In this paper, we present a sharper version of the results in the paper Dimension independent bounds for general shallow networks; Neural Networks, \textbf{123} (2020), 142-152. Let $\mathbb{X}$ and $\mathbb{Y}$ be compact metric spaces. We consider approximation of functions of the form $ x\mapsto\int_{\mathbb{Y}} G( x, y)d\tau( y)$, $ x\in\mathbb{X}$, by $G$-networks of the form $ x\mapsto \sum_{k=1}^n a_kG( x, y_k)$, $ y_1,\cdots, y_n\in\mathbb{Y}$, $a_1,\cdots, a_n\in\mathbb{R}$. Defining the dimensions of $\mathbb{X}$ and $\mathbb{Y}$ in terms of covering numbers, we obtain dimension independent bounds on the degree of approximation in terms of $n$, where also the constants involved are all dependent at most polynomially on the dimensions. Applications include approximation by power rectified linear unit networks, zonal function networks, certain radial basis function networks as well as the important problem of function extension to higher dimensional spaces.

翻译：本文给出了论文《一般浅层网络的维度无关界》（Neural Networks, \textbf{123} (2020), 142-152）中结果的一个更优版本。设$\mathbb{X}$和$\mathbb{Y}$是紧致度量空间。我们考虑形如$ x\mapsto\int_{\mathbb{Y}} G( x, y)d\tau( y)$（$ x\in\mathbb{X}$）的函数，通过形如$ x\mapsto \sum_{k=1}^n a_kG( x, y_k)$（$ y_1,\cdots, y_n\in\mathbb{Y}$，$a_1,\cdots, a_n\in\mathbb{R}$）的$G$网络进行逼近。利用覆盖数定义$\mathbb{X}$和$\mathbb{Y}$的维度，我们得到了关于$n$的逼近程度的维度无关界，其中所有涉及的常数最多以多项式方式依赖于维度。应用包括：幂修正线性单元网络、分区函数网络、特定径向基函数网络，以及函数向高维空间延拓这一重要问题。