Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Deep neural network approximation theory for high-dimensional functions   The purpose of this article is to develop machinery to study the capacity of
deep neural networks (DNNs) to approximate high-dimensional functions. In
particular, we show that DNNs have the expressive power to overcome the curse
of dimensionality in the approximation of a large class of functions. More
precisely, we prove that these functions can be approximated by DNNs on compact
sets such that the number of parameters necessary to represent the
approximating DNNs grows at most polynomially in the reciprocal $1/\varepsilon$
of the approximation accuracy $\varepsilon>0$ and in the input dimension $d\in
\mathbb{N} =\{1,2,3,\dots\}$. To this end, we introduce certain approximation
spaces, consisting of sequences of functions that can be efficiently
approximated by DNNs. We then establish closure properties which we combine
with known and new bounds on the number of parameters necessary to approximate
locally Lipschitz continuous functions, maximum functions, and product
functions by DNNs. The main result of this article demonstrates that DNNs have
sufficient expressiveness to approximate certain sequences of functions which
can be constructed by means of a finite number of compositions using locally
Lipschitz continuous functions, maxima, and products without the curse of
dimensionality.

30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin