Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples

TL;DR

This paper establishes convergence guarantees for Gaussian-smoothed optimal transport distances under broad conditions, including sub-gamma distributions and dependent samples, enhancing understanding and applicability of the GOT framework.

Contribution

It provides the first convergence analysis of GOT distances for sub-gamma distributions and dependent samples, with explicit dependence on dimension and scale parameters.

Findings

Convergence guarantees under minimal moment conditions.

Quantification of dimension dependence for sub-gamma distributions.

Extension of convergence results to dependent samples with covariance conditions.

Abstract

The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.