Gaussian graphical models are useful to analyze and visualize conditional dependence relationships between interacting units. Motivated from network analysis under different experimental conditions, such as gene networks for disparate cancer subtypes, we model structural changes over multiple networks with possible heterogeneities. In particular, we 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 nonconvex method is proposed to seek a sparse representation for each matrix and identify clusters of the entries across the matrices. Computationally, we develop an efficient method 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.
Bibliographical noteFunding Information:
Yunzhang Zhu (E-mail: firstname.lastname@example.org), and Xiaotong Shen (E-mail: email@example.com), School of Statistics, University of Minnesota, Minneapolis, MN 55455. Wei Pan, Division of Biostatistics, University of Minnesota, Minneapolis, MN 55455 (E-mail: firstname.lastname@example.org). This research was supported in part by the National Science Foundation Grant DMS-1207771 and National Institutes of Health Grants R01GM081535, HL65462, and R01HL105397. The authors thank the editors and the reviewers for helpful comments and suggestions.
- Multiple networks
- Signaling network inference
- Simultaneous pursuit of sparseness and clustering