Computing distances and means on manifolds with a metric-constrained Eikonal approach
This addresses a challenging computational geometry problem with applications in physics, statistics, and machine learning, offering a new method for distance-based tasks on manifolds where existing approaches are computationally prohibitive.
The paper tackles the problem of computing distances on Riemannian manifolds by introducing a metric-constrained Eikonal solver that provides continuous, differentiable distance representations, enabling direct computation of globally length-minimizing paths and applications like Fréchet mean calculation and unsupervised clustering.
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.