# Function approximation by deep neural networks with parameters 0,± frac12, ± 1, 2

arXiv: https://arxiv.org/abs/2103.08659 · HTML: https://mlsanity.com/paper/2103.08659/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 13 references that are in the map.
- First posted on arXiv 15 Mar 2021.

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

- [Learning both Weights and Connections for Efficient Neural Networks](https://mlsanity.com/paper/1506.02626/) (2015)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [The State of Sparsity in Deep Neural Networks](https://mlsanity.com/paper/1902.09574/) (2019)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Statistical Guarantees for Regularized Neural Networks](https://mlsanity.com/paper/2006.00294/) (2020)
- [Nonconvex sparse regularization for deep neural networks and its optimality](https://mlsanity.com/paper/2003.11769/) (2020)

## Built on by (most cited first, 2 of 2)

- [Statistical guarantees for sparse deep learning](https://mlsanity.com/paper/2212.05427/) (2022)
- [Neural networks with superexpressive activations and integer weights](https://mlsanity.com/paper/2105.09917/) (2021)

## Abstract (arXiv)

In this paper it is shown that Cβ-smooth functions can be approximated by deep neural networks with ReLU activation function and with parameters 0,± frac12, ± 1, 2. The l0 and l1 parameter norms of considered networks are thus equivalent. The depth, width and the number of active parameters of the constructed networks have, up to a logarithmic factor, the same dependence on the approximation error as the networks with parameters in [-1,1]. In particular, this means that the nonparametric regression estimation with the constructed networks attains the same convergence rate as with sparse networks with parameters in [-1,1].