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

Riemannian-geometry-based modeling and clustering of network-wide non-stationary time series: The brain-network case