Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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]

27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol