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

Adrien Vacher 1, 2, 3 Boris Muzellec 4, 5 Alessandro Rudi 4, 5 Francis Bach 4, 5 François-Xavier Vialard 1, 2 
3 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
CEREMADE - CEntre de REcherches en MAthématiques de la DEcision, Inria de Paris
4 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique - ENS Paris, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
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 infinitedimensional sum-of-squares representation, we derive a statistical estimator of smooth optimal transport which achieves a precision ε fromÕ(ε −2) independent and identically distributed samples from the distributions, for a computational cost ofÕ(ε −4) when the smoothness increases, hence yielding dimension-free statistical and computational rates, with potentially exponentially dimension-dependent constants.
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https://hal.archives-ouvertes.fr/hal-03454237
Contributor : Boris Muzellec Connect in order to contact the contributor
Submitted on : Monday, November 29, 2021 - 10:03:13 AM
Last modification on : Wednesday, June 8, 2022 - 12:50:06 PM

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  • HAL Id : hal-03454237, version 1
  • ARXIV : 2101.05380

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Adrien Vacher, Boris Muzellec, Alessandro Rudi, Francis Bach, François-Xavier Vialard. A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation. COLT 2021 - 34th Annual Conference on Learning Theory, Aug 2021, Boulder, United States. ⟨hal-03454237⟩

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