Uniform Confidence Band for Optimal Transport Map on One-Dimensional Data

TL;DR

This paper introduces a statistical inference method for optimal transport maps on real numbers, providing uniform confidence bands and bootstrap validation to enhance visualization and interpretation of distribution distances.

Contribution

It derives a limit distribution for the estimation error of OT maps and develops a bootstrap method with kernel smoothing for confidence band construction.

Findings

Derived a limit distribution for the uniform norm of OT map estimation error.

Developed a bootstrap method with kernel smoothing for confidence bands.

Validated asymptotic coverage probability of the confidence bands.

Abstract

We develop a statistical inference method for an optimal transport map between distributions on real numbers with uniform confidence bands. The concept of optimal transport (OT) is used to measure distances between distributions, and OT maps are used to construct the distance. OT has been applied in many fields in recent years, and its statistical properties have attracted much interest. In particular, since the OT map is a function, a uniform norm-based statistical inference is significant for visualization and interpretation. In this study, we derive a limit distribution of a uniform norm of an estimation error for the OT map, and then develop a uniform confidence band based on it. In addition to our limit theorem, we develop a bootstrap method with kernel smoothing, then also derive its validation and guarantee on an asymptotic coverage probability of the confidence band. Our proof is based on the functional delta method and the representation of OT maps on the reals.