A study on fast optimization method based on Frank-Wolfe algorithm for relaxed optimal transport

Abstract

Optimal transport, which expresses the distance between probability distributions, have been applied to various applications. In optimal transport, it is necessary to solve a linear programming problem which has the tight mass conversation constraints, but it is known that optimal transport is difficult to solve fast. To improve this problem, relaxed optimal transport loosing the constraints has been proposed. It develop the fast methods and, moreover, some papers report that there are applications (color transfer, etc) which relaxed constraints are effective for. In this paper, we focus on the convex relaxed optimal transport, propose our new fast method and analyze it. Concretely, we propose the fast optimization method using Frank-Wolfe algorithm and prove the upper-bound of the worst convergence iterations. Finally, numerical evaluations show that our proposed method converges more fast than other existing methods.