  Human brain functional connectivity (FC) is often measured as the similarity
of functional MRI responses across brain regions when a brain is either resting
or performing a task. This paper aims to statistically analyze the dynamic
nature of FC by representing the collective time-series data, over a set of
brain regions, as a trajectory on the space of covariance matrices, or
symmetric-positive definite matrices (SPDMs). We use a recently developed
metric on the space of SPDMs for quantifying differences across FC
observations, and for clustering and classification of FC trajectories. To
facilitate large scale and high-dimensional data analysis, we propose a novel,
metric-based dimensionality reduction technique to reduce data from large SPDMs
to small SPDMs. We illustrate this comprehensive framework using data from the
Human Connectome Project (HCP) database for multiple subjects and tasks, with
task classification rates that match or outperform state-of-the-art techniques.
