the update rule for the measure $\bm{\NU}$. is not differentiable w.r.t. $\bm{\NU}$. Still, we follow [@cuturi2014fast; @mi2018regularized] and give the critical point and include the derivation in Appendix. $$\label{eq:vwb_dv}
\begin{split}
    \NUUL{*}{\IdxCentroid}
    = \frac{1}{N}\sum_{i = 1}^{N} \int_{\RegionLU{i, \IdxCentroid}{*}} d\MUL{i}(\x)
    \approx \frac{1}{\N} \sum_{i = 1}^{\N} \sum_{ \x \in \RegionLU{i, \IdxCentroid}{*}} \MUL{i}(\x), \nonumber
\end{split}$$ where $\RegionLU{i, \IdxCentroid}{*} = \{\x\in\XL{i}\ |\ \|\x-\yL{\IdxCentroid}\|_{2}^{2} < \|\x-\yL{\IdxCentroidSecond}\|_{2}^{2}\ \forall \IdxCentroidSecond \neq \IdxCentroid\}$. $\NULU{\IdxCentroid}{*}$ coincides with the result of Lloyd’s K-means algorithm in which the measure on each centroid accumulates all its assigned empirical measures.

Now that we have derived the rules for updating $\T$ and $\NU$, we summarize our algorithm for computing the VWB of a collection of measures $\{\MUL{i}\}_{i}$ in Appendix. As for the initial guess of the barycenter, if not specified, we can either run Lloyd’s algorithm on all the measures as a whole and adopt the resulting $\K$ centroids or uniformly sample the space $\Y$. The choice of the measure on the centroids depends on the specific application. A ubiquitous choice is uniform Dirac measures, i.e. $\NUL{\IdxCentroid} = \frac{1}{\K} \delta[\yL{\IdxCentroid}]$. Figure \[fig:vwb\] suggests that by regarding the WD as the metric, we can find a mean shape on the same manifold, if there exists one.

