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 $\binom{d+n}{d}$ many 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-瓦瑟斯坦度量，用于衡量将一个概率分布转化为另一个分布所需的最小“工作量”。该距离可自然推广至度量任意数量（设为 $d$ 个）分布之间的“距离”。在先前的研究（2021年）中，我们基于标准 $n$-单纯形上的均匀分布假设，得到了广义EMD期望值的一个递归计算公式。然而该递归计算代价高昂，需要 $\binom{d+n}{d}$ 次迭代。本文的主要结果是给出了该期望值的一个非递归公式，其表达式为一个次数不超过 $dn$ 的特定多项式的积分。作为次要结果，我们通过给出广义EMD基于两两EMD的表达式，解决了一个悬而未决的问题；该公式可视为凯莱-门格尔行列式公式的类比——后者用单纯形各边长表示其超体积。