  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 OT problems within the framework of the Riemannian
manifold optimization. The basis of this is the manifold of doubly stochastic
matrices (and its generalization). Even though the manifold geometry is not
new, surprisingly, its usefulness for solving OT problems has not been
considered. 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
https://github.com/SatyadevNtv/MOT.
