Mean convergence rates for Gaussian-smoothed Wasserstein distances and classical Wasserstein distances
Andrea Cosso, Mattia Martini, Laura Perelli
TL;DR
The paper advances the understanding of Wasserstein-distance convergence by analyzing Gaussian-smoothed distances $\mathcal{W}_p^{(\sigma)}$ between an arbitrary distribution $\mu$ with finite $q$-th moment ($q>p$) and its empirical counterpart $\mu_N$. It proves upper bounds for $\mathbb{E}[({\mathcal W}_p^{(\sigma)}(\mu_N,\mu))^p]$ with two complementary proofs: a Carlson-type inequality combined with Marcinkiewicz-Zygmund, and a dyadic partitioning approach, establishing rates that depend on $q$: $O(1/\sqrt{N})$ if $q \ge 2p+d$ and $O(N^{-(q-p)/(q+d)})$ for $p<q\le 2p+d$. The work also shows how to transfer these results to the classical ${\mathcal W}_p$ by letting $\sigma\to 0$ and provides a moment-free bound via a truncation argument. Additionally, it develops a third upper bound for ${\mathcal W}_p^{(\sigma)}$ to derive a bound for ${\mathcal W}_p$ itself without moment assumptions, albeit at the cost of optimality when moments exist. These results contribute practical, dimension-robust guarantees for Wasserstein-distance estimation from finite samples.
Abstract
We establish upper bounds for the expected Gaussian-smoothed $p$-Wasserstein distance between a probability measure $μ$ and the corresponding empirical measure $μ_N$, whenever $μ$ has finite $q$-th moments for any $q>p$. This generalizes recent results that were valid only for $q>2p+2d$. We provide two distinct proofs of such a result. We also use a third upper bound for the Gaussian-smoothed $p$-Wasserstein distance to derive an upper bound for the classical $p$-Wasserstein distance. Although the latter upper bound is not optimal when $μ$ has finite $q$-th moment with $q>p$, this bound does not require imposing such a moment condition on $μ$, as it is usually done in the literature.