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In many high-impact applications, it is important to ensure the quality of the output of a machine learning algorithm as well as its reliability in comparison to the complexity of the algorithm used. In this paper, we have initiated a mathematically rigorous theory to decide which models (algorithms applied on data sets) are close to each other in terms of certain metrics, such as performance and the complexity level of the algorithm. This involves creating a grid on the hypothetical spaces of data sets and algorithms so as to identify a finite set of probability distributions from which the data sets are sampled and a finite set of algorithms. A given threshold metric acting on this grid will express the nearness (or statistical distance) of each algorithm and data set of interest to any given application. A technically difficult part of this project is to estimate the so-called metric entropy of a compact subset of functions of \textbf{infinitely many variables} that arise in the definition of these spaces. 

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.