Mu Niu (University of Glasgow) – Data-Driven Riemannian Geometry for Statistical Modelling on Point Clouds

In this talk, I will present a data-driven framework for incorporating Riemannian geometry into statistical modelling, with a particular focus on Gaussian process (GP) regression. High-dimensional data encountered in practice—such as point clouds, remote sensing measurements, or image collections—often concentrate near lower-dimensional manifolds with non-Euclidean geometry. Standard Euclidean GPs ignore this structure, leading to poor predictive performance and misleading uncertainty quantification. Our approach constructs GPs on complex or unknown manifolds by first learning a probabilistic atlas of the latent geometry, using tools such as autoencoders and latent variable models, and then defining stochastic processes that respect this geometry. This perspective connects ideas from stochastic differential equations on manifolds with statistical learning, allowing principled modelling of manifold-valued data. I will illustrate the method through simulations on the torus and applications to remote sensing of chlorophyll concentration in the Aral Sea, and Image point clouds. The talk will give an overview of how data-driven Riemannian geometry can inform statistical modelling more broadly, including directions towards diffusion based generative modelling and finding the shortest path on point cloud, while highlighting the role of stochastic processes in bridging geometry, statistics, and machine learning.