两两多边际最优输运与地球移动器距离嵌入
Pairwise Multi-marginal Optimal Transport and Embedding for Earth Mover's Distance
摘要：
研究了两两多边际最优输运问题，即给定波兰空间$\mathcal{X}$上概率分布$\{P_α\}$的集合，寻找一个耦合的$\{X_α\}$，$X_α\sim P_α$，使得对于所有的$α,β$，有$\mathbf{E}[c(X_α,X_β)]\le r\inf_{X\sim P_α,Y\sim P_β}\mathbf{E}[c(X,Y)]$，其中$c$是一个代价函数，$r\ge1$ .换句话说，每一对$(X_α,X_β)$的期望成本最多是其最低可能值的$r$的一个因子。这可以被视为概率分布的局部敏感散列函数，并且具有诸如运输计划的鲁棒和分布式计算的应用。它也可以被认为是一个双Lipschitz嵌入的概率分布的集合到空间的随机变量取值于$\mathcal{X}$。对于$c(x,y)=\Vert x-y\Vert_2^q$在$\mathbb{R}^n$，其中$q>0$，我们证明了一个有限的$r$是可达到的当且仅当$n=1$或$0<q<1$。作为$n\to\infty$，最小可能的$r$的增长率正好是$Θ(n^{q/2})$，如果$0<q<1$。因此，具有有限$q$阶绝对矩$0<q<1$的$\mathbb{R}^n$上的概率分布的度量空间，以及推土机相对于雪花度量$c(x,y)=\Vert x-y\Vert_2^q$的距离（或1-Wasserstein距离），是双Lipschitz可嵌入到具有失真$O(n^{q/2})$的$L_1$中。如果我们考虑$c(x,y)=\Vert x-y\Vert_2$（即，$q=1$）而不是$\mathbb{R}^n$，则$r=O(\sqrt{n}\log s)$是可达的，这意味着$[0..s]^n$上的概率分布空间可嵌入到具有失真$O(\sqrt{n}\log s)$的$L_1$中，改进了Indyk和Thaper的$O(n\log s)$结果.离散度量成本$c(x,y)=\mathbf{1}\{x\neq y\}$和更一般的度量和超度量成本的情况下也进行了研究。
英文摘要：
We investigate the problem of pairwise multi-marginal optimal transport, that is, given a collection of probability distributions $\{P_α\}$ on a Polish space $\mathcal{X}$, to find a coupling $\{X_α\}$, $X_α\sim P_α$, such that $\mathbf{E}[c(X_α,X_β)]\le r\inf_{X\sim P_α,Y\sim P_β}\mathbf{E}[c(X,Y)]$ for all $α,β$, where $c$ is a cost function and $r\ge1$. In other words, every pair $(X_α,X_β)$ has an expected cost at most a factor of $r$ from its lowest possible value. This can be regarded as a locality sensitive hash function for probability distributions, and has applications such as robust and distributed computation of transport plans. It can also be considered as a bi-Lipschitz embedding of the collection of probability distributions into the space of random variables taking values on $\mathcal{X}$. For $c(x,y)=\Vert x-y\Vert_2^q$ on $\mathbb{R}^n$, where $q>0$, we show that a finite $r$ is attainable if and only if either $n=1$ or $0<q<1$. As $n\to\infty$, the growth rate of the smallest possible $r$ is exactly $Θ(n^{q/2})$ if $0<q<1$. Hence, the metric space of probability distributions on $\mathbb{R}^n$ with finite $q$-th absolute moments, $0<q<1$, with the earth mover's distance (or 1-Wasserstein distance) with respect to the snowflake metric $c(x,y)=\Vert x-y\Vert_2^q$, is bi-Lipschitz embeddable into $L_1$ with distortion $O(n^{q/2})$. If we consider $c(x,y)=\Vert x-y\Vert_2$ (i.e., $q=1$) on the grid $[0..s]^n$ instead of $\mathbb{R}^n$, then $r=O(\sqrt{n}\log s)$ is attainable, which implies the embeddability of the space of probability distributions on $[0..s]^n$ into $L_1$ with distortion $O(\sqrt{n}\log s)$, and improves upon the $O(n\log s)$ result by Indyk and Thaper. The case of the discrete metric cost $c(x,y)=\mathbf{1}\{x\neq y\}$ and more general metric and ultrametric costs are also investigated.
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