Gaussian Mixture Graphical Lasso with Application to Edge Detection in Brain Networks

Sparse inverse covariance estimation (i.e., edge de-tection) is an important research problem in recent years, wherethe goal is to discover the direct connections between a set ofnodes in a networked system based upon the observed nodeactivities. Existing works mainly focus on unimodal distributions,where it is usually assumed that the observed activities aregenerated from asingleGaussian distribution (i.e., one graph).However, this assumption is too strong for many real-worldapplications. In many real-world applications (e.g., brain net-works), the node activities usually exhibit much more complexpatterns that are difficult to be captured by one single Gaussiandistribution. In this work, we are inspired by Latent DirichletAllocation (LDA) [4] and consider modeling the edge detectionproblem as estimating a mixture ofmultipleGaussian distribu-tions, where each corresponds to a separate sub-network. Toaddress this problem, we propose a novel model called GaussianMixture Graphical Lasso (MGL). It learns the proportionsof signals generated by each mixture component and theirparameters iteratively via an EM framework. To obtain moreinterpretable networks, MGL imposes a special regularization,called Mutual Exclusivity Regularization (MER), to minimize theoverlap between different sub-networks. MER also addresses thecommon issues in read-world data sets,i.e., noisy observationsand small sample size. Through the extensive experiments onsynthetic and real brain data sets, the results demonstrate thatMGL can effectively discover multiple connectivity structuresfrom the observed node activities