Neural Networks on Groups

Although neural networks traditionally are typically used to approximate functions defined over $\mathbb{R}^n$, the successes of graph neural networks, point-cloud neural networks, and manifold deep learning among other methods have demonstrated the clear value of leveraging neural networks to approximate functions defined over more general spaces. The theory of neural networks has not kept up however,and the relevant theoretical results (when they exist at all) have been proven on a case-by-case basis without a general theory or connection to classical work. The process of deriving new theoretical backing for each new type of network has become a bottleneck to understanding and validating new approaches. In this paper we extend the definition of neural networks to general topological groups and prove that neural networks with a single hidden layer and a bounded non-constant activation function can approximate any $\mathcal{L}^p$ function defined over any locally compact Abelian group. This framework and universal approximation theorem encompass all of the aforementioned contexts. We also derive important corollaries and extensions with minor modification, including the case for approximating continuous functions on a compact subset, neural networks with ReLU activation functions on a linearly bi-ordered group, and neural networks with affine transformations on a vector space. Our work obtains as special cases the recent theorems of Qi et al. [2017], Sennai et al. [2019], Keriven and Peyre [2019], and Maron et al. [2019]