Sub-Riemannian geometry in probabilistic geometric statistics

Abstract: Geometric statistics, the statistical analysis of manifold and Lie group valued data, can be approached from a probabilistic viewpoint where families of parametric probability distributions are fitted to data.

This likelihood-based approach gives one way to generalize Euclidean statistical procedures to the non-linear manifold context. Stochastic processes here play an important role in providing geometrically natural ways of defining probability distributions. In the talk, I will discuss such constructions and how they lead to new geometric evolution equations for the most probable paths to observed data. In particular, we will see how such paths for an anisotropically scaled Brownian motion arise as geodesics of a sub-Riemannian metric on the frame bundle of the manifold.

differential geometryprobability

Audience: advanced learners

Young Researchers between Geometry and Stochastic Analysis 2021