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.