Manifold optimization for non-linear optimal transport problems
This work provides a computational framework for researchers and practitioners in machine learning dealing with optimal transport, though it is incremental as it applies existing manifold geometry to a new context.
The authors tackled the computational challenge of solving general non-linear optimal transport problems by applying Riemannian manifold optimization to the manifold of doubly stochastic matrices, resulting in the development of an open-source repository (MOT) with codes in Python and Matlab.
Optimal transport (OT) has recently found widespread interest in machine learning. It allows to define novel distances between probability measures, which have shown promise in several applications. In this work, we discuss how to computationally approach general non-linear OT problems within the framework of Riemannian manifold optimization. The basis of this is the manifold of doubly stochastic matrices (and their generalization). Even though the manifold geometry is not new, surprisingly, its usefulness for solving general non-linear OT problems has not been popular. To this end, we specifically discuss optimization-related ingredients that allow modeling the OT problem on smooth Riemannian manifolds by exploiting the geometry of the search space. We also discuss extensions where we reuse the developed optimization ingredients. We make available the Manifold optimization-based Optimal Transport, or MOT, repository with codes useful in solving OT problems in Python and Matlab. The codes are available at \url{https://github.com/SatyadevNtv/MOT}.