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Learning Common Harmonic Waves on Stiefel Manifold -- A New Mathematical Approach for Brain Network Analyses

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Document pages: 25 pages

Abstract: Converging evidence shows that disease-relevant brain alterations do notappear in random brain locations, instead, its spatial pattern follows largescale brain networks. In this context, a powerful network analysis approachwith a mathematical foundation is indispensable to understand the mechanism ofneuropathological events spreading throughout the brain. Indeed, the topologyof each brain network is governed by its native harmonic waves, which are a setof orthogonal bases derived from the Eigen-system of the underlying Laplacianmatrix. To that end, we propose a novel connectome harmonic analysis frameworkto provide enhanced mathematical insights by detecting frequency-basedalterations relevant to brain disorders. The backbone of our framework is anovel manifold algebra appropriate for inference across harmonic waves thatovercomes the limitations of using classic Euclidean operations on irregulardata structures. The individual harmonic difference is measured by a set ofcommon harmonic waves learned from a population of individual Eigen systems,where each native Eigen-system is regarded as a sample drawn from the Stiefelmanifold. Specifically, a manifold optimization scheme is tailored to find thecommon harmonic waves which reside at the center of Stiefel manifold. To thatend, the common harmonic waves constitute the new neuro-biological bases tounderstand disease progression. Each harmonic wave exhibits a uniquepropagation pattern of neuro-pathological burdens spreading across brainnetworks. The statistical power of our novel connectome harmonic analysisapproach is evaluated by identifying frequency-based alterations relevant toAlzheimer s disease, where our learning-based manifold approach discovers moresignificant and reproducible network dysfunction patterns compared to Euclidianmethods.

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