A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation

Adrien Vacher, Boris Muzellec, Alessandro Rudi, Francis Bach, Francois-Xavier Vialard
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:4143-4173, 2021.

Abstract

It is well-known that plug-in statistical estimation of optimal transport suffers from the curse of dimensionality. Despite recent efforts to improve the rate of estimation with the smoothness of the problem, the computational complexity of these recently proposed methods still degrade exponentially with the dimension. In this paper, thanks to an infinite-dimensional sum-of-squares representation, we derive a statistical estimator of smooth optimal transport which achieves a precision $\varepsilon$ from $\tilde{O}(\varepsilon^{-2})$ independent and identically distributed samples from the distributions, for a computational cost of $\tilde{O}(\varepsilon^{-4})$ when the smoothness increases, hence yielding dimension-free statistical \emph{and} computational rates, with potentially exponentially dimension-dependent constants.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-vacher21a, title = {A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation}, author = {Vacher, Adrien and Muzellec, Boris and Rudi, Alessandro and Bach, Francis and Vialard, Francois-Xavier}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {4143--4173}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/vacher21a/vacher21a.pdf}, url = {https://proceedings.mlr.press/v134/vacher21a.html}, abstract = {It is well-known that plug-in statistical estimation of optimal transport suffers from the curse of dimensionality. Despite recent efforts to improve the rate of estimation with the smoothness of the problem, the computational complexity of these recently proposed methods still degrade exponentially with the dimension. In this paper, thanks to an infinite-dimensional sum-of-squares representation, we derive a statistical estimator of smooth optimal transport which achieves a precision $\varepsilon$ from $\tilde{O}(\varepsilon^{-2})$ independent and identically distributed samples from the distributions, for a computational cost of $\tilde{O}(\varepsilon^{-4})$ when the smoothness increases, hence yielding dimension-free statistical \emph{and} computational rates, with potentially exponentially dimension-dependent constants.} }
Endnote
%0 Conference Paper %T A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation %A Adrien Vacher %A Boris Muzellec %A Alessandro Rudi %A Francis Bach %A Francois-Xavier Vialard %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-vacher21a %I PMLR %P 4143--4173 %U https://proceedings.mlr.press/v134/vacher21a.html %V 134 %X It is well-known that plug-in statistical estimation of optimal transport suffers from the curse of dimensionality. Despite recent efforts to improve the rate of estimation with the smoothness of the problem, the computational complexity of these recently proposed methods still degrade exponentially with the dimension. In this paper, thanks to an infinite-dimensional sum-of-squares representation, we derive a statistical estimator of smooth optimal transport which achieves a precision $\varepsilon$ from $\tilde{O}(\varepsilon^{-2})$ independent and identically distributed samples from the distributions, for a computational cost of $\tilde{O}(\varepsilon^{-4})$ when the smoothness increases, hence yielding dimension-free statistical \emph{and} computational rates, with potentially exponentially dimension-dependent constants.
APA
Vacher, A., Muzellec, B., Rudi, A., Bach, F. & Vialard, F.. (2021). A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:4143-4173 Available from https://proceedings.mlr.press/v134/vacher21a.html.

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