The role of fibration symmetries in geometric deep learning

This work extends the current framework of Geometric Deep Learning to incorporate local symmetries, specifically fibration symmetries, which are more commonly found in real-world data. By introducing these local symmetries, we improve the expressiveness and computational efficiency of Graph Neural Networks, which are widely used in machine learning for tasks involving structured data. This mathematical framework has implications that extend beyond graphs, providing a foundation for models of more complex structures, such as manifolds and grids. Our results provide insights that could enhance the generalization ability of machine learning models, making them more robust for diverse scientific applications. Geometric Deep Learning (GDL) unifies a broad class of machine learning techniques from the perspectives of symmetries, offering a framework for introducing problem-specific inductive biases like Graph Neural Networks (GNNs). However, the current formulation of GDL is limited to global symmetries. We propose to relax GDL to allow for local symmetries, specifically fibration symmetries, which only require isomorphic input trees—a property that is much more common in real-world graphs. We show that GNNs apply the inductive bias of fibration symmetries and derive a tighter upper bound for their expressive power. Additionally, by identifying symmetries in networks, we compress network nodes, thereby increasing their computational efficiency during both inference and training of deep neural networks. The mathematical extension introduced here applies beyond graphs to manifolds, bundles, and grids for the development of models with inductive biases induced by local symmetries that can lead to better generalization.