  Optimal transport (OT) is a powerful tool for measuring the distance between
two defined probability distributions. In this paper, we develop a new manifold
named the coupling matrix manifold (CMM), where each point on CMM can be
regarded as the transportation plan of the OT problem. We firstly explore the
Riemannian geometry of CMM with the metric expressed by the Fisher information.
These geometrical features of CMM have paved the way for developing numerical
Riemannian optimization algorithms such as Riemannian gradient descent and
Riemannian trust-region algorithms, forming a uniform optimization method for
all types of OT problems. The proposed method is then applied to solve several
OT problems studied by previous literature. The results of the numerical
experiments illustrate that the optimization algorithms that are based on the
method proposed in this paper are comparable to the classic ones, for example,
the Sinkhorn algorithm, while outperforming other state-of-the-art algorithms
without considering the geometry information, especially in the case of
non-entropy optimal transport.
