Kolmogorov-Smirnov distance and discrepancies versus Wasserstein distances

TL;DR

This paper derives inequalities comparing Wasserstein distances with discrepancies like the Kolmogorov-Smirnov distance, providing bounds and applications in probability measure comparisons.

Contribution

It establishes sharp bounds relating p-Wasserstein distances to discrepancies such as uniform discrepancy and KS distance, extending to measures supported on different spaces.

Findings

Upper bounds of Wasserstein distances via discrepancies for measures on [0,1]^d.

Retrieval of the Proinov Theorem as an application.

Reverse inequalities involving densities and L^s integrability.

Abstract

We establish inequalities that compare the p-Wasserstein distance to distances which are built as suprema of box measures. More precisely, when the measures are supported on $[0,1]^d$, we obtain sharp upper-bounds of the $p$-Wasserstein distance by (powers of) the (uniform) discrepancy. As an application, we retrieve the Pro\''inov Theorem. When the two distributions are supported {by the whole} $R^d$, {their} $p$-Wasserstein distance is upper bounded by the product of a (power of) their Kolmogorov-Smirnov (KS) distance with the sum of their $p$-moments. Reverse inequalities are established when one of the two distributions has a density, depending on its ${\cal L}^s$-integrability with respect to the Lebesgue measure for some $s>1$.