# Shallow neural network representation of polynomials

arXiv: https://arxiv.org/abs/2208.08138 · HTML: https://mlsanity.com/paper/2208.08138/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 24 references that are in the map.
- First posted on arXiv 17 Aug 2022.

## Builds on (most cited first, 7 of 7)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [Risk Bounds for High-dimensional Ridge Function Combinations Including Neural Networks](https://mlsanity.com/paper/1607.01434/) (2016)
- [Deep nets for local manifold learning](https://mlsanity.com/paper/1607.07110/) (2016)
- [Analytic function approximation by path norm regularized deep networks](https://mlsanity.com/paper/2104.02095/) (2021)

## Abstract (arXiv)

We show that d-variate polynomials of degree R can be represented on [0,1]d as shallow neural networks of width 2(R+d)d. Also, by SNN representation of localized Taylor polynomials of univariate Cβ-smooth functions, we derive for shallow networks the minimax optimal rate of convergence, up to a logarithmic factor, to unknown univariate regression function.