the Euclidean manifold. Panel (\ref{['fig:euclidean:pred_distance']}) shows the predicted distance around a point, panel (\ref{['fig:euclidean:true_distance']}) shows the corresponding true distance from the point, and panel (\ref{['fig:euclidean:dimension_study']}) shows the mean relative $\ell^2$ error across five repeats for increasing dimensions $n \in \{2, 5, 10, 15, 20\}$.
• Figure 3: Results for the hypersphere manifold. Panel (\ref{['fig:hypersphere:pred_distance']}) shows the predicted distance around a point, panel (\ref{['fig:hypersphere:true_distance']}) shows the corresponding true distance from the point, and panel (\ref{['fig:hypersphere:dimension_study']}) shows the mean relative $\ell^2$ error across five repeats for dimensions $n \in \{ 2, 5, 10, 15, 20 \}$.
• Figure 4: Results for the single-point solution on the peaks manifold. Panel (\ref{['fig:single_point:curvature_sampling']}) shows the probability density function generated from the curvature-based sampling (Algorithm \ref{['applications:algorithm:sampling']}), with samples from the distribution overlaid. Panel (\ref{['fig:single_point:geodesics']}) shows the distance field around a point in the centre of the domain with arrows depicting the geodesic flow overlaid. The yellow curves denote length-minimising geodesics obtained from the geodesic flow, while the red curves denotes the corresponding valid geodesics. Panel (\ref{['fig:single_point:distance']}) shows results for the symmetricity test.
• Figure 5: Comparison of results from standard parameterised curve approach against results from the metric-constrained Eikonal solver. Results are shown only in the cases where the parameterised curve approach converges. We observe good agreement between the proposed metric-constrained Eikonal approach, and the standard length-minimising curve approach.