The Topos of Transformer Networks

The transformer neural network has significantly out-shined all other neural network architectures as the engine behind large language models. We provide a theoretical analysis of the expressivity of the transformer architecture through the lens of topos theory. From this viewpoint, we show that many common neural network architectures, such as the convolutional, recurrent and graph convolutional networks, can be embedded in a pretopos of piecewise-linear functions, but that the transformer necessarily lives in its topos completion. In particular, this suggests that the two network families instantiate different fragments of logic: the former are first order, whereas transformers are higher-order reasoners. Furthermore, we draw parallels with architecture search and gradient descent, integrating our analysis in the framework of cybernetic agents.

Any Deep ReLU Network is Shallow

We constructively prove that every deep ReLU network can be rewritten as a functionally identical three-layer network with weights valued in the extended reals. Based on this proof, we provide an algorithm that, given a deep ReLU network, finds the explicit weights of the corresponding shallow network. The resulting shallow network is transparent and used to generate explanations of the model s behaviour.

Unwrapping All ReLU Networks

Deep ReLU Networks can be decomposed into a collection of linear models, each defined in a region of a partition of the input space. This paper provides three results extending this theory. First, we extend this linear decompositions to Graph Neural networks and tensor convolutional networks, as well as networks with multiplicative interactions. Second, we provide proofs that neural networks can be understood as interpretable models such as Multivariate Decision trees and logical theories. Finally, we show how this model leads to computing cheap and exact SHAP values. We validate the theory through experiments with on Graph Neural Networks.