We develop the theory of a metric, which we call the $\nu$-based Wasserstein metric and denote by $W_\nu$, 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 $\nu$ and is relevant in particular for the case when $\nu$ is singular with respect to $m$-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $\nu$-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $\nu$; 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 $\nu$, the $\nu$-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance and a metric associated with the uniquely defined generalized geodesics obtained when $\nu$ is sufficiently regular. When $\nu$ concentrates on a lower dimensional submanifold of $\mathbb{R}^m$, we prove that the variational problem in the definition of the $\nu$-based Wasserstein distance has a unique solution. We establish geodesic convexity of the usual class of functionals and of the set of source measures $\mu$ such that optimal transport between $\mu$ and $\nu$ satisfies a strengthening of the generalized nestedness condition introduced in \cite{McCannPass20}. We also present two applications of the ideas introduced here. First, our dual metric is used to prove convergence of an iterative scheme to solve a variational problem arising in game theory. We also use the multi-marginal formulation to characterize solutions to the multi-marginal problem by an ordinary differential equation, yielding a new numerical method for it.
翻译:我们开发了一种度量的理论, 我们称之为以美元为基数的以美元为基数的基地德概念略微细化, 特别是当美元为单数时, 以美元为基数的奥氏变异度为基数, 以美元为基数的瓦塞斯坦度为基数, 以美元为基数, 以美元为基数, 以美元为基数, 以美元为基数的莱贝斯格度为基数, 也与线性最佳运输的概念密切相关。 以美元为基数的瓦塞斯坦度为基数, 以美元为基数的奥氏度为基数, 以美元为基数的瓦塞斯坦度度为基数, 以美元为基数的奥数, 以美元为基数的奥基数为基数, 以美元为基数的运算法, 以美元为基数, 以美元为基数的奥基数为基数, 以美元为基数的运算法, 。