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 according to
Riemannian multi-manifold modeling. A novel sequential-clustering scheme
of Riemannian features is also established: feature points are first
sampled in a non-random way to reveal the underlying geometric
information, 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.