1-Wasserstein Distance on the Standard Simplex

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 $\Omega \times \Omega$.