Statistical Optimal Transport and its Entropic Regularization: Compared and Contrasted

by Shayan Max Ahmad Hundrieser
Cumulative thesis
Date of Examination:2024-02-07
Date of issue:2024-04-26
Advisor:Prof. Dr. Axel Munk
Referee:Prof. Dr. Axel Munk
Referee:Prof. Dr. Stephan F. Huckemann
Persistent Address: http://dx.doi.org/10.53846/goediss-10471

 

 

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Abstract

English

In recent years, statistical methodology based on optimal transport witnessed a considerable increase in practical and theoretical interest. A central reason for this trend is the ability of optimal transport to compare data in a manner that is consistent with the geometry of the domain. This development was further amplified by computational advances spurred by the introduction of entropy regularized optimal transport. The presented doctoral thesis delves into statistical aspects of empirical optimal transport and its entropy regularized surrogate. It focuses on the statistical performance and uncertainty of empirical plug-in estimators based on sampling from unknown distributions, and collects four research articles which contribute to these topics. The first article analyzes the performance of empirical optimal transport under different population measures and uncovers that for increasing sample size the convergence rate is governed by the less complex measure, a phenomenon termed lower complexity adaptation. The second article provides a unifying approach to distributional limits for the empirical optimal transport cost. The third article extends upon this and establishes distributional limits for empirical optimal transport when additionally the cost function is estimated. Notably, the second and third article both align with the principle of lower complexity adaptation and also showcase that similar limits do not hold in high-dimensional settings. Finally, the fourth article derives distributional limits for empirical entropy regularized surrogates and confirms their validity in arbitrary high-dimensional settings. Altogether, this doctoral thesis highlights key similarities and differences between empirical optimal transport and its entropic regularization, offering comprehensive insights into strengths and limitations of transport-based methodologies in statistical contexts.
Keywords: Wasserstein distance; Sinkhorn divergence; Empirical measures; Lower complexity adaptation; Curse of dimensionality; Central limit theorem
 
Statistik