Generalizing Wasserstein Barycenters to discretize gradient flows – BARYFLOW

Wasserstein barycenters are a powerful tool for analyzing complex data represented by probability measures. They enable the interpolation of different types of objects, such as images, texts, or distributions of physical quantities, while taking into account their global geometric structure. However, in many applications, including statistical regression tasks or the design of reduced-order models, interpolation alone is not sufficient, as there is often a need to extrapolate based on the available information. Yet, there are currently no computational methods for data extrapolation that are fully consistent with the Wasserstein geometry. This project addresses this gap by developing a framework for the analysis and numerical computation of Wasserstein barycenters with negative weights. This is achieved by building on a recently discovered connection between such problems and Weak Optimal Transport theory, which allows to bypass their inherent non-convexity.

We will exploit this framework for three main applications. First, we will develop computational tools for Wasserstein regression, extending standard linear regression techniques to measure-valued data. Second, we aim to improve methods for computing Wasserstein barycenters with positive weights, by incorporating extrapolation problems as a form of regularization, preventing the diffuse solutions that arise from traditional entropy-based regularization methods.

Finally, we will use generally-weighted Wasserstein barycenters to construct accurate particle discretizations of Wasserstein gradient flows. These systems describe the evolution of a measure following the steepest descent direction of a given energy with respect to an Optimal Transport metric. They are ubiquitous models in physics, providing a unifying framework for a wide range of dissipative phenomena. They have also recently become crucial tools for designing and analyzing particle-based inference methods in statistics. We will use our framework to generalize standard numerical integration techniques for these models and design particle methods with high-order accuracy to simulate them. We will explore the use of such schemes both for the simulation of physical systems and for accelerating particle-based variational inference techniques.