Fast Sequential Clustering in Riemannian Manifolds for Dynamic and Time-Series-Annotated Multilayer Networks

Abstract

This work exploits Riemannian manifolds to build a sequential-clustering framework able to address a wide variety of clustering tasks in dynamic multilayer (brain) networks via the information extracted from their nodal time-series. The discussion follows a bottom-up path, starting from feature extraction from time-series and reaching up to Riemannian manifolds (feature spaces) to address clustering tasks such as state clustering, community detection (a.k.a. network-topology identification), and subnetwork-sequence tracking. Kernel autoregressive-moving-average modeling and kernel (partial) correlations serve as case studies of generating features in the Riemannian manifolds of Grassmann and positive-(semi)definite matrices, respectively. Feature point-clouds form clusters which are viewed as submanifolds through the Riemannian multi-manifold modeling. A novel sequential-clustering scheme of Riemannian features is also established: landmark points are first identified in a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">non-random</i> way to reveal the underlying geometric information of the feature point-cloud, and, then, a fast sequential-clustering scheme is brought forth that takes advantage of Riemannian distances and the angular information on tangent spaces. By virtue of the landmark points and the sequential processing of the Riemannian features, the computational complexity of the framework is rendered free from the length of the available time-series data. The effectiveness and computational efficiency of the proposed framework is validated by extensive numerical tests against several state-of-the-art manifold-learning and (brain-)network-clustering schemes on synthetic as well as real functional-magnetic-resonance-imaging (fMRI) and electro-encephalogram (EEG) data.