Estimating Common Harmonic Waves of Brain Networks on Stiefel Manifold

Network neuroscience has been widely studied in understanding brain functions as well as the neurobiological underpinnings of cognition and behavior that are related to the development of neuro-disorders. Since the network organization is inherently governed by the harmonic waves (Eigensystem) of the underlying Laplacian matrix, discovering the harmonic-like alterations emerges as a new research interest in understanding the factors behind brain developmental and neurodegenerative diseases, where an unbiased reference space of harmonic waves is often required to quantify the difference across individuals with standard measurement. However, simple arithmetic averaging over the individual harmonic waves is commonly used in current studies, despite that such Euclidean operation might break down the intrinsic data geometry of harmonic waves. To overcome this limitation, we propose a novel manifold optimization framework to find the group mean (aka. common harmonic waves), where each set of harmonic waves from the individual subject is treated as a data sample residing on the Stiefel manifold. To further improve the robustness of learned common harmonic waves to possible outliers, we promote the common harmonic waves to the setting of a geometric median on Stiefel manifold, instead of Frechet mean, by optimizing towards the \( \ell_{1} \)-norm shortest overall geodesic distance on the manifold. We have compared our proposed method with the existing methods on both synthetic and real network data. The experimental results indicate that our proposed approach shows improvements in accuracy and statistical power.