2022/06/17 by Federico Barbero, Barbero, Federico, Cristian Bodnar +9 · 6 citations
Computer Science · Engineering · Neuroscience · #Advanced Numerical Analysis Techniques #Algebraic Topology (math.AT) #Brain Tumor Detection and Classification #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Neural Networks and Applications
paper · pdf · doi:10.48550/arxiv.2206.08702
openalex publication_date 2022/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Sheaf Neural Network (SNN) is a type of Graph Neural Network (GNN) that operates on a sheaf, an object that equips a graph with vector spaces over its nodes and edges and linear maps between these spaces. SNNs have been shown to have useful theoretical properties that help tackle issues arising from heterophily and over-smoothing. One complication intrinsic to these models is finding a good sheaf for the task to be solved. Previous works proposed two diametrically opposed approaches: manually constructing the sheaf based on domain knowledge and learning the sheaf end-to-end using gradient-based methods. However, domain knowledge is often insufficient, while learning a sheaf could lead to overfitting and significant computational overhead. In this work, we propose a novel way of computing sheaves drawing inspiration from Riemannian geometry: we leverage the manifold assumption to compute manifold-and-graph-aware orthogonal maps, which optimally align the tangent spaces of neighbouring data points. We show that this approach achieves promising results with less computational overhead when compared to previous SNN models. Overall, this work provides an interesting connection between algebraic topology and differential geometry, and we hope that it will spark future research in this direction.