As interest in graph data has grown in recent years, the computation of various geometric tools has become essential. In some area such as mesh processing, they often rely on the computation of geodesics and shortest paths in discretized manifolds. A recent example of such a tool is the computation of Wasserstein barycenters (WB), a very general notion of barycenters derived from the theory of Optimal Transport, and their entropic-regularized variant. In this paper, we examine how WBs on discretized meshes relate to the geometry of the underlying manifold. We first provide a generic stability result with respect to the input cost matrices. We then apply this result to random geometric graphs on manifolds, whose shortest paths converge to geodesics, hence proving the consistency of WBs computed on discretized shapes.

翻译：随着近年来对图数据兴趣的增长，各类几何工具的计算变得至关重要。在网格处理等领域，这些工具通常依赖于离散流形上测地线及最短路径的计算。近期一个典型例子是Wasserstein重心（WB）的计算——源于最优传输理论的广义重心概念，及其熵正则化变体。本文旨在探讨离散网格上的Wasserstein重心如何反映底层流形的几何特性。我们首先给出一个关于输入代价矩阵的通用稳定性结果，随后将该结果应用于流形上的随机几何图（其最短路径收敛于测地线），从而证明了离散形状上计算的Wasserstein重心的一致性。