Connectomics has emerged as a powerful tool in neuroimaging and has spurred recent advancements in statistical and machine learning methods for connectivity data. Despite connectomes inhabiting a matrix manifold, most analytical frameworks ignore the underlying data geometry. This is largely because simple operations, such as mean estimation, do not have easily computable closed-form solutions. We propose a geometrically aware neural framework for connectomes, i.e., the mSPD-NN, designed to estimate the geodesic mean of a collections of symmetric positive definite (SPD) matrices. The mSPD-NN is comprised of bilinear fully connected layers with tied weights and utilizes a novel loss function to optimize the matrix-normal equation arising from Fr\'echet mean estimation. Via experiments on synthetic data, we demonstrate the efficacy of our mSPD-NN against common alternatives for SPD mean estimation, providing competitive performance in terms of scalability and robustness to noise. We illustrate the real-world flexibility of the mSPD-NN in multiple experiments on rs-fMRI data and demonstrate that it uncovers stable biomarkers associated with subtle network differences among patients with ADHD-ASD comorbidities and healthy controls.

翻译：连接组学已成为神经影像学中的强大工具，并推动了连接数据统计与机器学习方法的近期进展。尽管连接组数据位于矩阵流形上，但大多数分析框架忽视了其底层数据几何结构，这主要是因为均值估计等简单操作缺乏易于计算的闭式解。我们提出了一种面向连接组的几何感知神经框架——mSPD-NN，旨在估计对称正定矩阵集合的测地线均值。该框架由具有权重绑定的双线性全连接层构成，并利用新型损失函数优化来自弗雷歇均值估计的矩阵-正态方程。通过在合成数据上的实验，我们证明了mSPD-NN相较于常见的SPD均值估计替代方法具有更优性能，在可扩展性和噪声鲁棒性方面表现出竞争力。我们通过多项静息态功能磁共振成像实验展示了mSPD-NN在真实场景中的灵活性，并揭示其能够识别与ADHD-ASD共病及健康对照组之间细微网络差异相关的稳定生物标志物。