# Graph Convolutional Neural Networks as Parametric CoKleisli morphisms

arXiv: https://arxiv.org/abs/2212.00542 · HTML: https://mlsanity.com/paper/2212.00542/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 18 references that are in the map.
- First posted on arXiv 1 Dec 2022.

## Builds on (most cited first, 9 of 9)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Recent Advances in Convolutional Neural Networks](https://mlsanity.com/paper/1512.07108/) (2015)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Reverse derivative categories](https://mlsanity.com/paper/1910.07065/) (2019)

## Abstract (arXiv)

We define the bicategory of Graph Convolutional Neural Networks GCNNn for an arbitrary graph with n nodes. We show it can be factored through the already existing categorical constructions for deep learning called Para and Lens with the base category set to the CoKleisli category of the product comonad. We prove that there exists an injective-on-objects, faithful 2-functor GCNNn → Para(mathsfCoKl(ℝn × n × -)). We show that this construction allows us to treat the adjacency matrix of a GCNN as a global parameter instead of a a local, layer-wise one. This gives us a high-level categorical characterisation of a particular kind of inductive bias GCNNs possess. Lastly, we hypothesize about possible generalisations of GCNNs to general message-passing graph neural networks, connections to equivariant learning, and the (lack of) functoriality of activation functions.