Title: Geometry and Statistics: New Developments in Statistics on Manifolds

Abstract: With the increasing prevalence of modern complex data in non-Euclidean (e.g., manifold) forms, there is a growing need for developing models and theory for  inference of  non-Euclidean data. This talk first presents some recent advances in nonparametric inference on manifolds and other non-Euclidean spaces.   The initial focus is on nonparametric inference based on Fréchet means.  In particular, we present omnibus central limit theorems for Fréchet means for inference, which can be applied to general metric spaces including  stratified spaces, greatly expanding the current scope of inference.  A robust framework based on the classical idea of median-of-means is then proposed which yields  estimates with provable robustness and improved concentration. In addition to inferring i.i.d data, we also consider nonparametric regression problems where predictors or responses lie on manifolds. Various simulated or real data examples are considered