The earth mover’s distance (EMD), also called the first Wasserstein distance, can be naturally
extended to compare arbitrarily many probability distributions, rather than only two, on
the set
.
We present the details for this generalization, along with a highly efficient algorithm
inspired by combinatorics; it turns out that in the special case of three distributions,
the EMD is half the sum of the pairwise EMDs. Extending the methods of Bourn and
Willenbring (2020), we compute the expected value of this generalized EMD on
random tuples of distributions, using a generating function which coincides with the
Hilbert series of the Segre embedding. We then use the EMD to analyze a real-world
data set of grade distributions.
