  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 several 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, experimental results with
both synthetic and real data are presented to demonstrate the flexibility and
scalability of the proposed approach.
