# Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks

arXiv: https://arxiv.org/abs/2307.15285 · HTML: https://mlsanity.com/paper/2307.15285/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 54 references that are in the map.
- First posted on arXiv 28 Jul 2023.

## Builds on (most cited first, 11 of 11)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [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)
- [Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces](https://mlsanity.com/paper/2211.14400/) (2022)
- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problems](https://mlsanity.com/paper/2105.01228/) (2021)
- [Optimal rates of approximation by shallow ReLUk neural networks and applications to nonparametric regression](https://mlsanity.com/paper/2304.01561/) (2023)
- [Function approximation with zonal function networks with activation functions analogous to the rectified linear unit functions](https://mlsanity.com/paper/1709.08174/) (2017)
- [Dimension independent bounds for general shallow networks](https://mlsanity.com/paper/1908.09880/) (2019)

## Built on by (most cited first, 11 of 11)

- [On the rates of convergence for learning with convolutional neural networks](https://mlsanity.com/paper/2403.16459/) (2024)
- [Approximation Rates for Shallow ReLUk Neural Networks on Sobolev Spaces via the Radon Transform](https://mlsanity.com/paper/2408.10996/) (2024)
- [Approximation Bounds for Recurrent Neural Networks with Application to Regression](https://mlsanity.com/paper/2409.05577/) (2024)
- [Condition Numbers and Eigenvalue Spectra of Shallow Networks on Spheres](https://mlsanity.com/paper/2511.02625/) (2025)
- [Shallow ReLUs Networks in Lp-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization](https://mlsanity.com/paper/2605.18468/) (2026)
- [Universality of reservoir systems with recurrent neural networks](https://mlsanity.com/paper/2403.01900/) (2024)
- [Configuration-Dependent Lower Bounds for Approximation by Shallow ReLUk Networks on the Sphere](https://mlsanity.com/paper/2510.04060/) (2025)
- [Do Neural Networks Really Beat the Curse of Dimensionality? A Bit-Complexity View](https://mlsanity.com/paper/2608.01357/) (2026)
- [From Approximation Rates to Loss-Landscape Barrier Decay in Shallow ReLU Networks](https://mlsanity.com/paper/2602.17596/) (2026)
- [Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks](https://mlsanity.com/paper/2609.19937/) (2026)
- [Minimax rates for learning spectral Barron functions by deep ReLU neural networks](https://mlsanity.com/paper/2609.39020/) (2026)

## Abstract (arXiv)

We study the following two related problems. The first is to determine to what error an arbitrary zonoid in ℝd+1 can be approximated in the Hausdorff distance by a sum of n line segments. The second is to determine optimal approximation rates in the uniform norm for shallow ReLUk neural networks on their variation spaces. The first of these problems has been solved for dneq 2,3, but when d=2,3 a logarithmic gap between the best upper and lower bounds remains. We close this gap, which completes the solution in all dimensions. For the second problem, our techniques significantly improve upon existing approximation rates when k≥ 1, and enable uniform approximation of both the target function and its derivatives.