Sebastian Reich (University of Potsdam)

Title: Statistical inverse problems and affine-invariant gradient flow structures in the space of probability measures
Abstract: Statistical inverse problems lead to complex optimisation and/or Monte Carlo sampling problems. Gradient descent and Langevin samplers provide examples of widely used algorithms. In my talk, I will discuss recent results on sampling algorithms, which can be viewed as interacting particle systems, and their mean-field limits. I will highlight the geometric structure of these mean-field equations within the, so called, Otto calculus, that is, a gradient flow structure in the space of probability measures. Affine invariance is an important outcome of recent work on the subject, a property shared by Newton’s method but not by gradient descent or ordinary Langevin samplers. The emerging affine invariant gradient flow structures allow us to discuss coupling-based Bayesian inference methods, such as the ensemble Kalman filter, as well as invariance-of-measure-based inference methods, such as preconditioned Langevin dynamics, within a common mathematical framework. The talk is based joint research work with Nik Nüsken, Simon Weißmann, Andrew Stuart, Daniel Zhengyu Huang, Jiaoyang Huang, Zifan Chen, and Edoardo Calvello. Most results have been summarised in a recent survey paper available from arXiv2209.11371.