and show in Figure \[fig:uvwb\] the resulting barycenter of two Gaussian’s of different samples, $5$k vs. $1$k. We choose $\lambda = \infty$ in . We can also see that Monge maps are absolutely *binary* and *sparse*.

![Interpolating two Gaussian’s of different number of samples by computing the VWB results in a mean isotropic Gaussian.[]{data-label="fig:uvwb"}](4_4_2.pdf){width="0.9\columnwidth"}

On Spherical Domains {#sec:svwb}
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Optimal transport on geometric domains other than the Euclidean domain extends its applications [@solomon2015convolutional; @staib2017parallel; @cui2019spherical]. [@cui2019spherical] relates *spherical power Voronoi diagram* to OT on unit spheres. Inspired by that, we study our VWB on spherical domains and its metric properties.

Let us define a new ground metric on a unit sphere, $\mathbb{S}^2 \times \mathbb{S}^2 \rightarrow \mathbb{R}^{\geq 0}$, as $\Dist(\x, \yL{\IdxCentroid}) = -\ln\langle\x , \yL{\IdxCentroid}\rangle$ and the OT distance: $$\WL{1}' = \underset{\T \in \PSL{T}(\MU,\NU)}{\inf} \EnergyL{8}[\PI] \eqdef -\int_{\mathbb{S}^2} \ln\langle x , T(x) \rangle d\MU(\x)$$ s.t. $\int_{\mathbb{S}^2}(\psi \circ T)d\MU(x) = \int_{\mathbb{S}^2}\psi d\NU(y)$ for all non-negative $\psi$.

![Interpolating two Gaussian distributions on a sphere w.r.t. the VWD. By using the VWB, we can build sparse connection between the two domains via a few discrete relays.[]{data-label="fig:sphere"}](sphere.pdf){width="0.7\columnwidth"}

Following [@cui2019spherical], we define the *power distance* on a sphere as $c'(\x, \yL{\IdxCentroid}) = -\ln\langle \x , \yL{\IdxCentroid} \rangle 