$L_1$-distortion of Wasserstein metrics: a tale of two dimensions

TL;DR

This paper introduces a new dimensionality perspective on Wasserstein metrics, showing how graph properties influence their $L_1$-distortion, and applies this to specific graph sequences like diamond graphs.

Contribution

It provides a novel interpretation of Wasserstein metric distortion in terms of graph dimensions and computes these dimensions for certain graph sequences, answering open questions.

Findings

Wasserstein $L_1$-distortion lower bounds relate to graph dimensions.

Diamond graphs have isoperimetric and spectral dimensions equal to 2.

Sequence of diamond graphs' Wasserstein metrics are not $L_1$-embeddable.

Abstract

By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid $\{0,1,\dots n\}^2$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log n}$. We provide a new "dimensionality" interpretation of Kislyakov's argument, showing that, if $\{G_n\}_{n=1}^\infty$ is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number $\delta \in [2,\infty)$, then the 1-Wasserstein metric over $G_n$ has $L_1$-distortion bounded below by a constant multiple of $(\log |G_n|)^{\frac{1}{\delta}}$. We proceed to compute these dimensions for $\oslash$-powers of certain graphs. In particular, we get that the sequence of diamond graphs $\{\mathsf{D}_n\}_{n=1}^\infty$ has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over $\mathsf{D}_n$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log| \mathsf{D}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of $L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not $L_1$-embeddable.