Semidiscrete Optimal Transport Maps: Stability, Limit Theorems, and Asymptotic Efficiency

We study statistical inference for the optimal transport (OT) map (also known as the Brenier map) from a known absolutely continuous reference distribution onto an unknown finitely discrete target distribution. We derive limit distributions for the integral and linear functionals of the empirical OT map, together with their moment convergence. The former has a non-Gaussian limit, whose explicit density is derived, while the latter attains asymptotic normality. For both cases, we also establish consistency of the nonparametric bootstrap. The derivation of our limit theorems relies on new stability estimates of functionals of the OT map with respect to the dual potential vector, which may be of independent interest. We also discuss applications of our limit theorems to the construction of confidence sets for the OT map and inference for a maximum tail correlation. Finally, we discuss asymptotic efficiency of the empirical OT map in an infinite dimensional setting.

About Kengo Kato

Most recently a member of the Faculty of Economics at The University of Tokyo, Kato’s research fields are mathematical statistics, econometrics, and economic statistics with a focus on high dimensional statistical and econometric models. He was a visiting scholar in the Department of Economics at MIT and is the associate editor of the Japanese Economic Review and the Journal of Statistical Planning and Inference.