Gaussian processes in the age of big data and AI

Abstract:

Gaussian processes constitute one of the most prominent mathematical concepts to model and predict phenomena with uncertainty, with applications ranging from the analysis of climate or forestry data in the environmental sciences to financial modelling and medical imaging. In addition, they have received an increased attention in the machine learning community, since they provide a practical probabilistic approach to learning in kernel machines.

The technological progress in data collection and storage capacities in the past decades has caused a tremendous growth in volume and variety of data, which poses new challenges for statistical modelling and computational methodologies. Classical approaches to statistical inference resort to Gaussian processes indexed by the Euclidean space; however, e.g., for global data, models on the sphere or on networks (rivers, streets, etc.) as well as their spatiotemporal counterparts are needed. Moreover, the computational costs generally are cubic in the number of observations which quickly becomes intractable for inference from large data sets.

In this presentation I will first give an introduction to the theory of Gaussian processes and their usage for statistical modelling and learning. I will then discuss the benefits and limitations of current methodologies for large spatial or spatiotemporal data, with a focus on the Stochastic Partial Differential Equation (SPDE) approach which expresses Gaussian processes as solutions to SPDEs. This viewpoint has already proven very powerful in the past decade by enabling the usage of computational methods for partial differential equations in the context of statistical inference. Exploring its full potential is an emerging challenge for the Numerical Analysis and Spatial Statistics communities.