Strong equivalence between metrics of Wasserstein type

Abstract

The sliced Wasserstein metric W̶p and more recently max-sliced Wasserstein metric 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 Wp. Recently, Paty and Cuturi have proved in [14] the strong equivalence of W‾2 and W2. We show that the strong equivalence also holds for p=1, while the sliced Wasserstein metric does not share this nice property.

Topics & Concepts