Wasserstein distance and metric trees

TL;DR

This paper explores the embedding properties of the Wasserstein metric on probability measures over metric spaces, establishing connections with metric tree embeddings and providing new formulas and algorithms for finite metric trees.

Contribution

It demonstrates that Wasserstein spaces over finite metric spaces can embed into  with controlled distortion if the base space embeds into metric trees, and extends the Wasserstein formula to real trees with new proofs.

Findings

Wasserstein space P(X) embeds into  with distortion D if X embeds into metric trees with distortion D.

Provides an extended formula for Wasserstein metric on real trees.

Offers algorithmic and Banach space theory-based proofs for the Wasserstein formula extension.

Abstract

We study the Wasserstein (or earthmover) metric on the space $P(X)$ of probability measures on a metric space $X$. We show that, if a finite metric space $X$ embeds stochastically with distortion $D$ in a family of finite metric trees, then $P(X)$ embeds bi-Lipschitz into $\ell^1$ with distortion $D$. Next, we re-visit the closed formula for the Wasserstein metric on finite metric trees due to Evans-Matsen \cite{EvMat}. We advocate that the right framework for this formula is real trees, and we give two proofs of extensions of this formula: one making the link with Lipschitz-free spaces from Banach space theory, the other one algorithmic (after reduction to finite metric trees).