Wasserstein distance and metric trees (Q6113469)

Summary: 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 \textit{S. N. Evans} and \textit{F. A. Matsen} [J. R. Stat. Soc., Ser. B, Stat. Methodol. 74, No. 3, 569--592 (2012; Zbl 1411.62317)]. 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).