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This paper advances the mathematical foundations of neural networks by formalizing equivariance through Lie groupoids within a differentiable setting, bridging differential geometry and category theory. It establishes theoretical equivalences between specific convolutional layers and admissible category-equivariant structures, offering rigorous generalizations for global pooling operations without introducing new hardware or empirical benchmarks.
Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks
arXiv:2606.02758v2 Announce Type: replace-cross Abstract: We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.
Abstract:We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.
28 pages, 3 figures. Second version (fixed typos, added reference). Comments and criticism welcome!
Differential Geometry (math.DG); Machine Learning (cs.LG); Category Theory (math.CT)
arXiv:2606.02758 [math.DG]
(or arXiv:2606.02758v2 [math.DG] for this version)
https://doi.org/10.48550/arXiv.2606.02758 Focus to learn more arXiv-issued DOI via DataCite
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