# Expressivity of Shallow and Deep Neural Networks for Polynomial Approximation

arXiv: https://arxiv.org/abs/2303.03544 · HTML: https://mlsanity.com/paper/2303.03544/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 28 references that are in the map.
- First posted on arXiv 6 Mar 2023.

## Builds on (most cited first, 16 of 16)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Why does deep and cheap learning work so well?](https://mlsanity.com/paper/1608.08225/) (2016)
- [On the Expressive Power of Deep Learning: A Tensor Analysis](https://mlsanity.com/paper/1509.05009/) (2015)
- [On the number of response regions of deep feed forward networks with piece-wise linear activations](https://mlsanity.com/paper/1312.6098/) (2013)
- [Representation Benefits of Deep Feedforward Networks](https://mlsanity.com/paper/1509.08101/) (2015)
- [Why Deep Neural Networks for Function Approximation?](https://mlsanity.com/paper/1610.04161/) (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)
- [The power of deeper networks for expressing natural functions](https://mlsanity.com/paper/1705.05502/) (2017)
- [Depth Separations in Neural Networks: What is Actually Being Separated?](https://mlsanity.com/paper/1904.06984/) (2019)
- [Learning Functions: When Is Deep Better Than Shallow](https://mlsanity.com/paper/1603.00988/) (2016)
- [Depth separation beyond radial functions](https://mlsanity.com/paper/2102.01621/) (2021)

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

- [Approximation of Analytic Functions by ReLU Neural Networks with Adjustable Depth and Width](https://mlsanity.com/paper/2607.10589/) (2026)

## Abstract (arXiv)

This study explores the number of neurons required for a Rectified Linear Unit (ReLU) neural network to approximate multivariate monomials. We establish an exponential lower bound on the complexity of any shallow network approximating the product function over a general compact domain. We also demonstrate this lower bound doesn't apply to normalized Lipschitz monomials over the unit cube. These findings suggest that shallow ReLU networks experience the curse of dimensionality when expressing functions with a Lipschitz parameter scaling with the dimension of the input, and that the expressive power of neural networks is more dependent on their depth rather than overall complexity.