Asymptotics of smoothed Wasserstein distances in the small noise regime

TL;DR

This paper analyzes how Gaussian smoothing affects Wasserstein-2 distances between measures, revealing a phase transition where the smoothed distance approximates the original exponentially well in the small noise limit.

Contribution

It provides precise bounds and identifies a phase transition in the approximation quality of Gaussian-smoothed Wasserstein distances, depending on the noise level.

Findings

Exponential approximation of Wasserstein distance for small noise

Existence of a phase transition at a critical noise threshold

Quantitative bounds on the approximation error

Abstract

We study the behavior of the Wasserstein-$2$ distance between discrete measures $\mu$ and $\nu$ in $\mathbb{R}^d$ when both measures are smoothed by small amounts of Gaussian noise. This procedure, known as Gaussian-smoothed optimal transport, has recently attracted attention as a statistically attractive alternative to the unregularized Wasserstein distance. We give precise bounds on the approximation properties of this proposal in the small noise regime, and establish the existence of a phase transition: we show that, if the optimal transport plan from $\mu$ to $\nu$ is unique and a perfect matching, there exists a critical threshold such that the difference between $W_2(\mu, \nu)$ and the Gaussian-smoothed OT distance $W_2(\mu \ast \mathcal{N}_\sigma, \nu\ast \mathcal{N}_\sigma)$ scales like $\exp(-c /\sigma^2)$ for $\sigma$ below the threshold, and scales like $\sigma$ above it. These results establish that for $\sigma$ sufficiently small, the smoothed Wasserstein distance approximates the unregularized distance exponentially well.