# Approximation in Lp(μ) with deep ReLU neural networks

arXiv: https://arxiv.org/abs/1904.04789 · HTML: https://mlsanity.com/paper/1904.04789/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Apr 2019.

## Built on by (most cited first, 3 of 3)

- [A general approximation lower bound in Lp norm, with applications to feed-forward neural networks](https://mlsanity.com/paper/2206.04360/) (2022)
- [Super-fast Rates of Convergence for Neural Network Classifiers under the Hard Margin Condition](https://mlsanity.com/paper/2505.08262/) (2025)
- [Efficient Approximation to Analytic and Lp functions by Height-Augmented ReLU Networks](https://mlsanity.com/paper/2603.11128/) (2026)

## Abstract (arXiv)

We discuss the expressive power of neural networks which use the non-smooth ReLU activation function varrho(x) = max0,x by analyzing the approximation theoretic properties of such networks. The existing results mainly fall into two categories: approximation using ReLU networks with a fixed depth, or using ReLU networks whose depth increases with the approximation accuracy. After reviewing these findings, we show that the results concerning networks with fixed depth--- which up to now only consider approximation in Lp(λ) for the Lebesgue measure λ--- can be generalized to approximation in Lp(μ), for any finite Borel measure μ. In particular, the generalized results apply in the usual setting of statistical learning theory, where one is interested in approximation in L2(mathbbP), with the probability measure mathbbP describing the distribution of the data.