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.

翻译：相关矩阵是神经影像学中功能脑网络的核心表示。传统分析通常在欧几里得空间中独立处理成对相互作用，忽略了相关矩阵的内在几何结构。虽然早期的尝试已采纳相关流形的商几何，但仍受限于计算效率低下和数值不稳定性，尤其是在高维情境下。本文提出了一种新颖的几何框架，该框架采用微分同胚变换将相关矩阵嵌入到欧几里得空间中，保留了流形的显著特性并支持大规模分析。所提出的方法与成熟的学习算法——回归、降维和聚类——相集成，并能自然地扩展到脑网络的群体水平推断。仿真研究表明，与传统的基于流形的方法相比，该方法在计算速度和准确性上均有提升。此外，在真实神经影像场景中的应用展示了该框架的实用性，它提升了行为评分预测、静息态功能磁共振成像中的受试者指纹识别以及脑电图数据中的假设检验能力。我们提供了一个开源的MATLAB工具箱，以促进更广泛的采用，并推动相关几何在功能脑网络研究中的应用。