# On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups

arXiv: https://arxiv.org/abs/1802.03690 · HTML: https://mlsanity.com/paper/1802.03690/

## Key facts

- Cited by 300 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2018.

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## Abstract (arXiv)

Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.