论文标题
平滑的瓦斯尔斯坦距离的渐近学
Asymptotics of smoothed Wasserstein distances
论文作者
论文摘要
我们研究了高斯平滑下的$ \ mathbb {r}^d $上瓦斯恒星距离的收缩。众所周知，在正弯曲的流形上，热半群相对于瓦斯尔斯坦的距离是指数的。但是，在平坦的欧几里得空间上---热半群的对应于高斯卷积平滑措施 - 情况更加微妙。在欧几里得热半群的作用下，我们证明了$ 2 $ - 沃塞尔斯坦距离的精确渐近学，并表明，与正面弯曲的情况相反，收缩率总是多项式的，取决于指数的矩序列。我们为$ p $ wasserstein的距离建立了类似的结果，$ p \ neq 2 $以及$χ^2 $ divergence，相对熵和总变化距离。这些结果共同确定了匹配参数的核心作用在对高斯卷积平滑的度量分析中。
We investigate contraction of the Wasserstein distances on $\mathbb{R}^d$ under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive curvature; however, on flat Euclidean space---where the heat semigroup corresponds to smoothing the measures by Gaussian convolution---the situation is more subtle. We prove precise asymptotics for the $2$-Wasserstein distance under the action of the Euclidean heat semigroup, and show that, in contrast to the positively curved case, the contraction rate is always polynomial, with exponent depending on the moment sequences of the measures. We establish similar results for the $p$-Wasserstein distances for $p \neq 2$ as well as the $χ^2$ divergence, relative entropy, and total variation distance. Together, these results establish the central role of moment matching arguments in the analysis of measures smoothed by Gaussian convolution.
