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

Sparse inverse covariance estimation (i.e., edge detection) is an important research problem in recent years, where the goal is to discover the direct connections between a set of nodes in a networked system based upon the observed node activities. Existing works mainly focus on unimodal distributions, where it is usually assumed that the observed activities are generated from a single Gaussian distribution (i.e., one graph). However, this assumption is too strong for many real-world applications. In many real-world applications (e.g., brain networks), the node activities usually exhibit much more complex patterns that are difficult to be captured by one single Gaussian distribution. In this work, we are inspired by Latent Dirichlet Allocation (LDA) [4] and consider modeling the edge detection problem as estimating a mixture of multiple Gaussian distributions, where each corresponds to a separate sub-network. To address this problem, we propose a novel model called Gaussian Mixture Graphical Lasso (MGL). It learns the proportions of signals generated by each mixture component and their parameters iteratively via an EM framework. To obtain more interpretable networks, MGL imposes a special regularization, called Mutual Exclusivity Regularization (MER), to minimize the overlap between different sub-networks. MER also addresses the common issues in read-world data sets, i.e., noisy observations and small sample size. Through the extensive experiments on synthetic and real brain data sets, the results demonstrate that MGL can effectively discover multiple connectivity structures from the observed node activities.