Abstract
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.
Acknowledgments
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.
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. https://doi.org/10.1214/24-EJS2217
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