Order Theory in the Context of Machine Learning

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.