# Tractability of approximation by general shallow networks

arXiv: https://arxiv.org/abs/2308.03230 · HTML: https://mlsanity.com/paper/2308.03230/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 55 references that are in the map.
- First posted on arXiv 7 Aug 2023.

## Builds on (most cited first, 12 of 12)

- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [Deep vs. shallow networks : An approximation theory perspective](https://mlsanity.com/paper/1608.03287/) (2016)
- [Understanding neural networks with reproducing kernel Banach spaces](https://mlsanity.com/paper/2109.09710/) (2021)
- [Kernel based analysis of massive data](https://mlsanity.com/paper/2003.13226/) (2020)
- [A deep learning approach to diabetic blood glucose prediction](https://mlsanity.com/paper/1707.05828/) (2017)
- [Function approximation with zonal function networks with activation functions analogous to the rectified linear unit functions](https://mlsanity.com/paper/1709.08174/) (2017)
- [Local approximation of operators](https://mlsanity.com/paper/2202.06392/) (2022)
- [Dimension independent bounds for general shallow networks](https://mlsanity.com/paper/1908.09880/) (2019)
- [Radial Basis Function Approximation with Distributively Stored Data on Spheres](https://mlsanity.com/paper/2112.02499/) (2021)

## Built on by (most cited first, 4 of 4)

- [Approximation Rates for Shallow ReLUk Neural Networks on Sobolev Spaces via the Radon Transform](https://mlsanity.com/paper/2408.10996/) (2024)
- [Configuration-Dependent Lower Bounds for Approximation by Shallow ReLUk Networks on the Sphere](https://mlsanity.com/paper/2510.04060/) (2025)
- [Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples](https://mlsanity.com/paper/2608.06687/) (2026)
- [Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks](https://mlsanity.com/paper/2609.19937/) (2026)

## Abstract (arXiv)

In this paper, we present a sharper version of the results in the paper Dimension independent bounds for general shallow networks; Neural Networks, 123 (2020), 142-152. Let mathbbX and mathbbY be compact metric spaces. We consider approximation of functions of the form  xmapstoint_mathbbY G( x, y)dτ( y),  x∈mathbbX, by G-networks of the form  xmapsto sumk=1n akG( x, yk),  y1,cdots, yn∈mathbbY, a1,cdots, an∈ℝ. Defining the dimensions of mathbbX and mathbbY 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.