Scalable Geometric Learning with Correlation-Based Functional Brain Networks

2025-03-31

AI Summary

Traditional Euclidean approaches to analyzing functional brain network correlation matrices ignore their intrinsic manifold structure, leading to geometric distortion; existing manifold-based methods suffer from computational inefficiency and numerical instability in high dimensions. This paper proposes a geometric embedding framework based on diffeomorphic transformations, enabling the first structure-preserving, efficient, and numerically stable mapping from the correlation manifold to Euclidean space. The framework supports large-scale regression, dimensionality reduction, and clustering, and is compatible with group-level statistical inference. Evaluated on fMRI subject fingerprinting, behavioral score prediction, and EEG hypothesis testing, it significantly outperforms state-of-the-art manifold methods—achieving an order-of-magnitude speedup in computation and concurrent improvements in prediction accuracy. An open-source MATLAB toolbox accompanies the work, facilitating practical adoption of geometric modeling in neuroimaging.

Technology Category

• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

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.

Problem

• Overcoming Euclidean limitations in brain network analysis
• Addressing computational inefficiency in correlation manifold methods
• Enabling scalable geometric learning for functional neuroimaging

Innovation

• Diffeomorphic transformations embed correlation matrices
• Integrates regression, dimensionality reduction, clustering
• Open-source MATLAB toolbox for brain networks