# Order Theory in the Context of Machine Learning

arXiv: https://arxiv.org/abs/2412.06097 · HTML: https://mlsanity.com/paper/2412.06097/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 39 references that are in the map.
- First posted on arXiv 8 Dec 2024.

## Builds on (most cited first, 13 of 13)

- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Maxout Networks](https://mlsanity.com/paper/1302.4389/) (2013)
- [The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks](https://mlsanity.com/paper/2202.08658/) (2022)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [The staircase property: How hierarchical structure can guide deep learning](https://mlsanity.com/paper/2108.10573/) (2021)
- [Pooling Methods in Deep Neural Networks, a Review](https://mlsanity.com/paper/2009.07485/) (2020)
- [Quaternion Convolutional Neural Networks](https://mlsanity.com/paper/1903.00658/) (2019)
- [Quaternion Recurrent Neural Networks](https://mlsanity.com/paper/1806.04418/) (2018)
- [An Alternative Practice of Tropical Convolution to Traditional Convolutional Neural Networks](https://mlsanity.com/paper/2103.02096/) (2021)
- [Convolutional neural networks applied to modification of images](https://mlsanity.com/paper/2410.05680/) (2024)

## Abstract (arXiv)

The paper ``Tropical Geometry of Deep Neural Networks'' by L. Zhang et al. introduces an equivalence between integer-valued neural networks (IVNN) with ReLUt and tropical rational functions, which come with a map to polytopes. Here, IVNN refers to a network with integer weights but real biases, and ReLUt is defined as ReLUt(x)=max(x,t) for t∈ℝcup-∞. For every poset with n points, there exists a corresponding order polytope, i.e., a convex polytope in the unit cube [0,1]n whose coordinates obey the inequalities of the poset. We study neural networks whose associated polytope is an order polytope. We then explain how posets with four points induce neural networks that can be interpreted as 2× 2 convolutional filters. These poset filters can be added to any neural network, not only IVNN. Similarly to maxout, poset pooling filters update the weights of the neural network during backpropagation with more precision than average pooling, max pooling, or mixed pooling, without the need to train extra parameters. We report experiments that support our statements. We also define the structure of algebra over the operad of posets on poset neural networks and tropical polynomials. This formalism allows us to study the composition of poset neural network arquitectures and the effect on their corresponding Newton polytopes, via the introduction of the generalization of two operations on polytopes: the Minkowski sum and the convex envelope.