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.

本文首先说明深度学习中卷积神经网络在图像识别方面的成功，然后介绍了将此框架推广到其他领域的尝试，并给出了使用表示论和非交换调和分析概念的卷积和等变性的理论证明，证明了卷积结构不仅是充分而且也是必要的对于紧致群作用的等变性的条件，并推导了新的广义卷积公式。