On the reach of isometric embeddings into Wasserstein type spaces

TL;DR

This paper investigates the geometric property called reach of the natural isometric embedding of a metric space into its Wasserstein space, revealing conditions under which the reach is zero or infinite, with implications for regularity and stability.

Contribution

It characterizes the reach of the isometric embedding into Wasserstein and Orlicz–Wasserstein spaces, providing new insights into the geometric structure and stability of these embeddings.

Findings

Reach is zero if points can be joined by two minimizing geodesics.

Reach is infinite for CAT(0) spaces.

Reach is null for embeddings into the space of persistence diagrams.

Abstract

We study the reach (in the sense of Federer) of the natural isometric embedding $X\hookrightarrow W_p(X)$ of $X$ inside its $p$-Wasserstein space, where $(X,\operatorname{dist})$ is a geodesic metric space. We prove that if a point $x\in X$ can be joined to another point $y\in X$ by two minimizing geodesics, then $\operatorname{reach}(x, X\subset W_p(X)) = 0$. This includes the cases where $X$ is a compact manifold or a non-simply connected one. On the other hand, we show that $\operatorname{reach}(X\subset W_p(X)) = \infty$ when $X$ is a CAT(0) space. The infinite reach enables us to examine the regularity of the projection map. Furthermore, we replicate these findings by considering the isometric embedding $X\hookrightarrow W_\vartheta(X)$ into an Orlicz--Wasserstein space, a generalization by Sturm of the classical Wasserstein space. Lastly, we establish the nullity of the reach for the isometric embedding of $X$ into $\operatorname{Dgm}_\infty$, the space of persistence diagrams equipped with the bottleneck distance.