We propose a novel approach to the problem of multilevel clustering, which
aims to simultaneously partition data in each group and discover grouping
patterns among groups in a potentially large hierarchically structured corpus
of data. Our method involves a joint optimization formulation over several
spaces of discrete probability measures, which are endowed with Wasserstein
distance metrics. We propose a number of variants of this problem, which admit
fast optimization algorithms, by exploiting the connection to the problem of
finding Wasserstein barycenters. Consistency properties are established for the
estimates of both local and global clusters. Finally, experiment results with
both synthetic and real data are presented to demonstrate the flexibility and
scalability of the proposed approach.

本文提出一种新的方法来解决多层次聚类问题，该方法旨在同时将数据在每个组中分区，并在潜在的大型分层结构数据集中发现组间的分组模式。我们的方法涉及到多个离散概率测度空间上的联合优化方案，这些测度空间具有 Wasserstein 距离度量。通过利用与 Wasserstein barycenter 问题的联系，我们提出了该问题的许多变体，这些变体可以采用快速的优化算法。本文还建立了局部和全局聚类估计的一致性性质。最后，使用合成和真实数据展示了所提出方法的灵活性和可扩展性。