A Central Limit Theorem for Semidiscrete Wasserstein Distances

TL;DR

This paper proves a Central Limit Theorem for empirical optimal transport costs in the semi-discrete case, revealing the asymptotic distribution as a Gaussian process and providing new insights into Wasserstein distances.

Contribution

It establishes the first CLT for semi-discrete Wasserstein distances and analyzes the second derivative of the dual formulation for optimal transport potentials.

Findings

Asymptotic distribution is the supremum of a Gaussian process.

Central limit theorem holds for p-Wasserstein distances with p ≥ 1.

Provides control on the second derivative of the dual formulation.

Abstract

We address the problem of proving a Central Limit Theorem for the empirical optimal transport cost, $\sqrt{n}\{\mathcal{T}_c(P_n,Q)-\mathcal{W}_c(P,Q)\}$, in the semi discrete case, i.e when the distribution $P$ is finitely supported. We show that the asymptotic distribution is the supremun of a centered Gaussian process which is Gaussian under some additional conditions on the probability $Q$ and on the cost. Such results imply the central limit theorem for the $p$-Wassertein distance, for $p\geq 1$. Finally, the semidiscrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials.