# On the space of coefficients of a Feed Forward Neural Network

arXiv: https://arxiv.org/abs/2109.03362 · HTML: https://mlsanity.com/paper/2109.03362/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 24 references that are in the map.
- First posted on arXiv 7 Sep 2021.

## Builds on (most cited first, 10 of 10)

- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [End to End Learning for Self-Driving Cars](https://mlsanity.com/paper/1604.07316/) (2016)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Explainable Machine Learning in Deployment](https://mlsanity.com/paper/1909.06342/) (2019)
- [Explainable Machine Learning for Scientific Insights and Discoveries](https://mlsanity.com/paper/1905.08883/) (2019)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Neural network identifiability for a family of sigmoidal nonlinearities](https://mlsanity.com/paper/1906.06994/) (2019)
- [Equivalent and Approximate Transformations of Deep Neural Networks](https://mlsanity.com/paper/1905.11428/) (2019)

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

- [Semialgebraic Neural Networks: From roots to representations](https://mlsanity.com/paper/2501.01564/) (2025)
- [Forgetting is Everywhere](https://mlsanity.com/paper/2511.04666/) (2025)

## Abstract (arXiv)

We define and establish the conditions for `equivalent neural networks' - neural networks with different weights, biases, and threshold functions that result in the same associated function. We prove that given a neural network N with piece-wise linear activation, the space of coefficients describing all equivalent neural networks is given by a semialgebraic set. This result is obtained by studying different representations of a given piece-wise linear function using the Tarski-Seidenberg theorem.