Overlapping Decomposition for Gaussian Graphical Modeling

Correlation based graphical models are developed to detect the dependence relationships among random variables and provide intuitive explanations for these relationships in complex systems. Most of the existing works focus on learning a single correlation based graphical model for all the random variables. However, it is difficult to understand and interpret the massive dependencies of the variables learned from a single graphical model at a global level especially when the graph is large. In order to provide a clearer understanding for the dependence relationships among a large number of random variables, in this paper, we propose the problem of estimating an overlapping decomposition for the Gaussian graphical model of a large scale to generate overlapping sub-graphical models, where strong and meaningful correlations remain in each subgraph with a small scale. Specifically, we propose a greedy algorithm to achieve the overlapping decomposition for the Gaussian graphical model. A key technique of the algorithm is that the problem of solving a (k + 1)-node Gaussian graphical model can be approximately reduced to the problem of solving a one-step vector regularization problem based on a solved k-node Gaussian graphical model with theoretical guarantee. Based on this technique, a greedy expansion algorithm is proposed to generate the overlapping subgraphs. Moreover, we extend the proposed method to deal with dynamic graphs where the dependence relationships among random variables vary with the time. We evaluate the proposed methods on synthetic dataset and a real-life traffic dataset, and the experimental results show the superiority of the proposed methods.