# Towards an Algebraic Framework For Approximating Functions Using Neural Network Polynomials

arXiv: https://arxiv.org/abs/2402.01058 · HTML: https://mlsanity.com/paper/2402.01058/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 17 references that are in the map.
- First posted on arXiv 1 Feb 2024.

## Builds on (most cited first, 5 of 5)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Topological Approaches to Deep Learning](https://mlsanity.com/paper/1811.01122/) (2018)
- [Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory](https://mlsanity.com/paper/2310.20360/) (2023)
- [The signature and cusp geometry of hyperbolic knots](https://mlsanity.com/paper/2111.15323/) (2021)

## Abstract (arXiv)

We make the case for neural network objects and extend an already existing neural network calculus explained in detail in Chapter 2 on citebigbook. Our aim will be to show that, yes, indeed, it makes sense to talk about neural network polynomials, neural network exponentials, sine, and cosines in the sense that they do indeed approximate their real number counterparts subject to limitations on certain of their parameters, q, and varepsilon. While doing this, we show that the parameter and depth growth are only polynomial on their desired accuracy (defined as a 1-norm difference over ℝ), thereby showing that this approach to approximating, where a neural network in some sense has the structural properties of the function it is approximating is not entire intractable.