Minimax estimation of discontinuous optimal transport maps: The semi-discrete case

Aram-Alexandre Pooladian, Vincent Divol, Jonathan Niles-Weed
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:28128-28150, 2023.

Abstract

We consider the problem of estimating the optimal transport map between two probability distributions, $P$ and $Q$ in $\mathbb{R}^d$, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution $Q$ is a discrete measure supported on a finite number of points in $\mathbb{R}^d$. We study a computationally efficient estimator initially proposed by (Pooladian & Niles-Weed, 2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate $n^{-1/2}$, independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-pooladian23b, title = {Minimax estimation of discontinuous optimal transport maps: The semi-discrete case}, author = {Pooladian, Aram-Alexandre and Divol, Vincent and Niles-Weed, Jonathan}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {28128--28150}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/pooladian23b/pooladian23b.pdf}, url = {https://proceedings.mlr.press/v202/pooladian23b.html}, abstract = {We consider the problem of estimating the optimal transport map between two probability distributions, $P$ and $Q$ in $\mathbb{R}^d$, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution $Q$ is a discrete measure supported on a finite number of points in $\mathbb{R}^d$. We study a computationally efficient estimator initially proposed by (Pooladian & Niles-Weed, 2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate $n^{-1/2}$, independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.} }
Endnote
%0 Conference Paper %T Minimax estimation of discontinuous optimal transport maps: The semi-discrete case %A Aram-Alexandre Pooladian %A Vincent Divol %A Jonathan Niles-Weed %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-pooladian23b %I PMLR %P 28128--28150 %U https://proceedings.mlr.press/v202/pooladian23b.html %V 202 %X We consider the problem of estimating the optimal transport map between two probability distributions, $P$ and $Q$ in $\mathbb{R}^d$, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution $Q$ is a discrete measure supported on a finite number of points in $\mathbb{R}^d$. We study a computationally efficient estimator initially proposed by (Pooladian & Niles-Weed, 2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate $n^{-1/2}$, independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.
APA
Pooladian, A., Divol, V. & Niles-Weed, J.. (2023). Minimax estimation of discontinuous optimal transport maps: The semi-discrete case. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:28128-28150 Available from https://proceedings.mlr.press/v202/pooladian23b.html.

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