Comparison between $W_2$ distance and $\dot{H}^{-1}$ norm, and localisation of Wasserstein distance

TL;DR

This paper establishes non-asymptotic comparisons between the Wasserstein $W_2$ distance and the $\, ext{dot} ext{-}H^{-1}$ norm, demonstrating a localization property of Wasserstein distance under measure restrictions.

Contribution

It provides a non-asymptotic comparison between $W_2$ and the $H^{-1}$ norm, and applies this to show Wasserstein distance localization with measure truncation.

Findings

Derived explicit bounds relating $W_2$ and $H^{-1}$ norms.

Proved Wasserstein distance localization under measure restrictions.

Demonstrated practical bounds for localized Wasserstein distances.

Abstract

It is well known that the quadratic Wasserstein distance $W_2 (\mathord{\boldsymbol{\cdot}}, \mathord{\boldsymbol{\cdot}})$ is formally equivalent, for infinitesimally small perturbations, to some weighted $H^{-1}$ homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the $W_2$ distance exhibits some localisation phenomenon: if $\mu$ and $\nu$ are measures on $\mathbf{R}^n$ and $\varphi \colon \mathbf{R}^n \to \mathbf{R}_+$ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between $\varphi \cdot \mu$ and $\varphi \cdot \nu$ by an explicit multiple of $W_2 (\mu, \nu)$.