  Scientific studies increasingly collect multiple modalities of data to
investigate a phenomenon from several perspectives. In integrative data
analysis it is important to understand how information is heterogeneously
spread across these different data sources. To this end, we consider a
parametric clustering model for the subjects in a multi-view data set (i.e.
multiple sources of data from the same set of subjects) where each view
marginally follows a mixture model. In the case of two views, the dependence
between them is captured by a cluster membership matrix parameter and we aim to
learn the structure of this matrix (e.g. the zero pattern). First, we develop a
penalized likelihood approach to estimate the sparsity pattern of the cluster
membership matrix. For the specific case of block diagonal structures, we
develop a constrained likelihood formulation where this matrix is constrained
to be block diagonal up to permutations of the rows and columns. To enforce
block diagonal constraints we propose a novel optimization approach based on
the symmetric graph Laplacian. We demonstrate the performance of these methods
through both simulations and applications to data sets from cancer genetics and
neuroscience. Both methods naturally extend to multiple views.
