  High dimensional data often contain multiple facets, and several clustering
patterns can co-exist under different variable subspaces, also known as the
views. While multi-view clustering algorithms were proposed, the uncertainty
quantification remains difficult --- a particular challenge is in the high
complexity of estimating the cluster assignment probability under each view,
and sharing information among views. In this article, we propose an approximate
Bayes approach --- treating the similarity matrices generated over the views as
rough first-stage estimates for the co-assignment probabilities; in its
Kullback-Leibler neighborhood, we obtain a refined low-rank matrix, formed by
the pairwise product of simplex coordinates. Interestingly, each simplex
coordinate directly encodes the cluster assignment uncertainty. For multi-view
clustering, we let each view draw a parameterization from a few candidates,
leading to dimension reduction. With high model flexibility, the estimation can
be efficiently carried out as a continuous optimization problem, hence enjoys
gradient-based computation. The theory establishes the connection of this model
to a random partition distribution under multiple views. Compared to
single-view clustering approaches, substantially more interpretable results are
obtained when clustering brains from a human traumatic brain injury study,
using high-dimensional gene expression data.
  KEY WORDS: Co-regularized Clustering, Consensus, PAC-Bayes, Random Cluster
Graph, Variable Selection
