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arXiv · 2110.02115

Wasserstein distance and metric trees

Maxime Mathey-Prevot·Alain Valette
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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).

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Maxime Mathey-Prevot, Alain Valette. 2021-10-05. Wasserstein distance and metric trees. https://arxiv.org/abs/2110.02115

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