论文标题
连续最佳运输问题的空间正规化
Orlicz space regularization of continuous optimal transport problems
论文作者
论文摘要
在这项工作中，我们分析了所谓的Kantorovich形式的正则最佳运输问题，即给出了两个紧凑型集合的两种ra措施，目的是找到一个运输计划，这是集合产品的另一种ra措施，这两种措施具有这两种措施作为边缘的措施，并将一定的线性成本函数和常规化功能和常规化术语最小化。我们专注于正规化术语，其中年轻的功能应用于运输计划的（密度）相对于产品度量集成。这迫使运输计划属于某个Orlicz空间。提出了预先的问题，并提出了强大的二元性和正规化问题的原始解决方案的证明。对于$ p $ p \ geq 2 $的特殊情况，显示了（预 - ）双重解决方案的存在。此外，指出了有关$γ$ - 联合的两个结果：第一个与不在适当的空间中的边际有关，并保证在平滑边际时，$γ$ - 与原始的Kantorovich问题相关。第二个结果给出了正规化和离散的问题与未注册的连续问题的收敛性。
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 $Γ$-convergence are stated: The first is concerned with marginals that do not lie in the appropriate Orlicz space and guarantees $Γ$-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.
