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. By focusing on a joint group invariant function on the data-parameter domain, we present a systematic rule to find a dual group action on the parameter domain from a group action on the data domain. Further, we introduce generalized neural networks induced from the joint invariant functions, and present a new group theoretic proof of their universality theorems by using Schur's lemma. Since traditional universality theorems were demonstrated based on functional analytical methods, 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性质被认为编码在神经网络内部的数据表示中，但具体的编码规则研究较少。通过聚焦数据-参数域上的联合群不变函数，我们提出了一套从数据域群作用推导参数域对偶群作用的系统性规则。进一步，我们引入了由联合不变函数诱导的广义神经网络，并利用舒尔引理给出了其通用性定理的群论新证明。由于传统通用性定理基于泛函分析方法论证，本研究揭示了逼近理论的群论视角，将几何深度学习与抽象调和分析联系起来。