Riemannian geometry meets fMRI: the advantages of modeling correlation manifolds and eigenvector subspaces

• URL: http://arxiv.org/abs/2605.22334v1
• Date: Thu, 21 May 2026 11:22:31 GMT
• Title: Riemannian geometry meets fMRI: the advantages of modeling correlation manifolds and eigenvector subspaces
• Abstract summary: Correlation matrices are fundamental summaries of functional brain networks.<n>Existing geometric methods frequently lack closed-form operations or depend on arbitrary region ordering.<n>We introduce a scalable geometric framework with two components: the Off-log metric and Grassmannian subspace discrimination.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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