Transport Type Metrics on the Space of Probability Measures Involving Singular Base Measures

Abstract

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 metric between the conditional probabilities when measures are disintegrated with respect to optimal transport to $\nu$, and through limits of certain multi-marginal optimal transport problems. We also introduce a class of metrics which are dual in a certain sense to $W_\nu$, defined relative to a fixed based measure $\mu$, on the set of measures which are absolutely continuous with respect to a second fixed based measure $\sigma$. As we vary the base measure $\nu$, the $\nu$-based Wasserstein metric interpolates between the usual quadratic Wasserstein metric (obtained when $\nu$ is a Dirac mass) and a metric associated with the uniquely defined generalized geodesics obtained when $\nu$ is sufficiently regular (eg, absolutely continuous with respect to Lebesgue). 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 metric has a unique solution. We also 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 McCann and Pass (Arch Ration Mech Anal 238(3):1475–1520, 2020). 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.

Appendix A Hierarchical Metrics with Multiple Reference Measures

The $\nu$-based Wasserstein metric can be extended to align with several different probability measures in a hierarchical way. Given $\nu _1,\dots ,\nu _k \in {\mathcal {P}}(X)$, we define iteratively $\Pi ^1_{opt} =\Pi _{opt}(\nu _1,\mu _i)$ and $\Pi ^j_{opt}(\nu _1,\ldots ,\nu _j, \mu _i) \subseteq {\mathcal {P}}(X^{j+1})$ by

$$\begin{aligned}{} & {} \Pi ^j_{opt}(\nu _1,\ldots ,\nu _j, \mu _i) ={\textrm{argmin}}_{\begin{array}{c} \gamma _{y_1,\ldots y_{j-1}x_i} \in \Pi ^{j-1}_{opt}(\nu _1,\ldots , \nu _{j-1},\mu _i) \\ \gamma _{y_j}=\nu _j \end{array}}\\{} & {} \qquad \int _{X^{j+1}}|x_i-y_j|^2d\gamma (y_1,\ldots .,y_j,x_i). \end{aligned}$$

A natural analogue of $W_\nu$ is then defined by:

$$\begin{aligned}{} & {} W^2_{\nu _1,\ldots \nu _k}(\mu _0,\mu _1)=\inf _{\gamma _{y_1...y_kx_i} \in \Pi ^k_{opt}(\nu _1,\ldots ,\nu _k, \mu _i), i=0,1}\nonumber \\ {}{} & {} \int _{X^{k+2}}|x_0-x_1|^2d\gamma (y_1,\ldots ,y_k,x_0,x_1). \end{aligned}$$

(43)

As before, optimal couplings can be recovered as limits of multi-marginal problems, where the weights on the interaction terms reflect the hierarchy of the measures $\nu _1,\ldots ,\nu _k$ in the definition of $W_{\nu _1,\ldots \nu _k}$:

Proposition 40

Let $\nu _1,\ldots ,\nu _k,\mu _{0},\mu _1$ be probability measures on X. Consider the multi-marginal optimal transport problem:

$$\begin{aligned} \int _{X^{k+2}}c_\varepsilon (x_0,x_1,y_1,y_2,\ldots y_k)d\gamma \end{aligned}$$

(44)

where $c_\varepsilon =\varepsilon ^{n-1}|x_0-x_1|^2 +\varepsilon ^{n-2}(|x_0-y_{n-1}|^2 +|x_1-y_{n-1}|^2)+\varepsilon ^{n-3}(|x_0-y_{n-2}|^2 +|x_1-y_{n-2}|^2)+\cdots +\varepsilon (|x_0-y_2|^2 +|x_1-y_2|^2)+(|x_0-y_1|^2 +|x_1-y_1|^2)$. Then any weak limit ${\overline{\gamma }}$ of solutions $\gamma _\varepsilon$ is optimal in (43)

Proof

The proof is similar to the proof of the second part of Theorem 12. Letting $\gamma \in \Pi ^k_{opt}(\nu _1,\ldots ,\nu _k, \mu _i)$ for $i=0,1$, optimality of $\gamma _\varepsilon$ in (44) implies

$$\begin{aligned} \int _{X^{k+2}}c_\varepsilon (x_0,x_1,y_1,y_2,\ldots y_k)d \gamma _\varepsilon \le \int _{X^{k+2}}c_\varepsilon (x_0,x_1,y_1,y_2,\ldots y_k)d\gamma . \end{aligned}$$

(45)

Passing to the limit implies that ${\overline{\gamma }} \in \Pi _{opt}^1(\mu _i, \nu )$. Now, combining the fact that $\gamma \in \Pi _{opt}^1(\mu _i, \nu )$ with (45) implies

$$\begin{aligned}\begin{aligned}&\int _{X^{k+2}}c_\varepsilon (x_0,x_1,y_1,y_2,\ldots y_k) - (|x_0-y_1|^2 +|x_1-y_1|^2)d{\overline{\gamma }}_\varepsilon \\ {}&\quad \le \int _{X^{k+2}}c_\varepsilon (x_0,x_1,y_1,y_2,\ldots y_k)-(|x_0-y_1|^2 +|x_1-y_1|^2)d\gamma . \end{aligned} \end{aligned}$$

Dividing by $\varepsilon$ and taking the limit then implies that ${\overline{\gamma }} \in \Pi _{opt}(\nu _1,\nu _2, \mu _i)$ for $i=0,2$. Continuing inductively in this way yields the desired result. $\square$

As a consequence, we easily obtain the following result, which is similar to the main result in [6].

Corollary 41

Let $k=n-1$, and $\nu _1,\ldots \mu _{n-1}$ be probability measures concentrated on mutually orthogonal line segments, each absolutely continuous with respect to one dimensional Hausdorff measure. Then solutions $\gamma _\varepsilon$ to the multi-marginal optimal transport problem (44) converge weakly to $\gamma$, whose 1, 2 marginal is $(Id,G)_\#\mu _0$, where G is the Knothe–Rosenblatt transport from $\mu _0$ to $\mu _1$.