Open Access
2024 Limit theorems for entropic optimal transport maps and Sinkhorn divergence
Ziv Goldfeld, Kengo Kato, Gabriel Rioux, Ritwik Sadhu
Author Affiliations +
Electron. J. Statist. 18(1): 980-1041 (2024). DOI: 10.1214/24-EJS2217


We study limit theorems for entropic optimal transport (EOT) maps, dual potentials, and the Sinkhorn divergence. The key technical tool we use is a first and second-order Hadamard differentiability analysis of EOT potentials with respect to the marginals, which may be of independent interest. Given the differentiability results, the functional delta method is used to obtain central limit theorems for empirical EOT potentials and maps. The second-order functional delta method is leveraged to establish the limit distribution of the empirical Sinkhorn divergence under the null. Building on the latter result, we further derive the null limit distribution of the Sinkhorn independence test statistic and characterize the correct order. Since our limit theorems follow from Hadamard differentiability of the relevant maps, as a byproduct, we also obtain bootstrap consistency and asymptotic efficiency of the empirical EOT map, potentials, and Sinkhorn divergence.

Funding Statement

Ziv Goldfeld is supported by NSF grants CCF-1947801, CCF-2046018, and DMS-2210368, and the 2020 IBM Academic Award. Kengo Kato is supported by NSF grants DMS-1952306, DMS-2014636, and DMS-2210368. Gabriel Rioux is supported by the NSERC postgraduate fellowship PGSD-567921-2022.


The authors would like to thank Editor Grace Y. Yi, an associate editor, and two anonymous referees for their constructive comments that help improve the quality of the manuscript.


Download Citation

Ziv Goldfeld. Kengo Kato. Gabriel Rioux. Ritwik Sadhu. "Limit theorems for entropic optimal transport maps and Sinkhorn divergence." Electron. J. Statist. 18 (1) 980 - 1041, 2024.


Received: 1 July 2022; Published: 2024
First available in Project Euclid: 4 March 2024

Digital Object Identifier: 10.1214/24-EJS2217

Primary: 60F05 , 62G20
Secondary: 60F17

Keywords: Entropic map , Entropic optimal transport , Functional Delta Method , Hadamard differentiability , Sinkhorn divergence

Vol.18 • No. 1 • 2024
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