# Function approximation with zonal function networks with activation functions analogous to the rectified linear unit functions

arXiv: https://arxiv.org/abs/1709.08174 · HTML: https://mlsanity.com/paper/1709.08174/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2017.

## Built on by (most cited first, 8 of 8)

- [Rates of Approximation by ReLU Shallow Neural Networks](https://mlsanity.com/paper/2307.12461/) (2023)
- [Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks](https://mlsanity.com/paper/2307.15285/) (2023)
- [Local approximation of operators](https://mlsanity.com/paper/2202.06392/) (2022)
- [Toward L_∞-recovery of Nonlinear Functions: A Polynomial Sample Complexity Bound for Gaussian Random Fields](https://mlsanity.com/paper/2305.00322/) (2023)
- [Tractability of approximation by general shallow networks](https://mlsanity.com/paper/2308.03230/) (2023)
- [Learning on manifolds without manifold learning](https://mlsanity.com/paper/2402.12687/) (2024)
- [Approximation by non-symmetric networks for cross-domain learning](https://mlsanity.com/paper/2305.03890/) (2023)
- [Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples](https://mlsanity.com/paper/2608.06687/) (2026)

## Abstract (arXiv)

A zonal function (ZF) network on the q dimensional sphere mathbbSq is a network of the form xmapsto sumk=1n akφ(x·xk) where φ :[-1,1]→R is the activation function, xk∈mathbbSq are the centers, and ak∈ℝ. While the approximation properties of such networks are well studied in the context of positive definite activation functions, recent interest in deep and shallow networks motivate the study of activation functions of the form φ(t)=|t|, which are not positive definite. In this paper, we define an appropriate smoothess class and establish approximation properties of such networks for functions in this class. The centers can be chosen independently of the target function, and the coefficients are linear combinations of the training data. The constructions preserve rotational symmetries.