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 $W_2(\mathbb{P}_n * \gamma, \mathbb{P}*\gamma) = n^{-\alpha + o(1)}$ 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 [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 $D_{KL}(\mathbb{P}_n * \gamma \|\mathbb{P}*\gamma)$ 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, it follows that for $K>\sigma$ the log-Sobolev inequality (LSI) for the Gaussian mixture $\mathbb{P} * N(0, \sigma^{2})$ 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.

翻译：考虑由$d$维$K$-次高斯分布$\mathbb{P}$的$n$个独立同分布样本诱导的经验测度$\mathbb{P}_n$，并令$\gamma = N(0,\sigma^2 I_d)$为各向同性高斯测度。我们研究平滑Wasserstein距离$W_2(\mathbb{P}_n * \gamma, \mathbb{P}*\gamma) = n^{-\alpha + o(1)}$的收敛速度，其中$*$表示测度的卷积。对于$K<\sigma$且在任意维度$d\ge 1$下，我们证明$\alpha = {1\over2}$。对于$K>\sigma$在维度$d=1$的情形，我们证明收敛速率较慢且由$\alpha = {(\sigma^2 + K^2)^2\over 4 (\sigma^4 + K^4)} < 1/2$给出。这解决了[GGNWP20]中提出的若干开放问题，特别是精确识别了获得参数化速率所需的平滑量$\sigma$。此外，对于任意$d$维$K$-次高斯分布$\mathbb{P}$，我们还建立了$D_{KL}(\mathbb{P}_n * \gamma \|\mathbb{P}*\gamma)$在$K<\sigma$时具有$O(1/n)$的收敛速率，但在$K>\sigma$时仅减慢至$O({(\log n)^{d+1}\over n})$。$W_2^2$与KL散度行为的显著差异意味着当$\sigma < K$时$T_{2}$-传输不等式不成立。因此可得，对于$K>\sigma$，高斯混合分布$\mathbb{P} * N(0, \sigma^{2})$的对数Sobolev不等式（LSI）不可能成立。这解决了[WW+16]中的一个开放问题，该研究在$K<\sigma$条件下建立了LSI，并询问其界限能否被改进。