Limit Laws for Empirical Optimal Transport

by Marcel Klatt
Cumulative thesis
Date of Examination:2022-02-09
Date of issue:2022-02-17
Advisor:Prof. Dr. Axel Munk
Referee:Prof. Dr. Axel Munk
Referee:Prof. Dr. Dominic Schuhmacher
Persistent Address: http://dx.doi.org/10.53846/goediss-458

 

 

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Abstract

English

Optimal Transport (OT) has recently gained increasing attention in various fields ranging from biology to machine learning and mathematics. Especially OT based dissimilarity measures can be designed to respect the underlying ground metric structure and hence provide a meaningful notion to compare general probability distributions. In fact, this often results in visually appealing and well interpretable outcomes which explains the advancement of OT based data analysis. The central theme of this thesis revolves around statistical aspects of OT and its entropy regularized surrogates, and aims to contribute to a better understanding of empirical OT and their related empirical entropy regularized surrogates. Of special interest are distributional limit laws for empirical plug-in (regularized) OT estimators. Distributional limit laws capture the asymptotic fluctuation of empirical estimators around their population quantities after proper standardization. Indeed, these asymptotic statements are essential for statistical data analysis, e.g., for hypothesis testing and asymptotic confidence bands. The presented articles in this thesis are broadly characterized in two categories: Distributional limit laws for empirical (regularized) OT and probability measures supported (1) on finite spaces and (2) on more general spaces. In particular, the main theoretical results demonstrate the importance of two fundamental principles that seamlessly integrate within a statistical context: Duality for (in)finite dimensional linear programs and weak convergence of underlying empirical processes.
Keywords: Optimal Transport, Wasserstein Distance, Distributional Limit Laws, Colocalization
 
Statistik