# Statistical data analysis in the Wasserstein space

## TL;DR

This paper reviews recent advances in statistical inference for data modeled as probability measures, emphasizing Wasserstein distances and optimal transport tools like barycenters and geodesic PCA for analyzing geometric variations.

## Contribution

It highlights the benefits of Wasserstein-based methods such as barycenter and geodesic PCA in understanding geometric data variations and discusses emerging research directions in statistical optimal transport.

## Key findings

- Wasserstein distances effectively analyze probability measure data.
- Barycenter and geodesic PCA reveal principal geometric variations.
- The paper discusses future research perspectives in statistical optimal transport.

## Abstract

This paper is concerned by statistical inference problems from a data set whose elements may be modeled as random probability measures such as multiple histograms or point clouds. We propose to review recent contributions in statistics on the use of Wasserstein distances and tools from optimal transport to analyse such data. In particular, we highlight the benefits of using the notions of barycenter and geodesic PCA in the Wasserstein space for the purpose of learning the principal modes of geometric variation in a dataset. In this setting, we discuss existing works and we present some research perspectives related to the emerging field of statistical optimal transport.