🤖 AI Summary
This study addresses the limitations of conventional fMRI functional connectivity analyses, which overlook the non-Euclidean geometric structure of correlation matrices, thereby constraining statistical sensitivity and scalability. To overcome this, the authors propose a scalable geometric framework that maps correlation matrices to symmetric zero-diagonal matrices via the Off-log metric, enabling closed-form statistical modeling. Furthermore, they introduce a comparison of eigenspaces using principal angles on the Grassmann manifold, effectively resolving ambiguities arising from eigenvector sign and basis indeterminacy. Evaluated on Parkinson’s disease, psychiatric disorders, and three aging-related fMRI datasets, the method achieves classification performance comparable to or better than existing approaches, significantly enhances sensitivity in permutation testing, and accurately identifies disease-relevant brain networks.
📝 Abstract
Correlation matrices are fundamental summaries of functional brain networks, yet standard analyses often treat entries independently, ignoring the curved geometry of correlation space. Existing geometric methods frequently lack closed-form operations or depend on arbitrary region ordering, limiting scalability. We introduce a scalable geometric framework with two components: (i) the Off-log metric, a smooth transformation mapping correlation matrices to symmetric zero-diagonal matrices. This enables closed-form expressions for distances, Frechet means, and linear models, allowing standard statistical modeling without complex manifold optimization. (ii) Grassmannian subspace discrimination, which compares subjects via principal-angle distances between eigenvector subspaces, resolving inherent sign and basis ambiguities. Both components integrate into standard machine-learning workflows for inference, regression, and classification. Validated across two clinical cohorts (Parkinson's and psychosis) and three ageing fMRI datasets, the Off-log metric increased sensitivity in permutation tests and matched or exceeded Riemannian and Euclidean baselines in classification. Brain-age prediction performance was comparable, with Riemannian metrics excelling in two of three cohorts. The Grassmannian method consistently outperformed Euclidean baselines, highlighting disease-relevant networks. Overall, geometry-aware representations improve sensitivity and predictive performance while remaining straightforward to deploy at scale.