Vol. 12, No. 2, 2021

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A generalization for the expected value of the earth mover's distance

William Q. Erickson

Vol. 12 (2021), No. 2, 139–166
DOI: 10.2140/astat.2021.12.139

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 [n] = {1,,n}. 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.

earth mover's distance, generating functions, Wallach representations
Mathematical Subject Classification
Primary: 05E14, 13P25
Secondary: 05E40
Received: 26 October 2020
Revised: 5 April 2021
Accepted: 18 May 2021
Published: 13 December 2021
William Q. Erickson
Department of Mathematical Sciences
University of Wisconsin–Milwaukee
Milwaukee, WI
United States