Orlicz space regularization of continuous optimal transport problems

Abstract: In this work we analyze regularized optimal transport problems in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, the aim is to find a transport plan, which is another Radon measure on the product of the sets, that has these two measures as marginals and minimizes the sum of a certain linear cost function and a regularization term. We focus on regularization terms where a Young's function applied to the (density of the) transport plan is integrated against a product measure. This forces the transport plan to belong to a certain Orlicz space. The predual problem is derived and proofs for strong duality and existence of primal solutions of the regularized problem are presented. Existence of (pre-)dual solutions is shown for the special case of $L^p$ regularization for $p\geq 2$. Moreover, two results regarding $\Gamma$-convergence are stated: The first is concerned with marginals that do not lie in the appropriate Orlicz space and guarantees $\Gamma$-convergence to the original Kantorovich problem, when smoothing the marginals. The second results gives convergence of a regularized and discretized problem to the unregularized, continuous problem.