[Paper Review] Duality and quotient spaces of generalized Wasserstein spaces

Nhan‐Phu Chung, Thanh‐Son Trinh|arXiv (Cornell University)|Apr 29, 2019

Geometric Analysis and Curvature Flows15 references4 citations

TL;DR

This paper establishes Kantorovich duality for generalized Wasserstein distances $W_{1}^{a,b}$ on Polish metric spaces, proving that $W_{1}^{a,b}$ coincides with the flat metric and that $(\mathcal{M}(X), W_{1}^{a,b})$ is a geodesic space. It further shows that for isometric actions of compact groups, the quotient map induces an isometric isomorphism between $\mathcal{M}^G(X)$ and $\mathcal{M}(X/G)$, and proves Gromov-Hausdorff and equivariant Gromov-Hausdorff convergence results for generalized Wasserstein spaces.

ABSTRACT

In this article, using ideas of Liero, Mielke and Savaré in [21], we establish a Kantorovich duality for generalized Wasserstein distances $W_1^{a,b}$ on a generalized Polish metric space, introduced by Picolli and Rossi. As a consequence, we give another proof that $W_1^{a,b}$ coincide with flat metrics which is a main result of [25], and therefore we get a result of independent interest that $\left(\mathcal{M}(X), W^{a,b}_1 ight)$ is a geodesic space for every Polish metric space $X$. We also prove that $(\mathcal{M}^G(X),W_p^{a,b})$ is isometric isomorphism to $(\mathcal{M}(X/G),W_p^{a,b})$ for isometric actions of a compact group $G$ on a Polish metric space $X$; and several results of Gromov-Hausdorrf convergence and equivariant Gromov-Hausdorff convergence of generalized Wasserstein spaces. The latter results were proved for standard Wasserstein spaces in [22],[14] and [8] respectively.

Motivation & Objective

• To establish Kantorovich duality for generalized Wasserstein distances $W_{1}^{a,b}$ on Polish metric spaces.
• To reprove the coincidence of $W_{1}^{a,b}$ with the flat metric using duality, offering an alternative to the original proof in [PR16].
• To show that $(\mathcal{M}(X), W_{1}^{a,b})$ is a geodesic space for any Polish metric space $X$.
• To extend isometric quotient results from standard Wasserstein spaces to generalized $W_{p}^{a,b}$ spaces under compact group actions.
• To generalize Gromov-Hausdorff and equivariant Gromov-Hausdorff convergence results to the setting of generalized Wasserstein spaces.

Proposed method

• Derive Kantorovich duality for $W_{1}^{a,b}$ using a dual formulation involving bounded continuous functions $\varphi_1, \varphi_2$ satisfying $\varphi_1(x) + \varphi_2(y) \leq b d(x,y)$ and $\varphi_i(x) \geq -a$, with the dual value expressed via $I(\varphi) = \inf_{s \geq 0} (s\varphi + a|1-s|)$.
• Prove that the dual formulation recovers the Kantorovich-Rubinstein-type identity for $W_{1}^{a,b}(\mu, \nu) = \sup \left\{ \int f d(\mu - \nu) : f \in \mathbb{F} \right\}$, where $\mathbb{F}$ consists of $a$-bounded, $b$-Lipschitz functions.
• Establish isometric isomorphism between $({\mathcal{M}}^G(X), W_p^{a,b})$ and $({\mathcal{M}}(X/G), W_p^{a,b})$ via the pushforward map $p_\sharp$ under isometric group actions.
• Use the duality framework to characterize optimal plans $\gamma$ via conditions on dual potentials $\varphi_1, \varphi_2$ and decomposition of measures into absolutely continuous and singular parts.
• Prove Gromov-Hausdorff convergence of $({\mathcal{M}}_p^C(X_n), W_p^{a,b})$ to $({\mathcal{M}}_p^C(X), W_p^{a,b})$ when $X_n \to X$ in Gromov-Hausdorff topology.
• Establish equivariant Gromov-Hausdorff convergence of induced actions on generalized Wasserstein spaces when the group actions converge.

Experimental results

Research questions

• RQ1 Does Kantorovich duality hold for the generalized Wasserstein distance $W_{1}^{a,b}$ on Polish metric spaces despite the lack of superlinear entropy and compact sublevels?
• RQ2 Can the coincidence of $W_{1}^{a,b}$ with the flat metric be re-proven via duality, independent of the original proof in [PR16]?
• RQ3 Is the space $(\mathcal{M}(X), W_{1}^{a,b})$ geodesic for any Polish metric space $X$?
• RQ4 Does the pushforward map $p_\sharp$ induce an isometric isomorphism between $\mathcal{M}^G(X)$ and $\mathcal{M}(X/G)$ for $W_p^{a,b}$ under isometric compact group actions?
• RQ5 Do generalized Wasserstein spaces inherit Gromov-Hausdorff and equivariant Gromov-Hausdorff convergence from their base metric spaces?

Key findings

• The Kantorovich duality for $W_{1}^{a,b}$ is established via the dual formulation $W_{1}^{a,b}(\mu_1, \mu_2) = \sup_{(\varphi_1, \varphi_2) \in \Phi_W} \sum_i \int_X I(\varphi_i(x)) d\mu_i(x)$, where $I(\varphi) = \inf_{s \geq 0} (s\varphi + a|1-s|)$.
• The duality implies the Kantorovich-Rubinstein-type identity $W_{1}^{a,b}(\mu, \nu) = \sup \left\{ \int f d(\mu - \nu) : \|f\|_\infty \leq a, \|f\|_{\text{Lip}} \leq b \right\}$, re-proving the flat metric coincidence from [PR16].
• The space $(\mathcal{M}(X), W_{1}^{a,b})$ is geodesic for any Polish metric space $X$, a result of independent interest.
• For a compact group $G$ acting isometrically on a locally compact Polish metric space $X$, the pushforward map $p_\sharp: (\mathcal{M}^G(X), W_p^{a,b}) \to (\mathcal{M}(X/G), W_p^{a,b})$ is an isometry.
• The generalized Wasserstein space $({\mathcal{M}}_p^C(X_n), W_p^{a,b})$ converges in Gromov-Hausdorff topology to $({\mathcal{M}}_p^C(X), W_p^{a,b})$ whenever $X_n \to X$ in Gromov-Hausdorff topology.
• The induced action $\alpha_n\sharp$ on $({\mathcal{M}}_p^C(X_n), W_p^{a,b})$ converges in equivariant Gromov-Hausdorff topology to $\alpha_\sharp$ whenever $\alpha_n \to \alpha$ in the action topology.