aiqbench analysis
This paper formalizes Lie groupoid equivariant neural networks within a differentiable framework, establishing their equivalence to Lie algebroid-equivariant architectures and extending global pooling operations. By proving these layers are special cases of admissible category-equivariant structures, the work provides a rigorous mathematical foundation for leveraging advanced category theory in deep learning. The primary value for researchers lies in its theoretical unification of topological and differential geometric approaches to symmetry invariance, rather than immediate practical deployment.
Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks
arXiv:2606.02758v1 Announce Type: 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, 2 figures. Preliminary version. Comments and criticism welcome!
Differential Geometry (math.DG); Machine Learning (cs.LG); Category Theory (math.CT)
arXiv:2606.02758 [math.DG]
(or arXiv:2606.02758v1 [math.DG] for this version)
https://doi.org/10.48550/arXiv.2606.02758 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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