Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges (2021) (arxiv.org)

The 2021 paper "Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges" presents a unifying mathematical framework for deep learning that captures intrinsic geometric structures underlying diverse data types. By leveraging principles from geometry, the authors show how popular architectures like CNNs, RNNs, GNNs, and Transformers can be understood as instances of a broader geometric paradigm. This approach not only explains the success of these models but also offers a principled way to incorporate prior knowledge about physical symmetries and structures directly into neural network designs. This geometric perspective is significant because it addresses a fundamental challenge in high-dimensional learning: most real-world tasks exhibit inherent regularities or symmetries tied to their physical or relational context. Instead of treating data as generic high-dimensional vectors, geometric deep learning exploits these symmetries—ranging from grids and groups to graphs and manifolds—to improve learning efficiency, generalization, and interpretability. The paper builds on ideas reminiscent of Felix Klein’s Erlangen Program by categorizing neural architectures via their symmetry groups and invariances, providing a constructive blueprint to design new architectures tailored for complex structured data. Technically, this framework formalizes how geometric priors can guide feature learning and local gradient-based optimization, deepening understanding of why certain architectures excel in tasks like vision, natural language, and graph-based problems. By uniting these diverse models under a coherent theory, geometric deep learning opens pathways for the AI/ML community to systematically invent and analyze neural models that are more aligned with the inherent structure of their target domains.