Approximation rates of entropic maps in semidiscrete optimal transport

Entropic optimal transport offers a computationally tractable approximation to the classical problem. We study the approximation rate of the entropic optimal transport map (in approaching the Brenier map) when the regularization parameter ε tends to zero in the semidiscrete setting, where the input measure is absolutely continuous while the output is finitely discrete. Previous work shows that the approximation rate is O(ε) under the L2-norm with respect to the input measure. In this work, we establish faster, O(ε2) rates up to polylogarithmic factors, under the dual Lipschitz norm, which is weaker than the L2-norm. For the said dual norm, the O(ε2) rate is sharp. As a corollary, we derive a central limit theorem for the entropic estimator for the Brenier map in the dual Lipschitz space when the regularization parameter tends to zero as the sample size increases.