Statistical inverse problems and affine-invariant gradient flow structures in the space of probability measures

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. Applications include nonlinear and logistic regression.