论文标题
平滑经验剂量纪念碑的收敛速率
Rate of convergence of the smoothed empirical Wasserstein distance
论文作者
论文摘要
考虑一个经验度量$ \ mathbb {p} _n $由$ n $ iid样品从$ d $ d $二维$ k $ -k $ -subgaussian发行$ \ mathbb {p} $，让$γ= n（0，σ^2 i_d）$是等性竞技表量。我们研究平滑的Wasserstein距离的收敛速度$ W_2（\ Mathbb {p} _n*γ，\ Mathbb {p}*γ）= n^{ - α+ o（1）} $，$*$是度量的卷积。对于$ k <σ$，在任何维度$ d \ ge 1 $中，我们表明$α= {1 \ over2} $。对于$ k>σ$ in dimension $ d = 1 $，我们表明速率较慢，由$α= {（σ^2 + k^2）^2 \ of 4（σ^4 + k^4）} <1/2 $。这可以解决[GGNWP20]中的几个开放问题，尤其是精确地确定获得参数率所需的平滑$σ$。此外，对于任何$ d $二维$ k $ -subgaussian发行$ \ mathbb {p} $，我们还确定$ d_ {kl}（\ mathbb {p} _n * _n *γ\ | \ | \ mathbb {p} p} p} *γγ$ for $ o（1/n）$ o（1/n） n）^{d+1} \ fover n}）$ for $ k>σ$。 $ W_2^2 $和KL的行为的令人惊讶的差异意味着$ T_ {2} $ - 运输不平等的故障。因此，因此，对于高斯混合物$ \ mathbb {p} * n（0，σ^{2}）$，对于$ k>σ$ log-sobolev不等式（LSI）$。这关闭了[WW+16]中的一个开放问题，后者在$ k <σ$的情况下建立了LSI，并询问是否可以改善其绑定。
Consider an empirical measure $\mathbb{P}_n$ induced by $n$ iid samples from a $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$ and let $γ= N(0,σ^2 I_d)$ be the isotropic Gaussian measure. We study the speed of convergence of the smoothed Wasserstein distance $W_2(\mathbb{P}_n * γ, \mathbb{P}*γ) = n^{-α+ o(1)}$ with $*$ being the convolution of measures. For $K<σ$ and in any dimension $d\ge 1$ we show that $α= {1\over2}$. For $K>σ$ in dimension $d=1$ we show that the rate is slower and is given by $α= {(σ^2 + K^2)^2\over 4 (σ^4 + K^4)} < 1/2$. This resolves several open problems in [GGNWP20], and in particular precisely identifies the amount of smoothing $σ$ needed to obtain a parametric rate. In addition, for any $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$, we also establish that $D_{KL}(\mathbb{P}_n * γ\|\mathbb{P}*γ)$ has rate $O(1/n)$ for $K<σ$ but only slows down to $O({(\log n)^{d+1}\over n})$ for $K>σ$. The surprising difference of the behavior of $W_2^2$ and KL implies the failure of $T_{2}$-transportation inequality when $σ< K$. Consequently, it follows that for $K>σ$ the log-Sobolev inequality (LSI) for the Gaussian mixture $\mathbb{P} * N(0, σ^{2})$ cannot hold. This closes an open problem in [WW+16], who established the LSI under the condition $K<σ$ and asked if their bound can be improved.
