Vol. 11, No. 1, 2020

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Expected value of the one-dimensional earth mover's distance

Rebecca Bourn and Jeb F. Willenbring

Vol. 11 (2020), No. 1, 53–78

From a combinatorial point of view, we consider the earth mover’s distance (EMD) associated with a metric measure space. The specific case considered is deceptively simple: Let the finite set of integers [n] = {1,,n} be regarded as a metric space by restricting the usual Euclidean distance on the real numbers. The EMD is defined on ordered pairs of probability distributions on [n]. We provide an easy method to compute a generating function encoding the values of EMD in its coefficients, which is related to the Segre embedding from projective algebraic geometry. As an application we use the generating function to compute the expected value of EMD in this one-dimensional case. The EMD is then used in clustering analysis for a specific data set.

earth mover's distance, generating function, Segre embedding, spectral graph theory, clustering
Received: 3 May 2019
Revised: 17 October 2019
Accepted: 5 November 2019
Published: 1 October 2020
Rebecca Bourn
Department of Mathematical Sciences
University of Wisconsin - Milwaukee
3200 N. Cramer St.
Milwaukee, WI 53211
United States
Jeb F. Willenbring
Department of Mathematical Sciences
University of Wisconsin - Milwaukee
3200 North Cramer Street
Milwaukee, WI 53211
United States