论文标题
在Kantorovich-Rubinstein的不平等
On a Kantorovich-Rubinstein inequality
论文作者
论文摘要
Kantorovich-Rubinstein二元性的一个简单结果是：如果$ f：[0,1]^d \ rightarrow \ infty \ inpschitz是Lipschitz和$ \ left \ weft \ {x_1，\ dots，x_n \ dots，x_n \ right \ right \ right \ right \ oir \ int _ {[0,1]^d} f（x）dx- \ frac {1} {n} {n} \ sum_ {k = 1}^{n} {n} {f（x_k）} \ right | \ leq \ left \ | \ nabla f \ right \ | _ {l^{\ infty}}} \ cdot w_1 \ left（\ frac {1} {n} {n} \ sum_ {k = 1}^{n} {n} {n} {δ__移动的距离。我们证明了另一种在$ \ nabla f $和较大的Wasserstein距离上具有较小规范的不平等现象。当点非常规，即$ w _ {\ infty} \ sim n^{ - 1/d} $时，我们的不平等很清晰。这引发了一个问题，这两个不平等是否是整个基础估算家族的特定实例，从而捕获了运输距离和功能空间之间的双重性。
An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d \rightarrow \infty$ is Lipschitz and $\left\{x_1, \dots, x_N \right\} \subset [0,1]^d$, then $$ \left| \int_{[0,1]^d} f(x) dx - \frac{1}{N} \sum_{k=1}^{N}{f(x_k)} \right| \leq \left\| \nabla f \right\|_{L^{\infty}} \cdot W_1\left( \frac{1}{N} \sum_{k=1}^{N}{δ_{x_k}} , dx\right),$$ where $W_1$ denotes the $1-$Wasserstein (or Earth Mover's) Distance. We prove another such inequality with a smaller norm on $\nabla f$ and a larger Wasserstein distance. Our inequality is sharp when the points are very regular, i.e. $W_{\infty} \sim N^{-1/d}$. This prompts the question whether these two inequalities are specific instances of an entire underlying family of estimates capturing a duality between transport distance and function space.
