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Entropic Optimal Transport in Machine Learning: applications to distributional regression, barycentric estimation and probability matching

Luise, Giulia; (2021) Entropic Optimal Transport in Machine Learning: applications to distributional regression, barycentric estimation and probability matching. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

Regularised optimal transport theory has been gaining increasing interest in machine learning as a versatile tool to handle and compare probability measures. Entropy-based regularisations, known as Sinkhorn divergences, have proved successful in a wide range of applications: as a metric for clustering and barycenters estimation, as a tool to transfer information in domain adaptation, and as a fitting loss for generative models, to name a few. Given this success, it is crucial to investigate the statistical and optimization properties of such models. These aspects are instrumental to design new and principled paradigms that contribute to further advance the field. Nonetheless, questions on asymptotic guarantees of the estimators based on Entropic Optimal Transport have received less attention. In this thesis we target such questions, focusing on three major settings where Entropic Optimal Transport has been used: learning histograms in supervised frameworks, barycenter estimation and probability matching. We present the first consistent estimator for learning with Sinkhorn loss in supervised settings, with explicit excess risk bounds. We propose a novel algorithm for Sinkhorn barycenters that handles arbitrary probability distributions with provable global convergence guarantees. Finally, we address generative models with Sinkhorn divergence as loss function: we analyse the role of the latent distribution and the generator from a modelling and statistical perspective. We propose a method that learns the latent distribution and the generator jointly and we characterize the generalization properties of such estimator. Overall, the tools developed in this work contribute to the understanding of the theoretical properties of Entropic Optimal Transport and their versatility in machine learning.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Entropic Optimal Transport in Machine Learning: applications to distributional regression, barycentric estimation and probability matching
Event: UCL (University College London)
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Copyright © The Author 2021. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10120291
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