Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space

Quentin Mérigot, Alex Delalande, Frederic Chazal
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:3186-3196, 2020.

Abstract

This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space and is shown to be bi-Hölder continuous. It enables the direct use of generic supervised and unsupervised learning algorithms on measure data consistently w.r.t. the Wasserstein geometry.

Cite this Paper


BibTeX
@InProceedings{pmlr-v108-merigot20a, title = {Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space}, author = {M\'erigot, Quentin and Delalande, Alex and Chazal, Frederic}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {3186--3196}, year = {2020}, editor = {Silvia Chiappa and Roberto Calandra}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/merigot20a/merigot20a.pdf}, url = { http://proceedings.mlr.press/v108/merigot20a.html }, abstract = {This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space and is shown to be bi-Hölder continuous. It enables the direct use of generic supervised and unsupervised learning algorithms on measure data consistently w.r.t. the Wasserstein geometry.} }
Endnote
%0 Conference Paper %T Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space %A Quentin Mérigot %A Alex Delalande %A Frederic Chazal %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-merigot20a %I PMLR %P 3186--3196 %U http://proceedings.mlr.press/v108/merigot20a.html %V 108 %X This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space and is shown to be bi-Hölder continuous. It enables the direct use of generic supervised and unsupervised learning algorithms on measure data consistently w.r.t. the Wasserstein geometry.
APA
Mérigot, Q., Delalande, A. & Chazal, F.. (2020). Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:3186-3196 Available from http://proceedings.mlr.press/v108/merigot20a.html .

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