minimizing $\|\bm{\y} - H_{i}\bm{\alpha}_{i}\|_{2}^{2}\ \forall i$, subject to $H_i$ composing a rotation and a translation, i.e. $H_i = [R_i|t_i]$. This can be done by singular value decomposition (SVD) with minimum computational costs. After that, we average all the transformations by separately averaging rotations and translations. With the abuse of notation, we simply denote the process by $\widetilde{H} = \frac{1}{N} \sum_{i} H_i$, but as we know, we need to factorize the rotations into quaternions before averaging them. The final location for the supports $\bm{\y}$ is given by $\widetilde{\bm{\y}} = \widetilde{H}\bm{\y}$.

Vector Quantization and Data Compression {#sec:compress}
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Lloyd’s K-means algorithm was initially proposed for vector quantization and has been a fundamental choice for data compression. It centers at using fewer samples to approximate the entire distribution. In light of the connection between VWBs and K-means, we raise the problem of compressing multiple distributional data as a whole with Wasserstein barycenters and propose the VWB as a natural choice. It shares the same objective as the WB. Intuitively, we use sum of WDs to measure the compression error.

By using VOT, we obtain a surjection from each domain to the barycenter. Because we optimize over the height vector $\hhL{i}$ , given empirical 