  Converging evidence shows that disease-relevant brain alterations do not
appear in random brain locations, instead, its spatial pattern follows large
scale brain networks. In this context, a powerful network analysis approach
with a mathematical foundation is indispensable to understand the mechanism of
neuropathological events spreading throughout the brain. Indeed, the topology
of each brain network is governed by its native harmonic waves, which are a set
of orthogonal bases derived from the Eigen-system of the underlying Laplacian
matrix. To that end, we propose a novel connectome harmonic analysis framework
to provide enhanced mathematical insights by detecting frequency-based
alterations relevant to brain disorders. The backbone of our framework is a
novel manifold algebra appropriate for inference across harmonic waves that
overcomes the limitations of using classic Euclidean operations on irregular
data structures. The individual harmonic difference is measured by a set of
common harmonic waves learned from a population of individual Eigen systems,
where each native Eigen-system is regarded as a sample drawn from the Stiefel
manifold. Specifically, a manifold optimization scheme is tailored to find the
common harmonic waves which reside at the center of Stiefel manifold. To that
end, the common harmonic waves constitute the new neuro-biological bases to
understand disease progression. Each harmonic wave exhibits a unique
propagation pattern of neuro-pathological burdens spreading across brain
networks. The statistical power of our novel connectome harmonic analysis
approach is evaluated by identifying frequency-based alterations relevant to
Alzheimer's disease, where our learning-based manifold approach discovers more
significant and reproducible network dysfunction patterns compared to Euclidian
methods.
