WEAK LIMITS OF ENTROPY-REGULARIZED OPTIMAL TRANSPORT
JONATHAN-NILES WEED – NEW YORK UNIVERSITY
We study the asymptotic behavior of empirical estimators of entropy-regularized optimal transport couplings between compact probability measures. These couplings were first proposed in a thought experiment of Schrödinger as a model for diffusing particles observed at different times. Harchaoui, Liu, and Pal (2020) conjectured that empirical estimators of these couplings satisfy a functional CLT. We prove this conjecture and show that dual solutions to the entropy-regularized problem also enjoy a CLT in a suitable Hölder space. These CLTs also allow us to propose asymptotically valid goodness-of-fit tests based on the Sinkhorn divergence, a popular measure in machine learning.