Uncovering shape signatures of resting‐state functional connectivity by geometric deep learning on Riemannian manifold

Functional neural activities manifest geometric patterns, as evidenced by the evolving network topology of functional connectivities (FC) even in the resting state. In this work, we propose a novel manifold‐based geometric neural network for functional brain networks (called “Geo‐Net4Net” for short) to learn the intrinsic low‐dimensional feature representations of resting‐state brain networks on the Riemannian manifold. This tool allows us to answer the scientific question of how the spontaneous fluctuation of FC supports behavior and cognition. We deploy a set of positive maps and rectified linear unit (ReLU) layers to uncover the intrinsic low‐dimensional feature representations of functional brain networks on the Riemannian manifold taking advantage of the symmetric positive‐definite (SPD) form of the correlation matrices. Due to the lack of well‐defined ground truth in the resting state, existing learning‐based methods are limited to unsupervised methodologies. To go beyond this boundary, we propose to self‐supervise the feature representation learning of resting‐state functional networks by leveraging the task‐based counterparts occurring before and after the underlying resting state. With this extra heuristic, our Geo‐Net4Net allows us to establish a more reasonable understanding of resting‐state FCs by capturing the geometric patterns (aka. spectral/shape signature) associated with resting states on the Riemannian manifold. We have conducted extensive experiments on both simulated data and task‐based functional resonance magnetic imaging (fMRI) data from the Human Connectome Project (HCP) database, where our Geo‐Net4Net not only achieves more accurate change detection results than other state‐of‐the‐art counterpart methods but also yields ubiquitous geometric patterns that manifest putative insights into brain function.