# Approximation Rates of Shallow Neural Networks: Barron Spaces, Activation Functions and Optimality Analysis

arXiv: https://arxiv.org/abs/2510.18388 · HTML: https://mlsanity.com/paper/2510.18388/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 33 references that are in the map.
- First posted on arXiv 21 Oct 2025.

## Builds on (most cited first, 5 of 5)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [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)
- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [Sharp Representation Theorems for ReLU Networks with Precise Dependence on Depth](https://mlsanity.com/paper/2006.04048/) (2020)

## Built on by (most cited first, 1 of 1)

- [Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation](https://mlsanity.com/paper/2607.27781/) (2026)

## Abstract (arXiv)

This paper investigates the approximation properties of shallow neural networks with activation functions that are powers of exponential functions. It focuses on the dependence of the approximation rate on the dimension and the smoothness of the function being approximated within the Barron function space. We examine the approximation rates of ReLUk activation functions, proving that the optimal rate cannot be achieved under ℓ1-bounded coefficients or insufficient smoothness conditions. We also establish optimal approximation rates in various norms for functions in Barron spaces and Sobolev spaces, confirming the curse of dimensionality. Our results clarify the limits of shallow neural networks' approximation capabilities and offer insights into the selection of activation functions and network structures.