The role of fibration symmetries in geometric deep learning

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.