Open Access
2021 Strong equivalence between metrics of Wasserstein type
Erhan Bayraktar, Gaoyue Guo
Author Affiliations +
Electron. Commun. Probab. 26: 1-13 (2021). DOI: 10.1214/21-ECP383


The sliced Wasserstein metric p and more recently max-sliced Wasserstein metric Wp 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 W2 and W2. We show that the strong equivalence also holds for p=1, while the sliced Wasserstein metric does not share this nice property.

Funding Statement

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.


Download Citation

Erhan Bayraktar. Gaoyue Guo. "Strong equivalence between metrics of Wasserstein type." Electron. Commun. Probab. 26 1 - 13, 2021.


Received: 17 December 2019; Accepted: 19 February 2021; Published: 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1214/21-ECP383

Primary: 90C25

Keywords: Duality , max-sliced Wasserstein metric , Optimal transport , Wasserstein metric

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