  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}.
