A McKean-Pontrygin maximum principle for entropic-regularized optimal transport
This provides a theoretical framework for researchers in optimal transport, but it appears incremental as it builds on existing principles without clear empirical validation.
The authors tackled dynamic optimal transport problems by applying the McKean-Pontryagin maximum principle, resulting in a mean-field approach that avoids stochastic path sampling and unifies deterministic and stochastic cases.
This note outlines a mean-field approach to dynamic optimal transport problems based on the recently proposed McKean-Pontryagin maximum principle. Key aspects of the proposed methodology include i) avoidance of sampling over stochastic paths, ii) a fully variational approach leading to constrained Hamiltonian equations of motion, and iii) a unified treatment of deterministic and stochastic optimal transport problems. We also discuss connections to well-known dynamic formulations in terms of forward-backward stochastic differential equations and extensions beyond classical entropic-regularized transport problems.