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

arXiv: https://arxiv.org/abs/2605.22334 · HTML: https://mlsanity.com/paper/2605.22334/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 57 references that are in the map.
- First posted on arXiv 21 May 2026.

## Builds on (most cited first, 11 of 11)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Statistical exploration of the Manifold Hypothesis](https://mlsanity.com/paper/2208.11665/) (2022)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry](https://mlsanity.com/paper/2505.18193/) (2025)
- [A New Parametrization of Correlation Matrices](https://mlsanity.com/paper/2012.02395/) (2020)
- [Federated PCA on Grassmann Manifold for Anomaly Detection in IoT Networks](https://mlsanity.com/paper/2212.12121/) (2022)
- [Riemannian Geometry-Based EEG Approaches: A Literature Review](https://mlsanity.com/paper/2407.20250/) (2024)
- [Scalable Geometric Learning with Correlation-Based Functional Brain Networks](https://mlsanity.com/paper/2503.23653/) (2025)
- [Graph Laplacian Spectrum and Primary Frequency Regulation](https://mlsanity.com/paper/1803.03905/) (2018)

## Abstract (arXiv)

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.