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Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.

Let $X$ be a topological space (you may assume $X$ is compact) and $\mathcal C_b(X)$ be the space of bounded continuous functions on $X$. Finally, let $\mathcal P(X)$ be the set of probability measures on $X$. For a fixed subset $\mathcal F \subseteq \mathcal C_b(X^2)$, define the mapping $ d_{\mathcal F}: \mathcal P(X) \times \mathcal P(X) \rightarrow \mathbb R \cup \{+\infty\}$, by

$$ d_{\mathcal F}(\mu,\nu) := \sup_{f \in \mathcal F}\mathbb E_{(x,y) \sim \mu \otimes \nu}[f(x,y)]. $$ I wish to find choices for $\mathcal F \subseteq \mathcal C_b(X^2$) for which $d_{\mathcal F}$ is a metric on $\mathcal P(X)$.

Example: Let $\mathcal U = \mathcal C_b(X) \cap \operatorname{Lip}_1(X)$ be the space of bounded $1$-Lipschitz continuous functions on $X$ and take $\mathcal F_1 := \{\operatorname{grad}(u)| u \in \mathcal U\}$, where $\operatorname{grad}(u): X \times X \rightarrow \mathbb R$ is defined by $\operatorname{grad}(u)(x, y) := u(x) - u(y)$. Then $$d_{\mathcal F_1}(\mu,\nu) = W_1(\mu,\nu) := \sup_{u \in \mathcal U}\mathbb E_{x \sim \mu}[u(x)] - \mathbb E_{y \sim \nu}[u(y)] $$ is precisely the dual formulation of the well-known Wasserstein $W_1$ distance. One could consider similar constructions for Wasserstein $W_p$ distances using Kantorovich duality.

Consider the following (roughly) separate questions.

Question 1: Prescribe choices for $\mathcal F \subseteq \mathcal C_b(X^2)$ (other than the $\mathcal F_p$ above) for which $d_{\mathcal F}$ is a metric on $\mathcal P(X)$.

Question 2 (Topology induced by $d_{\mathcal F}$): If $\mu$ is fixed and we have a sequence $(\nu_n)_{n\in \mathbb N}$ with $d_{\mathcal F}(\mu,\nu_n) \rightarrow 0$, what can be said about the convergence $\nu_n \rightarrow \mu$ (in some standard sense) ?

Question 3: Prescribe choices for $\mathcal F \subseteq \mathcal C_b(X^2)$ such that the solution of $\inf_{\nu \in \mathcal P(X)} d_{\mathcal F}(\mu,\nu)$ unique (i.e $=\mu$) for every $\mu \in \mathcal P(X)$.

Context: I'm trying to build some theoretical understanding of the concept of "generative adversarial networks" (a flexible way to fit a parametric model to the distribution of observed data). This is an interesting field of research in certain islands of what is now generally called "machine learning". My ramblings have let me to consider the above problems. I've provided these details on the context in hope that some theorists here may perhaps find vocation in these fields (or not!).

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    $\begingroup$ It seems to me that in your definition of $d_\mathcal{F}$, you need to take the inf over all possible couplings? When you write $\mu \otimes \nu$ it looks like you are specifically using product measure, but then you won't have $d_\mathcal{F}(\mu,\mu) =0$. $\endgroup$
    – Nate Eldredge
    Commented Jun 25, 2018 at 16:19
  • $\begingroup$ @NateEldredge No you don't need to take such an inf over couplings (are you possibly confusing this with the primal formulation of Wasserstein distances ?). $\endgroup$
    – dohmatob
    Commented Jun 25, 2018 at 17:36
  • $\begingroup$ Well, the point is that for general functions $f$, the quantity $\mathbb{E}_{x \sim \mu, y \sim \nu} f(x,y)$ is not even well defined, because it depends on the particular coupling of $x,y$ being used. In the dual formulation of Wasserstein distance, $f$ is of the special form $f(x,y) = v(x) - v(y)$ and then this quantity is actually independent of the coupling chosen (linearity of expectation). So if you want to formulate your problem like this, you need restrictions on $\mathcal{F}$ just to get started. $\endgroup$
    – Nate Eldredge
    Commented Jun 25, 2018 at 18:02
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    $\begingroup$ Are you familiar with the more general Kantorovich duality, which tells you the appropriate $\mathcal{F}$ to recover $\inf \mathbb{E} c(x,y)$ for appropriate cost functions $c$, the inf taken over all couplings? For instance you can recover the $W_p$ Wasserstein distances this way. See for instance Theorem 5.10 of Villani's Optimal Transport book. $\endgroup$
    – Nate Eldredge
    Commented Jun 25, 2018 at 18:06
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    $\begingroup$ It certainly makes sense as a definition. But the requirement to get $d_{\mathcal{F}}(\mu,\mu)=0$ is going to place tight restrictions on $\mathcal{F}$. $\endgroup$
    – Nate Eldredge
    Commented Jun 25, 2018 at 19:16

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