Rate of convergence of the smoothed empirical Wasserstein distance

Consider an empirical measure $\mathbb{P}_n$ induced by $n$ iid samples from a $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$ and let $\gamma = \mathcal{N}(0,\sigma^2 I_d)$ be the isotropic Gaussian measure. We study the speed of convergence of the smoothed Wasserstein distance $W_2(\mathbb{P}_n * \gamma, \mathbb{P}*\gamma) = n^{-\alpha + o(1)}$ with $*$ being the convolution of measures. For $K<\sigma$ and in any dimension $d\ge 1$ we show that $\alpha = {1\over2}$. For $K>\sigma$ in dimension $d=1$ we show that the rate is slower and is given by $\alpha = {(\sigma^2 + K^2)^2\over 4 (\sigma^4 + K^4)} < 1/2$. This resolves several open problems in \cite{goldfeld2020convergence}, and in particular precisely identifies the amount of smoothing $\sigma$ needed to obtain a parametric rate. In addition, we also establish that $D_{KL}(\mathbb{P}_n * \gamma \|\mathbb{P}*\gamma)$ has rate $O(1/n)$ for $K<\sigma$ but only slows down to $O({(\log n)^{d+1}\over n})$ 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, the requirement $K<\sigma$ is necessary for validity of the log-Sobolev inequality (LSI) for the Gaussian mixture $\mathbb{P} * \mathcal{N}(0, \sigma^{2})$, closing an open problem in \cite{wang2016functional}, who established the LSI under precisely this condition.