Gromov-Hausdorff limit of Wasserstein spaces on point clouds
This work addresses stability issues for evolution equations on random point clouds as the number of points increases, which is incremental in the study of geometric analysis and machine learning applications.
The paper tackles the problem of proving the convergence of discrete Wasserstein spaces on random point clouds to a continuous limit, showing that under certain conditions on the connectivity parameter, the space of probability measures on the cloud converges in the Gromov-Hausdorff sense to the space of probability measures on the torus with the Wasserstein distance.
We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ 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 by Chow et al, Maas, and Mielke 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.