Scalable geometric learning with correlation-based functional brain networks.

Correlation matrices serve as fundamental representations of functional brain networks in neuroimaging. Conventional analyses often treat pairwise interactions independently within Euclidean space, neglecting the underlying geometry of correlation structures. Although recent efforts have leveraged the quotient geometry of the correlation manifold, they suffer from computational inefficiency and numerical instability, especially in high-dimensional settings. We propose a novel geometric framework that uses diffeomorphic transformations to embed correlation matrices into a Euclidean space while preserving critical manifold characteristics. This approach enables scalable, geometry-aware analyses and integrates seamlessly with standard machine learning techniques, including regression, dimensionality reduction, and clustering. Moreover, it facilitates population-level inference of brain networks. Simulation studies demonstrate significant improvements in both computational speed and predictive accuracy over existing manifold-based methods. Applications to real neuroimaging data further highlight the framework's versatility, improving behavioral score prediction, subject fingerprinting in resting-state fMRI, and hypothesis testing in EEG analyses. To support community adoption and reproducibility, we provide an open-source MATLAB toolbox implementing the proposed techniques. Our work opens new directions for efficient and interpretable geometric modeling in large-scale functional brain network research.