Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Overcoming the Curse of Dimensionality in Neural Networks   Let $A$ be a set and $V$ a real Hilbert space. Let $H$ be a real Hilbert
space of functions $f:A\to V$ and assume $H$ is continuously embedded in the
Banach space of bounded functions. For $i=1,\cdots,n$, let $(x_i,y_i)\in
A\times V$ comprise our dataset. Let $0<q<1$ and $f^*\in H$ be the unique
global minimizer of the functional \begin{equation*} u(f) = \frac{q}{2}\Vert
f\Vert_{H}^{2} + \frac{1-q}{2n}\sum_{i=1}^{n}\Vert f(x_i)-y_i\Vert_{V}^{2}.
\end{equation*}
  In this paper we show that for each $k\in\mathbb{N}$ there exists a two layer
network where the first layer has $k$ functions which are Riesz representations
in the Hilbert space $H$ of point evaluation functionals and the second layer
is a weighted sum of the first layer, such that the functions $f_k$ realized by
these networks satisfy \begin{equation*} \Vert f_{k}-f^*\Vert_{H}^{2} \leq
\Bigl( o(1) + \frac{C}{q^2} E\bigl[ \Vert Du_{I}(f^*)\Vert_{H^{*}}^{2} \bigr]
\Bigr)\frac{1}{k}. \end{equation*}
  %Let us note that $x_i$ do not need to be in a linear space and $y_i$ are in
a possibly infinite dimensional Hilbert space $V$. %The error estimate is
independent of the data size $n$ and in the case $V$ is finite dimensional %the
error estimate is also independent of the dimension of $V$.
  By choosing the Hilbert space $H$ appropriately, the computational complexity
of evaluating the Riesz representations of point evaluations might be small and
thus the network has low computational complexity.

37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
2
The categories of this paper are: math
26
The categories of this paper are: bayes-an