# Minimax estimation of smooth optimal transport maps

## TL;DR

This paper establishes the first minimax estimation rates for smooth optimal transport maps in general dimensions, using wavelet-based estimators and stability analysis, supported by numerical experiments.

## Contribution

It introduces a new estimator based on semi-dual optimal transport with wavelet truncation, achieving near minimax optimality under smoothness assumptions.

## Key findings

- Estimator achieves near minimax optimality.
- Numerical experiments support theoretical results.
- First minimax rates for transport maps in general dimension.

## Abstract

Brenier's theorem is a cornerstone of optimal transport that guarantees the existence of an optimal transport map $T$ between two probability distributions $P$ and $Q$ over $\mathbb{R}^d$ under certain regularity conditions. The main goal of this work is to establish the minimax estimation rates for such a transport map from data sampled from $P$ and $Q$ under additional smoothness assumptions on $T$. To achieve this goal, we develop an estimator based on the minimization of an empirical version of the semi-dual optimal transport problem, restricted to truncated wavelet expansions. This estimator is shown to achieve near minimax optimality using new stability arguments for the semi-dual and a complementary minimax lower bound. Furthermore, we provide numerical experiments on synthetic data supporting our theoretical findings and highlighting the practical benefits of smoothness regularization. These are the first minimax estimation rates for transport maps in general dimension.