This three-year project grant from the National Science Foundation's Mathematical and Physical Sciences program (CFDA 47.049) provides $378,158 to Carnegie Mellon University to develop mathematical tools for data science and signal processing applications. Specifically, the awardee will investigate transportation-based geometries and gradient flows to provide sampling methods that leverage data geometry and can be accurately approximated in high dimensions. They will study ensemble methods like Stein variational gradient descent to provide a particle-based sampling approach for Gibbs distributions with general potentials. Additionally, the project will examine geometries and gradient flows in Stein geometry and related models, as well as paths in probability measure spaces using nonlocal continuity equations. The awardee will also develop deformation-based geometries on signal spaces allowing for both transportation and intensity-based differences. This work is expected to result in new and more accurate ways to sample and represent data distributions to address challenging machine learning problems.