Center for Applied Mathematics Colloquium
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 underlying distributions. 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. This talk is based on joint work with Ziv Goldfeld, Gabriel Rioux, and Ritwik Sadhu.