Research Training Group 2088

Project A1: Wasserstein Metrics in Statistics: Inference

The Wasserstein and related transport distances play a prominent role in probability theory, analysis and related areas. More recently, they have been recognized as a powerful tool in statistics at its interface with various applications. They excel where conventional methods fail, in particular when variability is large in magnitude and irregular in nature. However, due to the lack of appropriate distributional results, rigorous statistical inference with Wasserstein distances remains mainly limited to measures on the real line.

This project aims to facilitate reliable statistical inference for broader classes of measures by providing distributional limits, deviation and concentration results. Methods recently developed by our group for the Gaussian case will be extended to elliptical distributions and as a long term goal to probability measures on graphs and networks. Theoretical work will be complemented by extensive simulations in close collaboration with project A2. Particular attention will be paid to the case of measures representing two- and three-dimensional images as the most important application in collaboration with groups from cell biology.

A long-term goal of this project is to investigate possible extensions to barycenters in the Wasserstein space and to related transport distances such as the Gromov-Wasserstein distance for structures involving measures and metrics (e.g. 3-D shapes).

Methods: Wasserstein distances, limit theorems, delta-method, bootstrap
Applications: statistical tests, image analysis, biochemical substrate transfer

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