Vol. 12, No. 1, 2021

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1-Wasserstein distance on the standard simplex

Andrew Frohmader and Hans Volkmer

Vol. 12 (2021), No. 1, 43–56
Abstract

Wasserstein distances provide a metric on a space of probability measures. We consider the space Ω of all probability measures on the finite set χ = {1,,n}, where n is a positive integer. The 1-Wasserstein distance, W1(μ,ν), is a function from Ω × Ω to [0,). This paper derives closed-form expressions for the first and second moments of W1 on Ω × Ω assuming a uniform distribution on Ω × Ω.

Keywords
Wasserstein, earth mover, probability simplex, simplex, standard simplex, moments, expected value, variance
Mathematical Subject Classification 2010
Primary: 28A33, 60B05
Milestones
Received: 12 December 2019
Revised: 8 September 2020
Accepted: 29 September 2020
Published: 9 April 2021
Authors
Andrew Frohmader
Department of Mathematical Sciences
University of Wisconsin
Milwaukee, WI
United States
Hans Volkmer
Department of Mathematical Sciences
University of Wisconsin
Milwaukee, WI
United States