College of Liberal Arts
Multiple graphical models have been widely used to describe structural changes of a network responding to certain experimental conditions, which are expressed in terms of conditional dependence between interacting units. For instance, time-varying functional connectivity in brain image analysis is described by node connectivity over a dynamic network, with each node corresponding to one region of interest. Motivated from network analysis under different experimental conditions, such as gene networks for disparate cancer subtypes, these researchers model structural changes over multiple networks with possible heterogeneities. In particular, they estimate multiple precision matrices describing dependencies among interacting units through maximum penalized likelihood. Of particular interest are homogeneous groups of similar entries across and zero-entries of these matrices, referred to as clustering and sparseness structures, respectively. A non-convex method is proposed to seek a sparse representation for each matrix and identify clusters of the entries across the matrices. An efficient method is developed on the basis of difference convex programming, the augmented Lagrangian method, and the blockwise coordinate descent method, which is scalable to hundreds of graphs of thousands nodes through a simple necessary and sufficient partition rule, which divides nodes into smaller disjoint subproblems excluding zero-coefficients nodes for arbitrary graphs with convex relaxation. Theoretically, a finite-sample error bound is derived for the proposed method to reconstruct the clustering and sparseness structures. This leads to consistent reconstruction of these two structures simultaneously, permitting the number of unknown parameters to be exponential in the sample size, and yielding the optimal performance of the oracle estimator as if the true structures were given a priori. Simulation studies suggest that the method enjoys the benefit of pursuing these two disparate kinds of structures, and compares favorably against its convex counterpart in the accuracy of structure pursuit and parameter estimation.