Orlicz Space Regularization of Continuous Optimal Transport Problems
Abstrak
AbstractIn 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 $$L^p$$ L p -regularization for $$p\ge 2$$ p ≥ 2 . Moreover, two results regarding $$\varGamma $$ Γ -convergence are stated: The first is concerned with marginals that do not lie in the appropriate Orlicz space and guarantees $$\varGamma $$ Γ -convergence to the original Kantorovich problem, when smoothing the marginals. The second result gives convergence of a regularized and discretized problem to the unregularized, continuous problem.
Penulis (2)
Dirk Lorenz
Hinrich Mahler
Akses Cepat
- Tahun Terbit
- 2022
- Bahasa
- en
- Total Sitasi
- 10×
- Sumber Database
- CrossRef
- DOI
- 10.1007/s00245-022-09826-7
- Akses
- Open Access ✓