论文标题
运输类型的指标在涉及单数基础措施的概率措施的空间
Transport type metrics on the space of probability measures involving singular base measures
论文作者
论文摘要
我们开发了一个公制的理论，我们称之为$ν$的Wasserstein Metric，并用$W_ν$表示，在域上的$ \ Mathcal p（x）$上，$w_ν$表示，$ x \ subseteq \ subseteq \ subseteq \ mathbb {r}^m $。该指标基于对基本度量$ν$的广义大地测量学概念的略微完善，特别是对于$ν$相对于$ M $ $ $二维的Lebesgue Measure的情况；它也与线性化最佳传输的概念密切相关。基于$ν$的Wasserstein度量是根据涉及最佳运输到$ν$的迭代变化问题来定义的；我们还根据有条件概率之间的经典Wasserstein距离的整合以及某些多边界最佳最佳运输问题的限制来表征它。随着我们的基本量度$ν$，基于$ν$的余地公制插座是通常的二次二次Wasserstein距离和与$ν$相当规律时获得的唯一定义的通用地理学的度量。当$ν$集中在$ \ mathbb {r}^m $的较低维submanifold上时，我们证明，基于$ν$的Wasserstein距离的定义中的变异问题具有独特的解决方案。我们建立了通常的一类功能类别和一组源指标的大地测量凸度，以使$ $ $ $和$ν$之间的最佳运输能够满足\ cite {mccannpass20}中引入的广义嵌套条件的增强。 理论。
We develop the theory of a metric, which we call the $ν$-based Wasserstein metric and denote by $W_ν$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure $ν$ and is relevant in particular for the case when $ν$ is singular with respect to $m$-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $ν$-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $ν$; we also characterize it in terms of integrations of classical Wasserstein distance between the conditional probabilities and through limits of certain multi-marginal optimal transport problems. As we vary the base measure $ν$, the $ν$-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance and a metric associated with the uniquely defined generalized geodesics obtained when $ν$ is sufficiently regular. When $ν$ concentrates on a lower dimensional submanifold of $\mathbb{R}^m$, we prove that the variational problem in the definition of the $ν$-based Wasserstein distance has a unique solution. We establish geodesic convexity of the usual class of functionals and of the set of source measures $μ$ such that optimal transport between $μ$ and $ν$ satisfies a strengthening of the generalized nestedness condition introduced in \cite{McCannPass20}.We finally introduce a slight variant of the dual metric mentioned above in order to prove convergence of an iterative scheme to solve a variational problem arising in game theory.
