Collaborative Research: CIF: Small: Generalized Optimal Transport Models: Theory and Computation

About This Grant

Understanding how to compare and interpolate complex data, such as images, shapes, and network structures, is a fundamental challenge across science and engineering, especially in the context of artificial intelligence. This project develops new mathematical and computational tools that extend the theory of optimal transport, a well-established framework for measuring distances between probability distributions. The proposed methods are tailored to settings which more closely reflect specialized real world data structures than those considered in classical optimal transport. These advances will enable more adequate quantitative analysis methods for medical images, dynamic crowd movements, and biological network structures. An integral outcome of the project will consist in the production of robust and open-source software packages, which will make these generalized optimal transport methods accessible to researchers and practitioners in biomedical imaging, machine learning, and network analysis. Importantly, these algorithms will be firmly grounded in mathematical theory. The project will also train graduate and postdoctoral researchers through cross disciplinary collaborations, foster community engagement via a workshop, and engage with the broader community via a coding-focused course and K-12 outreach activities. The project pursues three interlocking aims. First, it formulates a new Constrained Unbalanced Optimal Transport model for comparing general positive measures under integral and parametric constraints; this involves rigorous well posedness results and efficient numerical solvers, targeted at shape analysis and population/crowd modeling. Second, it introduces an Optimal Riemannian Metric Transport framework, which blends ideas from optimal transport and infinite-dimensional geometry to compare Riemannian metrics on a fixed manifold; this framework is anchored in geometric connections to the well-established Wasserstein-Fisher-Rao metric, which has been proven successful in applications in machine learning and data science. Third, it investigates the Alexandrov and Riemannian geometry of Gromov–Wasserstein distances; this will result in geometry driven computational tools for comparing structured data on different domains, with applications to the analysis of astrocyte cell morphology. Together, these efforts will yield new methodological insights and a suite of software libraries. 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.