Let $\mathcal X$ be a Polish space, and let $(N_x)_{x \in \mathcal X}$ be a system of closed neighborhoods in $\mathcal X$. Define $\Omega := \{(x,x') \in \mathcal X^2 \mid N_x \cap N_{x'} = \emptyset\}$, assumed to be open in $\mathcal X^2$. For example in a Banach space $\mathcal X$, an example would be $N_x = x + \varepsilon\mathbb B$ with $\varepsilon > 0$, and this would lead to $\Omega := \{(x,x') \in \mathcal X^2 \mid \|x-x'\| > 2\varepsilon\}$.

Now, let $\mu$ and $\nu$ be probability measures, and consider the quantity $c_\Omega(\mu,\nu)$ defined by

$$ c_\Omega(\mu,\nu) := \inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega). $$ This is nothing but the transportation distance between $\mu$ and $\nu$, for the binary ground cost given by $c_\Omega(x,x') = 1$ if $(x,x') \in \Omega$; $c_\Omega(x,x') = 0$ else.

Finally, define $$ d_N(\mu,\nu) := \inf_{a} \text{TV}(a_{\#}\mu,a_{\#}\nu), $$ where the infimum is take over all measurable $a:\mathcal X \rightarrow \mathcal X$ such that $a(x) \in N_x$ for all $x \in \mathcal X$, and $a_{\#}\mu$ denotes the pushforward of $\mu$ under $a$

Question

• Is there any relationship (equality, inequality, etc.) between $c_\Omega(\mu,\nu)$ and $d_N(\mu,\nu)$ ?

Observations

By data-processing inequality, $\text{TV}(a_{\#}\mu,a_{\#}\nu) \le \text{TV}(\mu,\nu)$ and so $d_N(\mu,\nu) \le \text{TV}(\mu,\nu)$.

Let $\mathcal X$ be a Polish space, and let $(N_x)_{x \in \mathcal X}$ be a system of closed neighborhoods in $\mathcal X$. Define $\Omega := \{(x,x') \in \mathcal X^2 \mid N_x \cap N_{x'} = \emptyset\}$, assumed to be open in $\mathcal X^2$. For example in a Banach space $\mathcal X$, an example would be $N_x = x + \varepsilon\mathbb B$ with $\varepsilon > 0$, and this would lead to $\Omega := \{(x,x') \in \mathcal X^2 \mid \|x-x'\| > 2\varepsilon\}$.

Now, let $\mu$ and $\nu$ be probability measures, and consider the quantity $c_\Omega(\mu,\nu)$ defined by

$$ c_\Omega(\mu,\nu) := \inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega). $$ This is nothing but the transportation distance between $\mu$ and $\nu$, for the binary ground cost given by $c_\Omega(x,x') = 1$ if $(x,x') \in \Omega$; $c_\Omega(x,x') = 0$ else.

Finally, define $$ d_N(\mu,\nu) := \inf_{a_1, a_2} \text{TV}({a_1}_{\#}\mu,{a_2}_{\#}\nu), $$ where the infimum is take over all measurable $a_1, a_2:\mathcal X \rightarrow \mathcal X$ such that $a_1(x), a_2(x) \in N_x$ for all $x \in \mathcal X$, and $a_{\#}\mu$ denotes the pushforward of $\mu$ under $a$

Question

• Is there any relationship (equality, inequality, etc.) between $c_\Omega(\mu,\nu)$ and $d_N(\mu,\nu)$ ?

Observations

By data-processing inequality, $\text{TV}(a_{\#}\mu,a_{\#}\nu) \le \text{TV}(\mu,\nu)$ and so $d_N(\mu,\nu) \le \text{TV}(\mu,\nu)$.

Let $\mathcal X$ be a Polish space, and let $(N_x)_{x \in \mathcal X}$ be a system of closed neighborhoods in $\mathcal X$. Define $\Omega := \{(x,x') \in \mathcal X^2 \mid N_x \cap N_{x'} = \emptyset\}$, assumed to be open in $\mathcal X^2$. For example in a Banach space $\mathcal X$, an example would be $N_x = x + \varepsilon\mathbb B$ with $\varepsilon > 0$, and this would lead to $\Omega := \{(x,x') \in \mathcal X^2 \mid \|x-x'\| > 2\varepsilon\}$.

Now, let $\mu$ and $\nu$ be probability measures, and consider the quantity $c_\Omega(\mu,\nu)$ defined by

$$ c_\Omega(\mu,\nu) := \inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega). $$ This is nothing but the transportation distance between $\mu$ and $\nu$, for the binary ground cost given by $c_\Omega(x,x') = 1$ if $(x,x') \in \Omega$; $c_\Omega(x,x') = 0$ else.

Finally, define $$ d_N(\mu,\nu) := \inf_{a} \text{TV}(a_{\#}\mu,a_{\#}\nu), $$ where the infimum is take over all measurable $a:\mathcal X \rightarrow \mathcal X$ such that $a(x) \in N_x$ for all $x \in \mathcal X$, and $a_{\#}\mu$ denotes the pushforward of $\mu$ under $a$

Question

• Is there any relationship (equality, inequality, etc.) between $c_\Omega(\mu,\nu)$ and $d_N(\mu,\nu)$ ?

Let $\mathcal X$ be a Polish space, and let $(N_x)_{x \in \mathcal X}$ be a system of closed neighborhoods in $\mathcal X$. Define $\Omega := \{(x,x') \in \mathcal X^2 \mid N_x \cap N_{x'} = \emptyset\}$, assumed to be open in $\mathcal X^2$. For example in a Banach space $\mathcal X$, an example would be $N_x = x + \varepsilon\mathbb B$ with $\varepsilon > 0$, and this would lead to $\Omega := \{(x,x') \in \mathcal X^2 \mid \|x-x'\| > 2\varepsilon\}$.

Now, let $\mu$ and $\nu$ be probability measures, and consider the quantity $c_\Omega(\mu,\nu)$ defined by

$$ c_\Omega(\mu,\nu) := \inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega). $$ This is nothing but the transportation distance between $\mu$ and $\nu$, for the binary ground cost given by $c_\Omega(x,x') = 1$ if $(x,x') \in \Omega$; $c_\Omega(x,x') = 0$ else.

Finally, define $$ d_N(\mu,\nu) := \inf_{a} \text{TV}(a_{\#}\mu,a_{\#}\nu), $$ where the infimum is take over all measurable $a:\mathcal X \rightarrow \mathcal X$ such that $a(x) \in N_x$ for all $x \in \mathcal X$, and $a_{\#}\mu$ denotes the pushforward of $\mu$ under $a$

Question

• Is there any relationship (equality, inequality, etc.) between $c_\Omega(\mu,\nu)$ and $d_N(\mu,\nu)$ ?

Observations

By data-processing inequality, $\text{TV}(a_{\#}\mu,a_{\#}\nu) \le \text{TV}(\mu,\nu)$ and so $d_N(\mu,\nu) \le \text{TV}(\mu,\nu)$.