Gromov-Hausdorff limit of Wasserstein spaces on point clouds
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We consider a point cloud X_n := { x_1, ..., x_n } uniformly distributed on the flat torus T^d : = R^d / Z^d , and construct a geometric graph on the cloud by connecting points that are within distance ϵ of each other. We let P(X_n) be the space of probability measures on X_n and endow it with a discrete Wasserstein distance W_n as introduced independently by Maas and Zhou et al. for general finite Markov chains. We show that as long as ϵ= ϵ_n decays towards zero slower than an explicit rate depending on the level of uniformity of X_n, then the space (P(X_n), W_n) converges in the Gromov-Hausdorff sense towards the space of probability measures on T^d endowed with the Wasserstein distance.
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