samples and the barycenter, we can fully recover the surjection by only using $\hhL{i}$ at the negligible expense of computing the power distance as in . In this way, for a barycenter of size $\K$ of $\N$ empirical distributions each having $M$ samples, we reduce the storage burden from $\mathcal{O}(\N M\K)$, as it would be for Sinkhorn distance-based methods, to $\mathcal{O}(\N\K)$. This is particularly useful when $M$ is large and when we need to store multiple interpolations between marginals.

Furthermore, with the VWB, we do not even need the original distributions to parameterize the compression maps because our method is based on the geometry of the data and given the height vector $\hhL{i}$ and barycenter supports $\bm{\y}$ we can uniquely partition each original domain with a power Voronoi diagram $\RRegionL{i}$; or, equivalently, the graph of the piece-wise linear function $\theta_{\bm{\h}}(\x) = \underset{\IdxCentroid}{\max} \{\x \yL{\IdxCentroid} + \hL{\IdxCentroid}\}$.

Applications {#sec:exp}
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We demonstrate the use of VWBs with point cloud interpolation and image compression.

Point Cloud Interpolation with Global Invariance {#sec:point_cloud}
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Shape interpolation is a typical application of Wasserstein barycenter techniques. We compute the barycenter that has the minimum weighted average WD to all the marginal shapes. When the marginals are 