1-Wasserstein Distance on the Standard Simplex

TL;DR

This paper derives explicit formulas for the first and second moments of the 1-Wasserstein distance on the space of probability measures over a finite set, assuming a uniform distribution.

Contribution

It provides closed-form expressions for the moments of the 1-Wasserstein distance on the standard simplex, advancing understanding of its statistical properties.

Findings

Closed-form formulas for the first moment of W_1

Closed-form formulas for the second moment of W_1

Analysis under uniform distribution on probability measures

Abstract

Wasserstein distances provide a metric on a space of probability measures. We consider the space $\Omega$ of all probability measures on the finite set $\chi = \{1, \dots ,n\}$ where $n$ is a positive integer. 1-Wasserstein distance, $W_1(\mu,\nu)$ is a function from $\Omega \times \Omega$ to $[0,\infty)$. This paper derives closed form expressions for the First and Second moment of $W_1$ on $\Omega \times \Omega$ assuming a uniform distribution on $\Omega \times \Omega$.