论文标题
Wasserstein的收敛速率越来越集中概率度量
Wasserstein convergence rates of increasingly concentrating probability measures
论文作者
论文摘要
对于$ \ ell \ colon \ mathbb {r}^d \ to [0，\ infty）$，我们考虑概率度量的顺序$ \ left（μ_n\ right）_ {n \ in \ mathbb {n}} $，其中$μ_n$由$ el ell at $ n $ \ fexport $ \ fexport $ \ expectional（其中）。只要它们形成有限的紧凑型歧管结合，我们允许无限的许多全球最小点（$ \ ell $）。在这种情况下，我们将$ p $ -wasserstein收敛的估计值（μ_n\ right）_ {n \ in \ mathbb {n}} $显示为限制度量。施加规律性条件，我们获得了$ n^{ - 1/（2p）} $的收敛速度，并增加了技术假设，我们可以将其提高到所有订单的$ 1/2 $ $ 1/2 $的$ p \ in \ Mathbb {n} $的$ 1/2 $。
For $\ell\colon \mathbb{R}^d \to [0,\infty)$ we consider the sequence of probability measures $\left(μ_n\right)_{n \in \mathbb{N}}$, where $μ_n$ is determined by a density that is proportional to $\exp(-n\ell)$. We allow for infinitely many global minimal points of $\ell$, as long as they form a finite union of compact manifolds. In this scenario, we show estimates for the $p$-Wasserstein convergence of $\left(μ_n\right)_{n \in \mathbb{N}}$ to its limit measure. Imposing regularity conditions we obtain a speed of convergence of $n^{-1/(2p)}$ and adding a further technical assumption, we can improve this to a $p$-independent rate of $1/2$ for all orders $p\in\mathbb{N}$ of the Wasserstein distance.
