On the Connection between Dynamical Optimal Transport and Functional Lifting
This work provides a theoretical connection between functional lifting and dynamical optimal transport, which is incremental for researchers in mathematical optimization and image processing.
The paper tackles the problem of approximating solutions to non-convex problems by embedding them into a space of probability measures, showing this approach generalizes dynamical optimal transport and connects to variational models with first-order and higher-order regularization.
Functional lifting methods provide a tool for approximating solutions of difficult non-convex problems by embedding them into a larger space. In this work, we investigate a mathematically rigorous formulation based on embedding into the space of pointwise probability measures over a fixed range $Γ$. Interestingly, this approach can be derived as a generalization of the theory of dynamical optimal transport. Imposing the established continuity equation as a constraint corresponds to variational models with first-order regularization. By modifying the continuity equation, the approach can also be extended to models with higher-order regularization.