Fast entropic regularized optimal transport using semidiscrete cost approximation

The optimal transportation theory was successfully applied to different tasks on geometric domains as images and triangle meshes. In these applications the transport problem is defined on a Riemannian manifold with geodesic distance d(x, y). Usually, the cost function used is the geodesic distance d or the squared geodesic distance d². These choices result in the 1-Wasserstein distance, also known as the earth mover's distance (EMD), or the 2-Wasserstein distance. The entropy regularized optimal transport problem can be solved using the Bregman projection algorithm. This algorithm can be implemented using only matrix multiplications of matrix exp(-C/ϵ) (pointwise exponent) and pointwise vector multiplications, where C is a cost matrix, and e is the regularization parameter. In this paper, we obtain a low-rank decomposition of this matrix and exploit it to accelerate the Bregman projection algorithm. Our low-rank decomposition is based on the semidiscrete approximation of the cost function, which is valid for a large family of cost functions, including d^p(x,y), where p ≥ 1. Our method requires the calculation of only a small portion of the distances.