2024/11/12 by Sadhu, Ritwik, Goldfeld, Ziv, Kato, Kengo
#49N15 #49Q22 #60F05 #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)
paper · doi:10.48550/arxiv.2411.07947
Entropic optimal transport offers a computationally tractable approximation to the classical problem. In this note, 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.