Multi-Scale Factor Analysis of High-Dimensional Functional Connectivity in Brain Networks

We consider the challenges in modeling and estimating high-dimensional dependence in real complex networks typically with large number of nodes arranged in a hierarchical and modular structure. We develop a multi-scale factor analysis (MSFA) model which partitions the massive spatio-temporal data defined over the complex networks into a finite set of regional clusters. To achieve further dimension reduction, the signals in each cluster are represented by a small number of latent factors. The correlation matrix for all nodes in the network can be approximated by lower-dimensional sub-structures derived from the cluster-specific factors. To estimate regional connectivity between numerous nodes (within each cluster), we apply principal components analysis (PCA) to extract factors which are derived as the optimal reconstruction of the observed signals under the squared error loss. Then, we estimate global connectivity (between clusters or sub-networks) based on the factors across regions using the RV-coefficient as the cross-dependence measure. This gives a reliable and computationally efficient multi-scale analysis of both regional and global dependencies of the large networks. Simulation results show that our proposed MSFA-based estimator improves accuracy of connectivity estimation in high-dimensional settings. The novel approach is applied to estimate brain connectivity networks using functional magnetic resonance imaging (fMRI) data. Results on resting-state fMRI reveal interesting modular and hierarchical organization of human brain networks during rest.