On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups
This foundational result addresses the problem of designing equivariant neural networks for broader domains like graphs and manifolds, offering a rigorous framework for researchers in machine learning.
The paper provides a theoretical proof that convolutional structure is both sufficient and necessary for neural network equivariance under any compact group action, generalizing beyond translations.
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.