1-Wasserstein distance on the standard simplex

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

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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

Vol. 12, No. 1, 2021

Andrew Frohmader and Hans Volkmer

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors