forward* $\MU$ to $\NU$, i.e. $\NU = \T\#\MU$; more rigorously, for any measurable set $B \subset \Y,\ \NU[B] = \MU[\TU{-1}(B)]$. We direct readers to [@villani2003topics; @peyre2019computational] for more on OT. In this paper, we focus on Monge OT. In particular, we narrow our discussion to $\X, \Y \subseteq \mathbb{R}^{n}$, $c(\x, \y)= \|\x - \y\|_{2}$, and $p = 2$ unless specified otherwise. Hence, we compute $\WL{2}$.

Variational Optimal Transportation {#sec:vot}
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Directly computing a Monge map is highly intractable and variational methods have been adopted by many researchers. [@de2012blue; @gu2013variational; @levy2015numerical] offer three variational formulations. We follow [@gu2013variational] and in this paper refer to it as *variational OT* or VOT.

Suppose $\NU$ is supported on $\K$ discrete atoms $\bm{\y} = \{\yL{\IdxCentroid}\}_{\IdxCentroid=1}^{\K} \subset \Y$. The problem becomes *semi-discrete OT*. VOT starts with a piece-wise linear function $\theta_{\bm{\h}}(\x) = \underset{\IdxCentroid}{\max} \{\x \yL{\IdxCentroid} + \hL{\IdxCentroid}\}$. Each $\yL{\IdxCentroid}$ associates with a *height* $\hL{\IdxCentroid}$. The gradient, $\nabla\theta_{\bm{\h}}(x) = \yL{\IdxCentroid}$ where $\IdxCentroid$ induces the maximum, serves as a map from $\X$ to $\Y$. It induces a graph: $ \bm{\Region}_{\bm{\h}} = \bigcup\limits_{\IdxCentroid=1}^{\K} \left(\RegionL{\bm{\h}}\right)_{\IdxCentroid}, \left(\RegionL{\bm{\h}}\right)_{\IdxCentroid} \eqdef \{\x \in \X\ |\ \x\yL{\IdxCentroid} + \hL{\IdxCentroid} \geq \x\yL{\IdxCentroidSecond} + \hL{\IdxCentroidSecond}, \forall \IdxCentroidSecond \neq \IdxCentroid\}$. For simplicity, we remove $\bm{\h}$ and use $\RegionL{\IdxCentroid}$ 