Entropic regularization of continuous optimal transport problems

Understand

We analyze continuous optimal transport problems in the so-called Kantorovich form, where we seek a transport plan between two marginals that are probability measures on compact subsets of Euclidean space.

• We consider the case of regularization with the negative entropy with respect to the Lebesgue measure, which has attracted attention because it can be solved by the very simple Sinkhorn algorithm.
• We first analyze the regularized problem in the context of classical Fenchel duality and derive a strong duality result for a predual problem in the space of continuous functions.
• However, this problem may not admit a minimizer, which prevents obtaining primal-dual optimality conditions.