Order Theory in the Context of Machine Learning

TL;DR

The paper develops a rigorous bridge between order theory and tropical geometry to inform machine learning architectures. It defines poset neural networks whose tropical polynomials correspond to order polytopes, and introduces poset pooling filters that can rival traditional pooling while preserving more structure. An operadic framework is established, enabling composition of poset-based architectures via the lexicographic sum and related constructions, with actions extended to polytopes and tropical polynomials through Minkowski sums and convex envelopes. Empirical results across CNNs, quaternion networks, and compact models demonstrate potential gains in accuracy and parameter efficiency, motivating broader exploration across datasets and architectures. The work provides a principled, geometry-driven lens for constructing and combining neural networks with explicit order-theoretic constraints, enriching both theory and practice.

Abstract

The paper ``Tropical Geometry of Deep Neural Networks'' by L. Zhang et al. introduces an equivalence between integer-valued neural networks (IVNN) with $\text{ReLU}_{t}$ and tropical rational functions, which come with a map to polytopes. Here, IVNN refers to a network with integer weights but real biases, and $\text{ReLU}_{t}$ is defined as $\text{ReLU}_{t}(x)=\max(x,t)$ for $t\in\mathbb{R}\cup\{-\infty\}$. 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 $2\times 2$ 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.

Figures (4)

• Figure 1: Different linearizations of the poset $\causets@Padding \causets@Padding$.
• Figure 2: Histogram of the image of the lattice points in the sphere under different posets transformations.
• Figure 3: A plot of accuracy/loss per epoch for train/validation data on the original SimpleNet architecture, with 700 epochs. The loss is multiplied by 50. There are learning rate changes at epochs 100, 190, 306, 390, 440, 540, which correspond to the main jumps in train/valid accuracy. All experiments returned similar plots.
• Figure 4: Images b), c), d), and e) have half the dimensions of image a).

Theorems & Definitions (89)

• Remark 1
• Example 1
• Definition 1
• Example 2
• Remark 2
• Definition 2
• Definition 3
• Example 3
• Remark 3
• Definition 4