We present a constructive universal approximation theorem for learning machines equipped with joint-group-equivariant feature maps, based on the group representation theory. ``Constructive'' here indicates that the distribution of parameters is given in a closed-form expression known as the ridgelet transform. Joint-group-equivariance encompasses a broad class of feature maps that generalize classical group-equivariance. Notably, this class includes fully-connected networks, which are not group-equivariant but are joint-group-equivariant. Moreover, our main theorem also unifies the universal approximation theorems for both shallow and deep networks. While the universality of shallow networks has been investigated in a unified manner by the ridgelet transform, the universality of deep networks has been investigated in a case-by-case manner.

翻译：基于群表示理论，我们为配备联合群等变特征映射的学习机器提出了一个构造性的通用逼近定理。此处的“构造性”指参数分布以闭式表达式（称为脊波变换）给出。联合群等变性涵盖了一大类推广经典群等变性的特征映射。值得注意的是，此类包括全连接网络，它们虽非群等变，但属于联合群等变。此外，我们的主定理还统一了浅层与深层网络的通用逼近定理。虽然浅层网络的普适性已通过脊波变换得到统一研究，但深层网络的普适性此前仅以个案方式探讨。