[Paper Review] Minimax rates of estimation for smooth optimal transport maps

Jan-Christian Hütter, Philippe Rigollet|arXiv (Cornell University)|May 14, 2019

Statistical Methods and Inference11 citations

TL;DR

This paper establishes the first minimax estimation rates for smooth optimal transport maps in general dimension by proposing a wavelet-based estimator that minimizes an empirical semi-dual optimal transport problem. The method achieves near minimax optimality through novel stability arguments and a complementary lower bound, validated by synthetic experiments demonstrating the benefits of smoothness regularization.

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.

Motivation & Objective

• To establish minimax estimation rates for optimal transport maps between probability distributions in general dimension under smoothness assumptions.
• To develop a practical and theoretically grounded estimator for transport maps from finite samples.
• To bridge the gap in theoretical understanding of estimation error for smooth transport maps in high-dimensional settings.

Proposed method

• Propose an estimator based on minimizing an empirical version of the semi-dual optimal transport problem.
• Restrict the search space to truncated wavelet expansions to enforce smoothness and enable computation.
• Establish new stability bounds for the semi-dual formulation under perturbations of the underlying measures.
• Derive a minimax lower bound to characterize the fundamental limits of estimation accuracy.
• Combine stability results with approximation theory to show near minimax optimality of the proposed estimator.
• Validate the theoretical findings with numerical experiments on synthetic data, demonstrating the impact of smoothness regularization.

Experimental results

Research questions

• RQ1 What are the minimax estimation rates for smooth optimal transport maps in general dimension?
• RQ2 Can a wavelet-based estimator achieve near minimax optimality in the estimation of smooth transport maps?
• RQ3 How do stability properties of the semi-dual optimal transport problem contribute to estimation error bounds?
• RQ4 What is the fundamental limit (lower bound) on estimation accuracy for smooth transport maps?
• RQ5 How does smoothness regularization affect the performance of transport map estimators in finite-sample regimes?

Key findings

• The proposed wavelet-based estimator achieves near minimax optimality for smooth optimal transport maps in $ dd$.
• New stability bounds for the semi-dual optimal transport problem are derived and used to control estimation error.
• A complementary minimax lower bound is established, showing that the estimator's rate is optimal up to logarithmic factors.
• Numerical experiments confirm the theoretical findings and highlight the practical advantages of incorporating smoothness regularization.
• This work provides the first minimax estimation rates for smooth transport maps in general dimension, filling a critical theoretical gap.