Neural Optimal Transport with Lagrangian Costs
Keywords: neural optimal transport, Lagrangian costs, Riemannian geometry, amortized optimization
Abstract: Computational efforts in optimal transport traditionally revolve around the squared-Euclidean cost. In this work, we choose to investigate the optimal transport problem between probability measures when the underlying metric space is non-Euclidean, or when the cost function is understood to satisfy a *least action principle*, also known as a *Lagrangian* cost. These two generalizations are useful when connecting observations from a physical system, where the transport dynamics are influenced by the geometry of the system, such as obstacles, and allows practitioners to incorporate *a priori* knowledge of the underlying system. Examples include barriers for transport, or enforcing a certain geometry, i.e., paths must be circular. We demonstrate the effectiveness of this formulation on existing synthetic examples in the literature, where we solve the optimal transport problems in the absence of regularization, which is novel in the literature. Our contributions are of computational interest, where we demonstrate the ability to efficiently compute geodesics and amortize spline-based paths. We demonstrate the effectiveness of this formulation on existing synthetic examples in the literature, where we solve the optimal transport problems in the absence of regularization.
Submission Number: 56