supports and measures. $\yL{\IdxCentroid}$ and $\NUL{\yL{\IdxCentroid}}$ specify the location and measure of the $\IdxCentroid$th Dirac measure.

Variational Wasserstein Barycenters {#sec:vwb}
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Solving the WB problem relies on alternatively solving $\N$ OT problems and updating the barycenter, $\NU$. Eventually, $\NU$ minimizes the average WD between empirical distributions and the barycenter. A discrete distribution $\NU$ consists of support and measure $(\bm{\y}, \bm{\NU}) = \{(\yL{\IdxCentroid}, \NUL{\IdxCentroid})\}_{\IdxCentroid=1}^{\K}$. Updating both of them, e.g., [@ye2017fast], is difficult and even troublesome in some cases (see Appendix). In this paper, we follow [@cuturi2014fast] and only update one of them while fixing the other throughout the optimization.

Discrete Barycenters via VOT {#sec:wbvot}
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We first solve $N$ VOT problems : $$\label{eq:vwb}
\begin{gathered}
    \min_{\{\bm{\hL{i}}\}_{i=1}^{\N}} \EnergyL{5}[\{\bm{\hL{i}}\}] \\
    \eqdef \frac{1}{\N} \sum_{i=1}^{\N} \left( \int_{\bm{0}}^{\bm{\hL{i}}} \sum_{\IdxCentroid=1}^{\K} \int_{\RegionL{i, \IdxCentroid}} d\MUL{i}(x) d\hL{i, \IdxCentroid} - \sum_{\IdxCentroid = 1}^{\K}\NUL{\IdxCentroid}\hL{i, \IdxCentroid} \right)\\ \nonumber
    \end{gathered}$$ Its derivative w.r.t. the VOT optimizer $\hL{i, \IdxCentroid}$ is $$\label{eq:vwb_dh}
    \nabla \EnergyL{5}[\bm{\hL{i}}] = \left\{\pdv{\EnergyL{5}}{\hL{i, \IdxCentroid}} = \int_{\RegionL{i, \IdxCentroid}} d\MUL{i}(\x) - \NUL{\IdxCentroid}\right\}_{\IdxCentroid = 1}^{\K},$$ which, in practice, can be replaced by its stochastic version, $$\label{eq:vwb_sdh}
    \pdv{\EnergyL{5}}{\hL{i, \IdxCentroid}} \approx \sum_{\x \in \RegionL{i, \IdxCentroid}} \MUL{i}(\x) - \NUL{\IdxCentroid}, \nonumber$$ where $x$’s are now Monte Carlo samples. Then, we can naturally adopt the gradient descent (GD) update: $$\label{eq:vwb_gd}
    \bm{\hL{i}}^{(t+1)} = 