Coupling matrix manifolds assisted optimization for optimaltransport problems

Optimal transport (OT) is a powerful tool for measuring the distance between two 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 a 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 an essential optimization method for all types of OT problems. The proposed method is then applied to solve several OT problems studied by recent literature. For the classic OT problem and its entropy regularized variant, the OT solution generated from our method is comparable to that from the classic algorithms (i.e. Linear programming and Sinkhorn algorithms), while for other types of non-entropy regularized OT problems our method outperforms other state-of-the-art algorithms which don’t incorporate the geometric information of the OT feasible space.