The fast rate of convergence of the smooth adapted Wasserstein distance

Abstract

Estimating a $d$-dimensional distribution $μ$ by the empirical measure $\hatμ_n$ of its samples is an important task in probability theory, statistics and machine learning. It is well known that $\mathbb{E}[\mathcal{W}_p(\hatμ_n, μ)]\lesssim n^{-1/d}$ for $d>2p$, where $\mathcal{W}_p$ denotes the $p$-Wasserstein metric. An effective tool to combat this curse of dimensionality is the smooth Wasserstein distance $\mathcal{W}^{(σ)}_p$, which measures the distance between two probability measures after having convolved them with isotropic Gaussian noise $\mathcal{N}(0,σ^2\text{I})$. In this paper we apply this smoothing technique to the adapted Wasserstein distance. We show that the smooth adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p^{(σ)}$ achieves the fast rate of convergence $\mathbb{E}[\mathcal{A}\mathcal{W}_p^{(σ)}(\hatμ_n, μ)]\lesssim n^{-1/2}$, if $μ$ is subgaussian. This result follows from the surprising fact, that any subgaussian measure $μ$ convolved with a Gaussian distribution has locally Lipschitz kernels.