The symmetry and geometry of input data are considered to be encoded in the internal data representation inside the neural network, but the specific encoding rule has been less investigated. In this study, we present a systematic method to induce a generalized neural network and its right inverse operator, called the ridgelet transform, from a joint group invariant function on the data-parameter domain. Since the ridgelet transform is an inverse, (1) it can describe the arrangement of parameters for the network to represent a target function, which is understood as the encoding rule, and (2) it implies the universality of the network. Based on the group representation theory, we present a new simple proof of the universality by using Schur's lemma in a unified manner covering a wide class of networks, for example, the original ridgelet transform, formal deep networks, and the dual voice transform. Since traditional universality theorems were demonstrated based on functional analysis, this study sheds light on the group theoretic aspect of the approximation theory, connecting geometric deep learning to abstract harmonic analysis.

翻译：输入数据的对称性与几何结构被认为编码在神经网络内部的表征中，但具体的编码规则尚未得到充分研究。本研究提出了一种系统方法，从数据-参数域上的联合群不变函数出发，诱导出广义神经网络及其右逆算子（称为脊波变换）。由于脊波变换具有逆性质：（1）它可以描述网络表示目标函数时的参数排布方式（即编码规则），（2）它蕴含了网络的通用性。基于群表示论，我们利用舒尔引理给出了一种统一的新颖通用性证明，该证明覆盖了包括原始脊波变换、形式化深度网络和对偶语音变换在内的广泛网络类别。传统通用性定理通常基于泛函分析进行论证，而本研究揭示了逼近理论中群论视角的崭新面貌，将几何深度学习与抽象调和分析联系起来。