Stability of Entropic Wasserstein Barycenters and application to random geometric graphs

Abstract: 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.