Novel Transportation-Based Geometries, Gradient Flows, and Applications to Data Science

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This project will develop mathematical tools to study data science and signal processing tasks. While differential equations and variational approaches provide useful models for data science tasks, the effectiveness of many of the present models diminishes in high dimensions due to computational challenges and a lack of statistical reliability. This project will provide approaches to machine learning tasks that take advantage of the geometry of the data and can be accurately approximated in high dimensions. The project will lead to new and more accurate ways to sample and represent data distributions. The project will provide opportunities for training a new generation of mathematicians who will gain knowledge of modern techniques of applied analysis and be aware of important questions arising in data science.Motivated by problems in data science, partial differential equations (PDE) on graphs, and tasks in signal analysis, the project will investigate several distinct settings. A major effort will be devoted to studying ensemble methods for samplings, such as the Stein Variational Gradient Descent and related models. The models will provide a deterministic particle-based method for sampling Gibbs distributions for general potentials. The project will investigate the geometry and gradient flows in Stein geometry and related models. More broadly, ensemble-based methods provide a promising avenue to address challenging sampling problems (multimodal, highly anisotropic energy landscapes). Their connection to PDE via mean-field limits also allows for analytical study and modeling. The investigator and collaborators will explore several models and develop both theoretical understanding and computational approaches. The project will also study paths in the spaces of probability measures based on the nonlocal continuity equation and the resulting nonlocal Wasserstein metrics. Regarding signal analysis, the project will study deformation-based geometries on the space of signals that allow for both transportation and intensity-based differences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.