Clustering Time-Varying Connectivity Networks By Riemannian Geometry: The Brain-Network Case

In response to the demand on data-analytic tools that monitor time varying connectivity patterns within brain networks, the present paper introduces a framework for clustering (unsupervised learning) of dynamically evolving connectivity states of networks. This work advocates learning of network dynamics on Riemannian manifolds, capitalizing on the well-known fact that popular features in statistics enjoy that structure: (Partial) correlations or covariances can be mapped to the manifold of positive (semi-)definite symmetric matrices, while low-rank linear subspaces can be considered as points of the Grassmannian. Sequences of such features, collected over time and across a network, are mapped to sequences of points on a Riemannian manifold, and a sequence that corresponds to a specific state of the network forms a cluster or submanifold. Geometry is exploited in a novel way to demonstrate the rich potential of the proposed learning method for monitoring time-varying network patterns by outperforming state-of-the-art techniques on synthetic brain-network data.