Central Limit Theorems for Semidiscrete Wasserstein Distances

TL;DR

This paper establishes a central limit theorem for empirical optimal transport costs in the semi-discrete setting, showing that the asymptotic distribution is Gaussian under certain conditions, and provides bounds and simulations to illustrate these results.

Contribution

It introduces the first CLT for the semi-discrete optimal transport cost and potentials, avoiding the curse of dimensionality for fixed N.

Findings

Asymptotic distribution is the supremum of a Gaussian process.

Bounds on the expected Wasserstein distance depending on sample size and N.

Simulations support the theoretical limits and bounds.

Abstract

We prove a Central Limit Theorem for the empirical optimal transport cost, $\sqrt{\frac{nm}{n+m}}\{\mathcal{T}_c(P_n,Q_m)-\mathcal{T}_c(P,Q)\}$, in the semi discrete case, i.e when the distribution $P$ is supported in $N$ points, but without assumptions on $Q$. 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 $p\geq 1$. This means that, for fixed $N$, the curse of dimensionality is avoided. To better understand the influence of such $N$, we provide bounds of $E|\mathcal{W}_1(P,Q_m)-\mathcal{W}_1(P,Q)|$ depending on $m$ and $N$. 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. The results are supported by simulations that help to visualize the given limits and bounds. We analyse also the cases where classical bootstrap works.