Empirical Optimal Transport on Discrete Spaces: Limit Theorems, Distributional Bounds and Applications

Abstract

Optimal Transport and especially distances based on optimal transport are a widely applied tool in different mathematical disciplines. Among others it is used in probability theory to study for example limit laws or derive concentration inequalities. Furthermore, the distance based on optimal transport (Wasserstein distance) is used by practitioners in different areas to analyse complex data sets. Examples include machine learning, finance and imaging. This thesis considers two different topics related to optimal transport. In the first part we provide distributional limits for the Wasserstein distance on countable spaces in the one- and the two-sample case in both settings of equality of the underlying measures and different underlying measures. All these distributional limits are given implicitly via an optimization problem. In case of the underlying space being a tree we are able to calculate the limit distribution explicitly. Additionally, based on the explicit results for trees we derive a distributional upper bound for the limit distribution on general countable spaces. In the second part we propose a method based on optimal transport to measure the spatial proximity of different proteins in images generated by the super-resolution technique STED. The quantification of the spatial proximity of different proteins is an essential question in biology to investigate protein networks.