  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.
