# Scalable Geometric Learning with Correlation-Based Functional Brain Networks

arXiv: https://arxiv.org/abs/2503.23653 · HTML: https://mlsanity.com/paper/2503.23653/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 93 references that are in the map.
- First posted on arXiv 31 Mar 2025.

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

- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels](https://mlsanity.com/paper/1412.0265/) (2014)
- [Hypothesis Testing For Network Data in Functional Neuroimaging](https://mlsanity.com/paper/1407.5525/) (2014)
- [Graph Neural Networks on SPD Manifolds for Motor Imagery Classification: A Perspective from the Time-Frequency Analysis](https://mlsanity.com/paper/2211.02641/) (2022)
- [Rdimtools: An R package for Dimension Reduction and Intrinsic Dimension Estimation](https://mlsanity.com/paper/2005.11107/) (2020)
- [Comparing multiple latent space embeddings using topological analysis](https://mlsanity.com/paper/2208.12435/) (2022)
- [Simplicial faces of the set of correlation matrices](https://mlsanity.com/paper/1801.00749/) (2018)
- [Comparison of brain connectomes using geodesic distance on manifold:a twin study](https://mlsanity.com/paper/1902.01395/) (2019)
- [Tangent space functional reconfigurations in individuals at risk for alcohol use disorder](https://mlsanity.com/paper/2405.15905/) (2024)

## Built on by (most cited first, 2 of 2)

- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [Riemannian geometry meets fMRI: the advantages of modeling correlation manifolds and eigenvector subspaces](https://mlsanity.com/paper/2605.22334/) (2026)

## Abstract (arXiv)

The correlation matrix is a central representation of functional brain networks in neuroimaging. Traditional analyses often treat pairwise interactions independently in a Euclidean setting, overlooking the intrinsic geometry of correlation matrices. While earlier attempts have embraced the quotient geometry of the correlation manifold, they remain limited by computational inefficiency and numerical instability, particularly in high-dimensional contexts. This paper presents a novel geometric framework that employs diffeomorphic transformations to embed correlation matrices into a Euclidean space, preserving salient manifold properties and enabling large-scale analyses. The proposed method integrates with established learning algorithms - regression, dimensionality reduction, and clustering - and extends naturally to population-level inference of brain networks. Simulation studies demonstrate both improved computational speed and enhanced accuracy compared to conventional manifold-based approaches. Moreover, applications in real neuroimaging scenarios illustrate the framework's utility, enhancing behavior score prediction, subject fingerprinting in resting-state fMRI, and hypothesis testing in electroencephalogram data. An open-source MATLAB toolbox is provided to facilitate broader adoption and advance the application of correlation geometry in functional brain network research.