Minimax estimation of smooth optimal transport maps

By Jan-Christian Hütter, Philippe Rigollet. 2021

135 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7423bb9b3bbc06421640a345ef509c71cbe9e9ab

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Tudor Manole
2. Larry A. Wasserman
3. Jonathan Niles-Weed
4. Sivaraman Balakrishnan

In Mathematical Physics

1. Larry A. Wasserman
2. Sivaraman Balakrishnan
3. Tudor Manole
4. Jonathan Niles-Weed

In Statistics and Probability

1. Sivaraman Balakrishnan
2. Larry A. Wasserman
3. Tudor Manole
4. Jonathan Niles-Weed
5. Mathias Drton
6. Tommaso Cai
7. E.J. Candes
8. Alessio Farcomeni

Citations in the corpus

• Sharp convergence rates for empirical optimal transport with smooth costs (2021, background): “Under similar conditions on µ and ν , it is also possible to construct appropriate smooth estimators of the optimal map between µ and ν (Hütter and Rigollet, 2021).”
• Plugin estimation of smooth optimal transport maps (2021, background): “The lower bound (3) is reminiscent of, but generally faster than, the classical n−2α/(2α+d) minimax rate of estimating an α-Hölder nonparametric regression function (Tsybakov, 2008), and is shown by Hütter and Rigollet (2021) to be achievable up to a polylogarithmic factor.”