论文标题
小噪声状态中平滑瓦斯汀距离的渐近造型
Asymptotics of smoothed Wasserstein distances in the small noise regime
论文作者
论文摘要
我们研究Wasserstein- $ 2 $ $ 2 $ $μ$和$ν$之间的$ 2 $的行为，当两种措施都通过少量高斯噪声平滑时，$ \ mathbb {r}^D $。该程序被称为高斯平滑的最佳运输，最近引起了人们对未注册瓦斯汀距离的统计吸引力替代方案的关注。 We give precise bounds on the approximation properties of this proposal in the small noise regime, and establish the existence of a phase transition: we show that, if the optimal transport plan from $μ$ to $ν$ is unique and a perfect matching, there exists a critical threshold such that the difference between $W_2(μ, ν)$ and the Gaussian-smoothed OT distance $W_2(μ\ast \ Mathcal {n}_σ，ν\ ast \ Mathcal {n}_σ）$ scales $ scales，例如$ \ exp（-c /σ^2）$ for the Threshold的$ $σ$，以及像$σ$之类的比例。这些结果表明，对于$σ$，平滑的瓦斯汀距离近似于未注册的距离，呈指数级别。
We study the behavior of the Wasserstein-$2$ distance between discrete measures $μ$ and $ν$ in $\mathbb{R}^d$ when both measures are smoothed by small amounts of Gaussian noise. This procedure, known as Gaussian-smoothed optimal transport, has recently attracted attention as a statistically attractive alternative to the unregularized Wasserstein distance. We give precise bounds on the approximation properties of this proposal in the small noise regime, and establish the existence of a phase transition: we show that, if the optimal transport plan from $μ$ to $ν$ is unique and a perfect matching, there exists a critical threshold such that the difference between $W_2(μ, ν)$ and the Gaussian-smoothed OT distance $W_2(μ\ast \mathcal{N}_σ, ν\ast \mathcal{N}_σ)$ scales like $\exp(-c /σ^2)$ for $σ$ below the threshold, and scales like $σ$ above it. These results establish that for $σ$ sufficiently small, the smoothed Wasserstein distance approximates the unregularized distance exponentially well.
