The earth mover's distance (EMD), also known as the 1-Wasserstein metric, measures the minimum amount of work required to transform one probability distribution into another. The EMD can be naturally generalized to measure the "distance" between any number (say $d$) of distributions. In previous work (2021), we found a recursive formula for the expected value of the generalized EMD, assuming the uniform distribution on the standard $n$-simplex. This recursion, however, was computationally expensive, requiring $\binom{d+n}{d}$ iterations. The main result of the present paper is a nonrecursive formula for this expected value, expressed as the integral of a certain polynomial of degree at most $dn$. As a secondary result, we resolve an unanswered problem by giving a formula for the generalized EMD in terms of pairwise EMDs; this can be viewed as an analogue of the Cayley-Menger determinant formula that gives the hypervolume of a simplex in terms of its edge lengths.

翻译：推土机距离（EMD），亦称1-Wasserstein度量，衡量将一个概率分布转换为另一个概率分布所需的最小工作量。该度量可自然推广至任意数量（记为$d$）分布之间的“距离”度量。在前期工作（2021）中，我们基于标准$n$单纯形上的均匀分布假设，发现了广义EMD期望值的递归公式。然而该递归计算复杂度较高，需进行$\binom{d+n}{d}$次迭代。本文的主要成果是给出该期望值的非递归公式，表示为最高次$dn$的特定多项式积分。作为次要成果，我们通过给出基于成对EMD的广义EMD公式，解决了此前悬而未决的问题；该公式可视为Cayley-Menger行列式公式的类比——后者通过边长计算单纯形的超体积。