  This paper advocates Riemannian multi-manifold modeling in the context of
network-wide non-stationary time-series analysis. Time-series data, collected
sequentially over time and across a network, yield features which are viewed as
points in or close to a union of multiple submanifolds of a Riemannian
manifold, and distinguishing disparate time series amounts to clustering
multiple Riemannian submanifolds. To support the claim that exploiting the
latent Riemannian geometry behind many statistical features of time series is
beneficial to learning from network data, this paper focuses on brain networks
and puts forth two feature-generation schemes for network-wide dynamic time
series. The first is motivated by Granger-causality arguments and uses an
auto-regressive moving average model to map low-rank linear vector subspaces,
spanned by column vectors of appropriately defined observability matrices, to
points into the Grassmann manifold. The second utilizes (non-linear)
dependencies among network nodes by introducing kernel-based partial
correlations to generate points in the manifold of positive-definite matrices.
Capitilizing on recently developed research on clustering Riemannian
submanifolds, an algorithm is provided for distinguishing time series based on
their geometrical properties, revealed within Riemannian feature spaces.
Extensive numerical tests demonstrate that the proposed framework outperforms
classical and state-of-the-art techniques in clustering brain-network
states/structures hidden beneath synthetic fMRI time series and brain-activity
signals generated from real brain-network structural connectivity matrices.
