  We study in this paper a variant of Wasserstein barycenter problem, which we
refer to as tree-Wasserstein barycenter, by leveraging a specific class of
ground metrics, namely tree metrics, for Wasserstein distance. Drawing on the
tree structure, we propose an efficient algorithmic approach to solve the
tree-Wasserstein barycenter and its variants. The proposed approach is not only
fast for computation but also efficient for memory usage. Exploiting the
tree-Wasserstein barycenter and its variants, we scale up multi-level
clustering and scalable Bayes, especially for large-scale applications where
the number of supports in probability measures is large. Empirically, we test
our proposed approach against other baselines on large-scale synthetic and real
datasets.
