Computing distances and means on manifolds with a metric-constrained
  Eikonal approach
Daniel Kelshaw Luca Magri
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.