# Revisiting Tropical Polynomial Division: Theory, Algorithms and Application to Neural Networks

arXiv: https://arxiv.org/abs/2306.15157 · HTML: https://mlsanity.com/paper/2306.15157/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 73 references that are in the map.
- First posted on arXiv 27 Jun 2023.

## Builds on (most cited first, 21 of 21)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [SNIP: Single-shot Network Pruning based on Connection Sensitivity](https://mlsanity.com/paper/1810.02340/) (2018)
- [What is the State of Neural Network Pruning?](https://mlsanity.com/paper/2003.03033/) (2020)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Towards Lower Bounds on the Depth of ReLU Neural Networks](https://mlsanity.com/paper/2105.14835/) (2021)
- [Learning Model Predictive Control for iterative tasks. A Data-Driven Control Framework](https://mlsanity.com/paper/1609.01387/) (2016)
- [Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums](https://mlsanity.com/paper/2104.08135/) (2021)
- [A Tropical Approach to Neural Networks with Piecewise Linear Activations](https://mlsanity.com/paper/1805.08749/) (2018)
- [On the Decision Boundaries of Neural Networks: A Tropical Geometry Perspective](https://mlsanity.com/paper/2002.08838/) (2020)
- [A Universal Approximation Result for Difference of log-sum-exp Neural Networks](https://mlsanity.com/paper/1905.08503/) (2019)
- [Deep Morphological Neural Networks](https://mlsanity.com/paper/1909.01532/) (2019)
- [Tropical Polynomial Division and Neural Networks](https://mlsanity.com/paper/1911.12922/) (2019)
- [Tropical Geometry and Piecewise-Linear Approximation of Curves and Surfaces on Weighted Lattices](https://mlsanity.com/paper/1912.03891/) (2019)
- [Data-driven Rollout for Deterministic Optimal Control](https://mlsanity.com/paper/2105.03116/) (2021)
- [Dynamical Systems On Weighted Lattices: General Theory](https://mlsanity.com/paper/1606.07347/) (2016)
- [Simple Policy Evaluation for Data-Rich Iterative Tasks](https://mlsanity.com/paper/1810.06764/) (2018)
- [Tropical Modeling of Weighted Transducer Algorithms on Graphs](https://mlsanity.com/paper/1811.00573/) (2018)
- [The Tropical Division Problem and the Minkowski Factorization of Generalized Permutahedra](https://mlsanity.com/paper/1908.00241/) (2019)

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

- [Relu and softplus neural nets as zero-sum turn-based games](https://mlsanity.com/paper/2512.20582/) (2025)
- [Maxout Polytopes](https://mlsanity.com/paper/2509.21286/) (2025)

## Abstract (arXiv)

Tropical geometry has recently found several applications in the analysis of neural networks with piecewise linear activation functions. This paper presents a new look at the problem of tropical polynomial division and its application to the simplification of neural networks. We analyze tropical polynomials with real coefficients, extending earlier ideas and methods developed for polynomials with integer coefficients. We first prove the existence of a unique quotient-remainder pair and characterize the quotient in terms of the convex bi-conjugate of a related function. Interestingly, the quotient of tropical polynomials with integer coefficients does not necessarily have integer coefficients. Furthermore, we develop a relationship of tropical polynomial division with the computation of the convex hull of unions of convex polyhedra and use it to derive an exact algorithm for tropical polynomial division. An approximate algorithm is also presented, based on an alternation between data partition and linear programming. We also develop special techniques to divide composite polynomials, described as sums or maxima of simpler ones. Finally, we present some numerical results to illustrate the efficiency of the algorithms proposed, using the MNIST handwritten digit and CIFAR-10 datasets.