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This research unifies geometric deep learning and topological approaches by introducing order-equivariant neural networks (OENN) that generalize both graph message passing and sheaf models. By proving universal approximation theorems for continuous equivariant maps, the work establishes a rigorous theoretical foundation capable of handling complex categorical symmetries in data structures. The framework offers significant advances for modeling systems where local-to-global relationships are governed by non-invertible or compositional symmetry constraints.
Foundations of Equivariant Deep Learning: Unifying Graph and Sheaf Neural Networks
arXiv:2607.03798v2 Announce Type: replace Abstract: Symmetry is everywhere in nature and society. Geometric deep learning builds architectures respecting group symmetries, whereas topological deep learning organizes computation through cells, incidence relations, and local-to-global structure. In this paper, we extend geometric deep learning beyond simple group actions and unify it with topological deep learning. Specifically, we develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant bundles over face posets (face categories). We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new results even when restricted to sheaf neural networks. We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well. In the appendix, we clarify the precise relationships between OENN and CENN (Category-Equivariant Neural Network), which gives the categorical general form of equivariant neural networks, allowing us to leverage categorical symmetry in data (e.g., non-invertible symmetries on multiple objects with compositional relations on those symmetries).
Abstract:Symmetry is everywhere in nature and society. Geometric deep learning builds architectures respecting group symmetries, whereas topological deep learning organizes computation through cells, incidence relations, and local-to-global structure. In this paper, we extend geometric deep learning beyond simple group actions and unify it with topological deep learning. Specifically, we develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant bundles over face posets (face categories). We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new results even when restricted to sheaf neural networks. We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well. In the appendix, we clarify the precise relationships between OENN and CENN (Category-Equivariant Neural Network), which gives the categorical general form of equivariant neural networks, allowing us to leverage categorical symmetry in data (e.g., non-invertible symmetries on multiple objects with compositional relations on those symmetries).
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computer Vision and Pattern Recognition (cs.CV)
arXiv:2607.03798 [cs.LG]
(or arXiv:2607.03798v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.03798 Focus to learn more arXiv-issued DOI via DataCite
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