Geometric Deep Learning: Understanding Graph Neural Networks through the Lens of Mathematics

Abstract: Geometric Deep Learning (GDL) extends traditional neural network paradigms to non-Euclidean data structures, enabling the effective processing of data that lies on manifolds or graphs. Among GDL techniques, Graph Neural Networks (GNNs) have emerged as powerful tools for modelling relational data by leveraging principles from graph theory and algebraic topology. This paper explores GNNs through the lens of mathematics, focusing on how geometric and topological insights drive the architecture and functionality of these networks. By framing GNNs in terms of graph signal processing and spectral theory, we illuminate how GNNs capture dependencies across nodes and edges, offering a structured approach to learning on graph-structured data. We further examine the theoretical underpinnings that make GNNs particularly suited for applications in social networks, molecular biology, and recommendation systems. In doing so, this study provides a mathematical perspective on the capabilities and limitations of GNNs, underscoring the role of invariance, equivariance, and generalization within graph-based learning models.

Geometric Deep Learning, Graph Neural Networks, Non-Euclidean Data, Algebraic Topology, Graph Theory

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