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
) 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
-simplex.
This recursion, however, was computationally expensive, requiring
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
. 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.
