# NN2Poly: A polynomial representation for deep feed-forward artificial neural networks

arXiv: https://arxiv.org/abs/2112.11397 · HTML: https://mlsanity.com/paper/2112.11397/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 64 references that are in the map.
- First posted on arXiv 21 Dec 2021.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Opening the Black Box of Deep Neural Networks via Information](https://mlsanity.com/paper/1703.00810/) (2017)
- [Deep Neural Networks and Tabular Data: A Survey](https://mlsanity.com/paper/2110.01889/) (2021)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [LassoNet: A Neural Network with Feature Sparsity](https://mlsanity.com/paper/1907.12207/) (2019)
- [Deep Polynomial Neural Networks](https://mlsanity.com/paper/2006.13026/) (2020)
- [On Expressivity and Trainability of Quadratic Networks](https://mlsanity.com/paper/2110.06081/) (2021)
- [Polynomial Regression As an Alternative to Neural Nets](https://mlsanity.com/paper/1806.06850/) (2018)
- [PolyGAN: High-Order Polynomial Generators](https://mlsanity.com/paper/1908.06571/) (2019)
- [Towards a mathematical framework to inform Neural Network modelling via Polynomial Regression](https://mlsanity.com/paper/2102.03865/) (2021)
- [Neural Networks and Polynomial Regression. Demystifying the Overparametrization Phenomena](https://mlsanity.com/paper/2003.10523/) (2020)
- [AMITE: A Novel Polynomial Expansion for Analyzing Neural Network Nonlinearities](https://mlsanity.com/paper/2007.06226/) (2020)

## Built on by (most cited first, 3 of 3)

- [HOPE: High-order Polynomial Expansion of Black-box Neural Networks](https://mlsanity.com/paper/2307.08192/) (2023)
- [A mathematical certification for positivity conditions in Neural Networks with applications to partial monotonicity and Trustworthy AI](https://mlsanity.com/paper/2406.08525/) (2024)
- [nn2poly: An R Package for Converting Neural Networks into Interpretable Polynomials](https://mlsanity.com/paper/2406.01588/) (2024)

## Abstract (arXiv)

Interpretability of neural networks and their underlying theoretical behavior remain an open field of study even after the great success of their practical applications, particularly with the emergence of deep learning. In this work, NN2Poly is proposed: a theoretical approach to obtain an explicit polynomial model that provides an accurate representation of an already trained fully-connected feed-forward artificial neural network (a multilayer perceptron or MLP). This approach extends a previous idea proposed in the literature, which was limited to single hidden layer networks, to work with arbitrarily deep MLPs in both regression and classification tasks. NN2Poly uses a Taylor expansion on the activation function, at each layer, and then applies several combinatorial properties to calculate the coefficients of the desired polynomials. Discussion is presented on the main computational challenges of this method, and the way to overcome them by imposing certain constraints during the training phase. Finally, simulation experiments as well as applications to real tabular data sets are presented to demonstrate the effectiveness of the proposed method.