Flows on Measure Spaces and Applications in Machine Learning

Abstract

Flows on measure spaces have long been examined in stochastic analysis and have recently attracted significant interest in machine learning, leading to intriguing research questions that often fall outside the scope of existing theory. Normalizing flows, score-based diffusion, and flow matching models are among the most powerful generative neural methods and rely on the geometry of measure spaces. In particular, the Wasserstein metric and optimal transport techniques have advanced the field in recent years. However, involving different Riemannian-like metrics on measure spaces, e.g., by the framework of right-invariant metrics on the group of diffeomorphisms and their action on objects, e.g., densities, and designing transport inference functionals with advanced properties like equivariance led to new neural models. Generative models can be conditioned on (degraded) data, which leads to new developments in the solution of Bayesian inverse problems. Viewing transformers as interacting particle systems introduced a new mathematical perspective on these complex systems and shed light on their clustering behavior. Finally, learning neural models comes with new challenges in (stochastic) optimization, such as accelerated optimization, operator splitting, and mirror descent on measure spaces, ensemble filtering methods, the treatment of high dimensions via slicing or Fourier random features, as well as scalability questions and related lifting to infinite-dimensional spaces. The workshop will bring together scientists interested in different aspects of flows on measure spaces to further understand and develop their analysis, in particular to address questions in deep generative learning and to develop improved optimization methods for measure spaces.