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Near-optimal estimation of smooth transport maps with kernel sums-of-squares

Muzellec, Boris; Vacher, Jonathan; Bach, Francis; Vialard, François-Xavier; Rudi, Alessandro (2021), Near-optimal estimation of smooth transport maps with kernel sums-of-squares. https://basepub.dauphine.psl.eu/handle/123456789/22823

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ArXiv__Convergence_of_potentials_(8).pdf (613.3Kb)
Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-03466696
Date
2021
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
28
Metadata
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Author(s)
Muzellec, Boris
Statistical Machine Learning and Parsimony [SIERRA]
Vacher, Jonathan
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Bach, Francis
Statistical Machine Learning and Parsimony [SIERRA]
Vialard, François-Xavier
Laboratoire d'Informatique Gaspard-Monge [LIGM]
Rudi, Alessandro cc
Statistical Machine Learning and Parsimony [SIERRA]
Abstract (EN)
It was recently shown that under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. However, rather than the distance itself, the object of interest for applications such as generative modeling is the underlying optimal transport map. Hence, computational and statistical guarantees need to be obtained for the estimated maps themselves. In this paper, we propose the first tractable algorithm for which the statistical L2 error on the maps nearly matches the existing minimax lower-bounds for smooth map estimation. Our method is based on solving the semi-dual formulation of optimal transport with an infinite-dimensional sum-of-squares reformulation, and leads to an algorithm which has dimension-free polynomial rates in the number of samples, with potentially exponentially dimension-dependent constants.

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