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In this work, we propose a novel manifold\u2010based geometric neural network for functional brain networks (called \u201cGeo\u2010Net4Net\u201d for short) to learn the intrinsic low\u2010dimensional feature representations of resting\u2010state 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\u2010dimensional feature representations of functional brain networks on the Riemannian manifold taking advantage of the symmetric positive\u2010definite (SPD) form of the correlation matrices. Due to the lack of well\u2010defined ground truth in the resting state, existing learning\u2010based methods are limited to unsupervised methodologies. To go beyond this boundary, we propose to self\u2010supervise the feature representation learning of resting\u2010state functional networks by leveraging the task\u2010based counterparts occurring before and after the underlying resting state. With this extra heuristic, our Geo\u2010Net4Net allows us to establish a more reasonable understanding of resting\u2010state 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\u2010based functional resonance magnetic imaging (fMRI) data from the Human Connectome Project (HCP) database, where our Geo\u2010Net4Net not only achieves more accurate change detection results than other state\u2010of\u2010the\u2010art counterpart methods but also yields ubiquitous geometric patterns that manifest putative insights into brain function.<\/jats:p>","DOI":"10.1002\/hbm.25897","type":"journal-article","created":{"date-parts":[[2022,5,11]],"date-time":"2022-05-11T05:50:15Z","timestamp":1652248215000},"page":"3970-3986","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Uncovering shape signatures of <scp>resting\u2010state<\/scp> functional connectivity by geometric deep learning on Riemannian 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