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oai:dspace.mit.edu:1721.1/1527752023-11-03T03:33:55Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022

Rigollet, Philippe Stromme, Austin J. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science 2023-11-02T20:15:21Z 2023-11-02T20:15:21Z 2023-09 2023-09-21T14:26:18.412Z https://hdl.handle.net/1721.1/152775 Optimal transport (OT) is a flexible framework for contrasting and interpolating probability measures which has recently been applied throughout science, including in machine learning, statistics, graphics, economics, biology, and more. In this thesis, we study several statistical problems at the forefront of applied optimal transport, prioritizing statistically and computationally practical results. We begin by considering one of the most popular applications of OT in practice, the barycenter problem, providing dimension-free rates of statistical estimation. In the Gaussian case, we analyze first-order methods for computing barycenters, and develop global, dimension-free rates of convergence despite the non-convexity of the problem. Extending beyond the Gaussian case, however, is challenging due to the fundamental curse of dimensionality for OT, which motivates the study of a regularized, and in fact more computationally feasible, form of optimal transport, dubbed entropic optimal transport (entropic OT). Recent work has suggested that entropic OT may escape the curse of dimensionality of un-regularized OT, and in this thesis we develop a refined theory of the statistical behavior of entropic OT by showing that entropic OT does attain truly dimension-free rates of convergence in the large regularization regime, as well as automatically adapts to the intrinsic dimension of the data in the small regularization regime. We also consider the rate of approximation of entropic OT in the semi-discrete case, and complement these results by considering the problem of trajectory reconstruction, proposing two practical methods based off both un-regularized and entropic OT. Ph.D. Massachusetts Institute of Technology In Copyright - Educational Use Permitted Copyright retained by author(s) https://rightsstatements.org/page/InC-EDU/1.0/ Statistical aspects of optimal transport Thesis application/pdf Doctoral Doctor of Philosophy Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="8659a3bf-43d2-4ca1-b8d7-8c8c142fb37d"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Title>Statistical aspects of optimal transport</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2023-09</PublicationDate> <Authors> <Author> <DisplayName>Stromme, Austin J.</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>https://rightsstatements.org/page/InC-EDU/1.0/</License> <Abstract>Optimal transport (OT) is a flexible framework for contrasting and interpolating probability measures which has recently been applied throughout science, including in machine learning, statistics, graphics, economics, biology, and more. In this thesis, we study several statistical problems at the forefront of applied optimal transport, prioritizing statistically and computationally practical results. We begin by considering one of the most popular applications of OT in practice, the barycenter problem, providing dimension-free rates of statistical estimation. In the Gaussian case, we analyze first-order methods for computing barycenters, and develop global, dimension-free rates of convergence despite the non-convexity of the problem. Extending beyond the Gaussian case, however, is challenging due to the fundamental curse of dimensionality for OT, which motivates the study of a regularized, and in fact more computationally feasible, form of optimal transport, dubbed entropic optimal transport (entropic OT). Recent work has suggested that entropic OT may escape the curse of dimensionality of un-regularized OT, and in this thesis we develop a refined theory of the statistical behavior of entropic OT by showing that entropic OT does attain truly dimension-free rates of convergence in the large regularization regime, as well as automatically adapts to the intrinsic dimension of the data in the small regularization regime. We also consider the rate of approximation of entropic OT in the semi-discrete case, and complement these results by considering the problem of trajectory reconstruction, proposing two practical methods based off both un-regularized and entropic OT.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>