with light color. Until $1500$ supports, ratio $0.3$, our algorithm produces stable approximations that have almost zero variance.

![WDs between two Gaussian’s vs. number of supports.[]{data-label="fig:ship2"}](ship_error.pdf){width="0.9\columnwidth"}

On Unbalanced Measures {#sec:unbalanced_ot}
----------------------

When measures are not probabilities or, more generally their integrals do not equal, we are solving *unbalanced OT*. [@benamou2003numerical] first explored the problem. Researchers since then have offered various formulations and perspectives to approach it, e.g. [@liero2018optimal] adding $f$-divergences as regularizers instead of constraints on the marginals. Here, we discuss VOT and VWBs for unbalanced measures. Without loss of generality, let us assume $\int_{\X}d\MU(\x) = w,\ \sum_{\IdxCentroid = 1}^{\K} \NUL{\IdxCentroid} = 1$. We denote the mass in each power Voronoi cell by $w_{\RegionL{\IdxCentroid}} = \int_{\RegionL{\IdxCentroid}}d\MU(\x)$. Inspired by the discussion in [@peyre2019computational], we propose to penalize the quadratic mismatch of the mass for each cell $\IdxCentroid$. $$\label{eq:unbalanced_vot}
\begin{gathered}
\min_{\bm{\Region}}\ \EnergyL{7}[\bm{\Region}] \eqdef 
\int_{\X}  \|\x - T(\x)\|^2_2  d\MU(\x) + \lambda \sum_{\IdxCentroid = 1}^{\K}  \left( w_{\IdxCentroid} - \NUL{\IdxCentroid}\right)^2,
\end{gathered}$$ s.t. $\sum_{\IdxCentroid} w_{\RegionL{\IdxCentroid}} = w$. If $\lambda \rightarrow \infty, w = 1$, reverses to .

In the following, we discuss in two cases: $\NUL{\IdxCentroid} = \frac{1}{\K}$ and a more general one, $\NUL{\IdxCentroid} \in (0,1),\ \sum_{\IdxCentroid = 1}^{\K} \NUL{\IdxCentroid} = 1$.

![Mass difference over iterations 