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

Area of Science:

• Neuroscience
• Machine Learning
• Computational Biology

Background:

• Functional brain networks are summarized by correlation matrices.
• Standard analyses often ignore the complex geometry of correlation space, limiting scalability and accuracy.
• Existing geometric methods can be computationally intensive and lack closed-form solutions.

Purpose of the Study:

• To introduce a scalable geometric framework for analyzing functional brain networks.
• To enable closed-form statistical modeling without complex manifold optimization.
• To improve sensitivity and predictive performance in machine learning tasks for neuroscience.

Main Methods:

• Developed the Off-log metric, transforming correlation matrices into symmetric zero-diagonal matrices for closed-form analysis.
• Utilized Grassmannian subspace discrimination for subject comparison via principal-angle distances.
• Integrated these components into standard machine learning workflows for inference, regression, and classification.

Main Results:

• The Off-log metric enhanced sensitivity in permutation tests and matched or exceeded baseline methods in classification.
• Grassmannian subspace discrimination consistently outperformed Euclidean baselines, identifying disease-relevant networks.
• Brain-age prediction performance was comparable to existing Riemannian metrics.

Conclusions:

• Geometry-aware representations of functional brain networks significantly improve sensitivity and predictive performance.
• The proposed framework offers a scalable and straightforward approach for neuroscience research and clinical applications.
• These methods advance the analysis of brain networks in conditions like Parkinson's disease and psychosis.