Strong equivalence between metrics of Wasserstein type

Abstract

The sliced Wasserstein metric ${\mathcal{W̶}}_p$ and more recently max-sliced Wasserstein metric ${\bar{\mathcal{W}}}_p$ have attracted abundant attention in data sciences and machine learning due to their advantages to tackle the curse of dimensionality, see e.g. [15], [6]. A question of particular importance is the strong equivalence between these projected Wasserstein metrics and the (classical) Wasserstein metric ${\mathcal{W}}_p$. Recently, Paty and Cuturi have proved in [14] the strong equivalence of ${\bar{\mathcal{W}}}_2$ and ${\mathcal{W}}_2$. We show that the strong equivalence also holds for $p=1$, while the sliced Wasserstein metric does not share this nice property.