The sliced Wasserstein metric and more recently max-sliced Wasserstein metric have attracted abundant attention in data sciences and machine learning due to their advantages to tackle the curse of dimensionality, see e.g. , . A question of particular importance is the strong equivalence between these projected Wasserstein metrics and the (classical) Wasserstein metric . Recently, Paty and Cuturi have proved in  the strong equivalence of and . We show that the strong equivalence also holds for , while the sliced Wasserstein metric does not share this nice property.
G. Guo is grateful for the support of CentraleSupélec, and in addition, to the University of Michigan and AMS Simons Travel Grant.
We thank the anonymous referees for the thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the publication.
"Strong equivalence between metrics of Wasserstein type." Electron. Commun. Probab. 26 1 - 13, 2021. https://doi.org/10.1214/21-ECP383