Contributions to the Theory of Statistical Optimal Transport

by Thomas Staudt
Doctoral thesis
Date of Examination:2022-06-08
Date of issue:2022-07-01
Advisor:Prof. Dr. Axel Munk
Referee:Prof. Dr. Axel Munk
Referee:Prof. Dr. Dominic Schuhmacher
Persistent Address: http://dx.doi.org/10.53846/goediss-9327

 

 

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Abstract

English

The application of optimal transport based methodologies for statistical purposes has experienced a surge of interest and activity in recent years. This doctoral thesis collects the work of four research articles on the topic of statistical optimal transport, each of which features a distinct contribution to the field. The first article, on the lower complexity adaptation of empirical optimal transport, explores that statistical properties of the empirical optimal transport cost between different probability measures are often governed by the simpler of the two measures, not the more complex one. The second article establishes a unified approach to central limit theorems in the context of empirical optimal transport. The third article focuses on the uniqueness of dual solutions of the general optimal transport problem and derives novel criteria that greatly enhance the scope of settings where uniqueness is understood to hold. Uniqueness of the dual solutions is crucial for Gaussian limits in the corresponding central limit theorems. Finally, the fourth article studies the application of optimal transport for the purpose of measuring the dependency between random variables, establishing the concept of transport dependency.
Keywords: Wasserstein distance; Lower complexity adaptation; Transport dependency; Kantorovich potentials; Curse of dimensionality; Empirical optimal transport
 
Statistik