The Role of Fibration Symmetries in Geometric Deep Learning
This work addresses the gap between theoretical symmetry frameworks and real-world problems in geometric deep learning, offering a mathematical extension with potential for better generalization across domains like graphs, manifolds, and grids.
The authors tackled the limitation of Geometric Deep Learning (GDL) to global symmetries by extending it to local fibration symmetries in graphs, showing that Graph Neural Networks (GNNs) apply this inductive bias and deriving a tighter upper bound for their expressive power while enabling computational efficiency through node collapse.
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 that are not often found in real-world problems. We propose to relax GDL to allow for local symmetries, specifically fibration symmetries in graphs, to leverage regularities of realistic instances. 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 collapse 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.