  We propose to compute Wasserstein barycenters (WBs) by solving for Monge maps
with variational principle. We discuss the metric properties of WBs and explore
their connections, especially the connections of Monge WBs, to K-means
clustering and co-clustering. We also discuss the feasibility of Monge WBs on
unbalanced measures and spherical domains. We propose two new problems --
regularized K-means and Wasserstein barycenter compression. We demonstrate the
use of VWBs in solving these clustering-related problems.
