We present a unified constructive universal approximation theorem covering a wide range of learning machines including both shallow and deep neural networks based on the group representation theory. Constructive here means that the distribution of parameters is given in a closed-form expression (called the ridgelet transform). Contrary to the case of shallow models, expressive power analysis of deep models has been conducted in a case-by-case manner. Recently, Sonoda et al. (2023a,b) developed a systematic method to show a constructive approximation theorem from scalar-valued joint-group-invariant feature maps, covering a formal deep network. However, each hidden layer was formalized as an abstract group action, so it was not possible to cover real deep networks defined by composites of nonlinear activation function. In this study, we extend the method for vector-valued joint-group-equivariant feature maps, so to cover such real networks.

基于群表示论，我们提出了一个统一的构造性通用逼近定理，涵盖了包括浅层和深层神经网络在内的广泛学习机。我们通过研究向量值联合群等变特征映射的方法，扩展了Sonoda等人最近发展的系统方法，从而对复合非线性激活函数定义的真实深层网络进行了形式化的通用逼近定理的证明。

通过舒尔引理的深度联合等变网络的构造性通用逼近定理