Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives
This is an incremental review paper summarizing existing work on statistical inference for optimal transport maps.
This paper reviews recent advances in estimating optimal transport maps and developing limit theorems for them using sample data, with the goal of providing practitioners with reliable inferential tools.
In many applications of optimal transport (OT), the object of primary interest is the optimal transport map. This map rearranges mass from one probability distribution to another in the most efficient way possible by minimizing a specified cost. In this paper we review recent advances in estimating and developing limit theorems for the OT map, using samples from the underlying distributions. We also review parallel lines of work that establish similar results for special cases and variants of the basic OT setup. We conclude with a discussion of key directions for future research with the goal of providing practitioners with reliable inferential tools.