🤖 AI Summary
This work investigates the construction of convolutional neural networks on smooth manifolds that exhibit equivariance under Lie groupoids and Lie algebroids. To this end, the authors propose a unified network architecture comprising Lie groupoid-lifted convolutions, Lie algebroid-equivariant layers, and groupoid-invariant global pooling, all formulated as natural transformations between continuous feature functors. This study establishes, for the first time, a systematic theoretical framework for Lie groupoid- and Lie algebroid-equivariant neural networks, demonstrating that each proposed component arises as a special case of admissible categorical equivariant layers. Furthermore, under suitable conditions, the equivalence between these two classes of networks is rigorously proven, thereby successfully extending topological categorical equivariance methods to the setting of differential geometry.
📝 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.