Central limit theorems for semi-discrete Wasserstein distances

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 sup 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 $p\ge 1$. This means that, for fixed N, the curse of dimensionality is avoided. To better understand the influence of such N, we provide bounds of $E|{\mathcal{W}}_p^p(P,Q_m)-{\mathcal{W}}_p^p(P,Q)|$ depending on m and N. Finally, the semi-discrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials and Laguerre cells. The results are supported by simulations that help to visualize the given limits and bounds. We analyse also the cases where classical bootstrap works.