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.

翻译：在黎曼流形上计算距离是一个具有挑战性的问题，其应用涵盖物理、统计和机器学习等多个领域。本文引入度量约束的Eikonal求解器以获得流形上距离函数的连续可微表示。这些表示的可微性质允许直接在流形上计算全局长度最小路径。我们展示了度量约束Eikonal求解器在多种流形上的应用，并演示了其实际价值。首先证明，通过采用具有解析解的高斯混合模型定义，该求解器可用于获取流形上的弗雷歇均值，以验证数值结果的准确性。其次，我们展示了如何利用所得距离函数在流形上进行无监督聚类——这是一项现有方法计算代价高昂的任务。本研究为流形上的距离计算开辟了新途径。