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：随着近年来对图表数据的兴趣增加,各种几何工具的计算变得至关重要,在网状处理等某些领域,这些工具往往依赖于大地测量学的计算,以及离散式元件中最短路径的计算,最近这种工具的一个实例是瓦塞斯坦采样中心(WB)的计算,Wasserstein采样中心(WB)是来自最佳运输理论的一个非常一般的采样概念,也是其昆虫常规变异。在本文中,我们研究了离散式介质上的世行与深层元件的几何测量有何关系。我们首先对输入成本矩阵提供了一种通用的稳定结果。然后,我们将这一结果应用于在各种元件上的随机几何图形,其最短路径与大地测量学汇合,从而证明了以离散式形状计算的世行的一致性。