Gromov–Hausdorff limit of Wasserstein spaces on point clouds

Abstract

We consider a point cloud $$X_n := \{ {\mathbf {x}}_1, \ldots , {\mathbf {x}}_n \}$$ uniformly distributed on the flat torus $${\mathbb {T}}^d : = \mathbb {R}^d / \mathbb {Z}^d $$, and construct a geometric graph on the cloud by connecting points that are within distance $$\varepsilon $$ of each other. We let $${\mathcal {P}}(X_n)$$ be the space of probability measures on $$X_n$$ and endow it with a discrete Wasserstein distance $$W_n$$ as introduced independently in Chow et al. (Arch Ration Mech Anal 203:969–1008, 2012), Maas (J Funct Anal 261:2250–2292, 2011) and Mielke (Nonlinearity 24:1329–1346, 2011) for general finite Markov chains. We show that as long as $$\varepsilon = \varepsilon _n$$ decays towards zero slower than an explicit rate depending on the level of uniformity of $$X_n$$, then the space $$({\mathcal {P}}(X_n), W_n)$$ converges in the Gromov–Hausdorff sense towards the space of probability measures on $${\mathbb {T}}^d$$ endowed with the Wasserstein distance. The analysis presented in this paper is a first step in the study of stability of evolution equations defined over random point clouds as the number of points grows to infinity.