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:随着对图形数据的兴趣在近年来的增长,计算各种几何工具已成为必要。在某些领域(如网格处理),它们经常依赖于离散流形上的测地线和最短路径的计算。最近一个这样的工具是计算Wasserstein barycenters(WB),这是从最优传输理论中导出的一种非常通用的重心概念,以及它们的熵正则化变量。在本文中,我们考察了离散网格上的WB如何与底层流形的几何相关联。我们首先提供了一个通用的稳定性结果,与输入成本矩阵有关。然后,我们将此结果应用于随机几何图形,这些图形在流形上的最短路径收敛于测地线,从而证明了计算离散形状的WB的一致性。