Johannes is a PostDoc at the Institute of Mathematics at TU Berlin in the group of Benjamin Gess. His research interests lie at the intersection of machine learning and applied mathematics, with a particular focus on the geometric structures appearing in reinforcement learning and scientific machine learning.
Abstract:
A novel advective Fisher–Rao metric is introduced for optimization tasks on paths of probability densities governed by the continuity equation, which covers recent generative models. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher–Rao metric on path measures, as the expected value of the second variation of the Freidlin-Wentzell large deviation rate functional, and as the Hessian of the Benamou-Brenier action functional from dynamic optimal transport. We supplement our geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher–Rao metric leads to the desired optimal fitting of probability densities, whereas the Gauss–Newton method leads to an optimal fitting of velocity fields.
