论文标题
熵Wasserstein Barycenters的稳定性和在随机几何图中的应用
Stability of Entropic Wasserstein Barycenters and application to random geometric graphs
论文作者
论文摘要
随着近年来对图形数据的兴趣，各种几何工具的计算变得至关重要。在某些领域（例如网状处理）中，它们通常依赖于离散流形中的测量学和最短路径的计算。这种工具的最新示例是计算Wasserstein Barycenters（WB），这是一种源自最佳运输理论及其熵调节变体的barycenters的概念。在本文中，我们研究了离散网格上的WBS与基础歧管的几何形状有关。我们首先就输入成本矩阵提供了通用稳定性结果。然后，我们将此结果应用于歧管上的随机几何图，其最短路径会收敛到大地测量学，因此证明了根据离散形状计算的WBS的一致性。
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.
