\times \Y$. Then, we denote by $\PS(\MU,\NU) = \{\PI \in \MUS(\X \times \Y)\ |\ \int_{\X}d\PI(\x, \y) = d\NU(\y), \int_{\Y}d\PI(\x, \y) = d\MU(\x) \}$ the set of all transportation maps $\PI$ between $\MU$ and $\NU$. Thus, $\PI$ is also the joint distribution of $\MU$ and $\NU$ and $d\PI(\x, \y)$ specifies the *mass* transported across $\x$ and $\y$. In addition, we use $\Dist(\x, \y): \X \times \Y \rightarrow \mathbb{R}^{\geq 0}$ to specify the transportation cost between $\x$ and $\y$.

Optimal Transportation {#sec:ot}
----------------------

The OT problem is to minimize the total transportation cost: $$\min_{\PI \in \PS(\MU,\NU)} \EnergyL{1}[\PI] = \int_{\X \times \Y} \Dist(\x, \y)^p d\PI(\x, \y), \nonumber$$ where $p \in [1, \infty)$ indicates the moment of the cost function. Then, we call this minimum the *p-Wasserstein distance*: $$\WL{p} = \underset{\PI \in \PS(\MU,\NU)}{\inf} \left(\EnergyL{1}[\PI]\right)^{1/p}. \nonumber$$ The above is the well-known Kantorovich’s OT formulation that allows a partial map that splits the measure $d\MU(\x)$ during transportation. In Monge’s original version, each location $\x$ has a unique correspondence $\y$. If we define such a map as $\T: \X \rightarrow \Y$, then we have $d\PIL{\T}(\x, \y) \equiv d\MU(x)\delta[\y = \T(\x)]$ and Monge OT: $$\label{eq:monge}
   \TU{*} = \argmin_{\PI_{\T} \in \PS(\MU,\NU)} \EnergyL{1}[\PI_{\T}] \equiv \int_{\X} \Dist(\x, \T(\x))^p d\MU(\x)$$ $\T$ *pushes 