Rate of convergence of the smoothed empirical Wasserstein distance

Abstract

Consider an empirical measure ${\mathbb{P}}_n$ induced by $n$ iid samples from a $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$ and let $\gamma =N(0,{\sigma }^2{I}_d)$ be the isotropic Gaussian measure. We study the speed of convergence of the smoothed Wasserstein distance $W_2({\mathbb{P}}_n\ast \gamma ,\mathbb{P}\ast \gamma )=n^{-\alpha +o(1)}$ with $\ast$ being the convolution of measures. For $K<\sigma$ and in any dimension $d\ge 1$ we show that $\alpha =\frac{1}{2}$. For $K>\sigma$ in dimension $d=1$ we show that the rate is slower and is given by $\alpha =\frac{({\sigma }^2+K^2{)}^2}{4({\sigma }^4+K^4)}<1/2$. This resolves several open problems in [GGNWP20], and in particular precisely identifies the amount of smoothing $\sigma$ needed to obtain a parametric rate. In addition, for any $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$, we also establish that $D_{KL}({\mathbb{P}}_n\ast \gamma \Vert \mathbb{P}\ast \gamma )$ has rate $O(1/n)$ for $K<\sigma$ but only slows down to $O(\frac{(\log n{)}^{d+1}}{n})$ for $K>\sigma$. The surprising difference of the behavior of $W_2^2$ and KL implies the failure of $T_2$-transportation inequality when $\sigma <K$. Consequently, it follows that for $K>\sigma$ the log-Sobolev inequality (LSI) for the Gaussian mixture $\mathbb{P}\ast N(0,{\sigma }^2)$ cannot hold. This closes an open problem in [WW+16], who established the LSI under the condition $K<\sigma$ and asked if their bound can be improved.