Sparsity Structure Estimation for Multi-View Mixture Models
Scientific studies increasingly collect multiple modalities of data to investigate a phenomenon from several perspectives (e.g. each subject in a study might have both clinical and genomic variables). In such integrative studies it is important to understand how information is shared across different data modalities. To this end, we consider a parametric clustering model for the subjects in a multi-view dataset (a fixed set of subjects and several disjoint sets of variables) where each data-view marginally follows a mixture model. In the case of two views, we assume the dependence between the views is captured by a “cluster membership matrix” parameter. Analyzing the structure of this matrix (e.g. the zero pattern) provides interpretable results about how the clusters in one view are related to the clusters in another view. First, we develop a penalized likelihood approach to estimate the sparsity structure of the cluster membership matrix. Next, we develop a constrained likelihood formulation where we assume this matrix is block diagonal up to permutations of the rows and columns. To enforce block diagonal constraints, we propose an optimization approach based on the symmetric, normalized graph Laplacian. We demonstrate the performance of these methods in a simulation study and show how they naturally extend to the general case of more than two views. Analysis of the motivating dataset from breast cancer histology is ongoing.