## Project A5: Topological Data Analysis from a Statistical Perspective

For the analysis of complex data that take values in a (non-linear) metric space, the Fréchet mean is among the most fundamental tools. A major effort in this RTG is the investigation of such data based on the Wasserstein metric. In this project, we will combine Fréchet means with Wasserstein spaces, resulting in what is known as the Wasserstein barycenter. It has been shown in simple examples that the Wasserstein barycenter provides promising results since it preserves important geometric features of the data in the resulting barycenter. This is in contrast to summary statistics, which do not take into account the geometry, e.g., coordinate based averages. Examples for the advantage of investigating barycenter-based data analysis have been given in different areas, such as biodemographics, texture analysis, deformation analysis, and imaging.

For the analysis of complex data that take values in a (non-linear) metric space, the Fréchet mean is among the most fundamental tools. A major effort in this RTG is the investigation of such data based on the Wasserstein metric. In this project, we will combine Fréchet means with Wasserstein spaces, resulting in what is known as the Wasserstein barycenter. It has been shown in simple examples that the Wasserstein barycenter provides promising results since it preserves important geometric features of the data in the resulting barycenter. This is in contrast to summary statistics, which do not take into account the geometry, e.g., coordinate based averages. Examples for the advantage of investigating barycenter-based data analysis have been given in different areas, such as biodemographics, texture analysis, deformation analysis, and imaging.

This project emerged from previous work in this project on topological data analysis, where topologically meaningful data descriptors have been investigated, such as persistent homology or modes.

**Methods:**barycenter, Frechet mean, Wasserstein distance, asymptotic analysis, nonlinear deformation

**Applications:**topological data analysis, mathematical imaging, cell biology