We identify hidden layers inside a deep neural network (DNN) with group actions on the data domain, and formulate a formal deep network as a dual voice transform with respect to the Koopman operator, a linear representation of the group action. Based on the group theoretic arguments, particularly by using Schur's lemma, we show a simple proof of the universality of DNNs.

翻译：我们将深度神经网络（DNN）中的隐藏层识别为作用于数据域上的群作用，并将形式化深度网络构建为关于Koopman算子的对偶声音变换——该算子作为群作用的线性表示。基于群论论证，特别是通过应用舒尔引理，我们展示了深度神经网络普适性的简洁证明。