Free complete Wasserstein algebras

TL;DR

This paper develops an algebraic framework for Wasserstein distances on complete metric spaces, connecting probabilistic measures with algebraic structures to facilitate reasoning about effects in programming languages.

Contribution

It introduces axioms for Wasserstein-algebraic structures parametrized by p and characterizes their free complete algebras as Radon probability measures with Wasserstein metrics.

Findings

Identifies axioms for Wasserstein algebras over metric spaces.

Characterizes free complete Wasserstein algebras as Radon probability measures.

Links algebraic structures with probabilistic metrics in programming semantics.

Abstract

We present an algebraic account of the Wasserstein distances $W_p$ on complete metric spaces, for $p \geq 1$. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in $p$, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance $W_p$. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order $p$, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations.