Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Canonical Transform for Strengthening the Local $L^p$-Type Universal
  Approximation Property   Most $L^p$-type universal approximation theorems guarantee that a given
machine learning model class $\mathscr{F}\subseteq
C(\mathbb{R}^d,\mathbb{R}^D)$ is dense in
$L^p_{\mu}(\mathbb{R}^d,\mathbb{R}^D)$ for any suitable finite Borel measure
$\mu$ on $\mathbb{R}^d$. Unfortunately, this means that the model's
approximation quality can rapidly degenerate outside some compact subset of
$\mathbb{R}^d$, as any such measure is largely concentrated on some bounded
subset of $\mathbb{R}^d$. This paper proposes a generic solution to this
approximation theoretic problem by introducing a canonical transformation which
"upgrades $\mathscr{F}$'s approximation property" in the following sense. The
transformed model class, denoted by $\mathscr{F}\text{-tope}$, is shown to be
dense in $L^p_{\mu,\text{strict}}(\mathbb{R}^d,\mathbb{R}^D)$ which is a
topological space whose elements are locally $p$-integrable functions and whose
topology is much finer than usual norm topology on
$L^p_{\mu}(\mathbb{R}^d,\mathbb{R}^D)$; here $\mu$ is any suitable
$\sigma$-finite Borel measure $\mu$ on $\mathbb{R}^d$. Next, we show that if
$\mathscr{F}$ is any family of analytic functions then there is always a strict
"gap" between $\mathscr{F}\text{-tope}$'s expressibility and that of
$\mathscr{F}$, since we find that $\mathscr{F}$ can never dense in
$L^p_{\mu,\text{strict}}(\mathbb{R}^d,\mathbb{R}^D)$. In the general case,
where $\mathscr{F}$ may contain non-analytic functions, we provide an abstract
form of these results guaranteeing that there always exists some function space
in which $\mathscr{F}\text{-tope}$ is dense but $\mathscr{F}$ is not, while,
the converse is never possible. Applications to feedforward networks,
convolutional neural networks, and polynomial bases are explored.

15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: atom-ph