# Deep neural network approximation of composite functions without the curse of dimensionality

arXiv: https://arxiv.org/abs/2304.05790 · HTML: https://mlsanity.com/paper/2304.05790/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 36 references that are in the map.
- First posted on arXiv 12 Apr 2023.

## Builds on (most cited first, 17 of 17)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Deep vs. shallow networks : An approximation theory perspective](https://mlsanity.com/paper/1608.03287/) (2016)
- [Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networks](https://mlsanity.com/paper/1610.09887/) (2016)
- [Depth Separation for Neural Networks](https://mlsanity.com/paper/1702.08489/) (2017)
- [Expressivity of Deep Neural Networks](https://mlsanity.com/paper/2007.04759/) (2020)
- [Learning Functions: When Is Deep Better Than Shallow](https://mlsanity.com/paper/1603.00988/) (2016)
- [Full error analysis for the training of deep neural networks](https://mlsanity.com/paper/1910.00121/) (2019)
- [Strong overall error analysis for the training of artificial neural networks via random initializations](https://mlsanity.com/paper/2012.08443/) (2020)
- [The necessity of depth for artificial neural networks to approximate certain classes of smooth and bounded functions without the curse of dimensionality](https://mlsanity.com/paper/2301.08284/) (2023)

## Abstract (arXiv)

In this article we identify a general class of high-dimensional continuous functions that can be approximated by deep neural networks (DNNs) with the rectified linear unit (ReLU) activation without the curse of dimensionality. In other words, the number of DNN parameters grows at most polynomially in the input dimension and the approximation error. The functions in our class can be expressed as a potentially unbounded number of compositions of special functions which include products, maxima, and certain parallelized Lipschitz continuous functions.