Year: 2025
Type: article
Abstract: Dynamical formulations of optimal transport (OT) frame the task of comparing distributions as a variational problem which searches for a path between distributions minimizing a kinetic energy functional. In applications, it is frequently natural to require paths of distributions to satisfy additional conditions. Inspired by this, we introduce a model for dynamical OT which incorporates constraints on the space of admissible paths into the framework of unbalanced OT, where the source and target measures are allowed to have a different total mass. Our main results establish, for several general families of constraints, the existence of solutions to the variational problem which defines this path constrained unbalanced OT framework. These results are primarily concerned with distributions defined on an Euclidean space, but we extend them to distributions defined over parallelizable Riemannian manifolds as well. We also consider metric properties of our framework, showing that, for certain types of constraints, our model defines a metric on the relevant space of distributions. This metric is shown to arise as a geodesic distance of a Riemannian metric, obtained through an analogue of Otto’s submersion in the classical OT setting.
Source: Nonlinearity
Institutions: Florida State University, University of Houston
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: CAREER: Shape Analysis in Submanifold Spaces: New Directions for Theory and Algorithms, Collaborative Research: Applications of Symplectic Geometry to Frame Theory and Signal Processing, Collaborative Research: CISE-ANR: RI: Small: 4DShape: 4D Human Shape Analysis and Synthesis, eMB: New Approaches for Interpreting Neural Responses to Behaviorally-Relevant Sensory Stimuli, G6490241812, G7135334059, Collaborative Research: Data-Driven Elastic Shape Analysis with Topological Inconsistencies and Partial Matching Constraints
OpenAlex ID: W4411346165