Graph Convolutional Neural Networks as Parametric CoKleisli morphisms
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We define the bicategory of Graph Convolutional Neural Networks ππππ_n for an arbitrary graph with n nodes. We show it can be factored through the already existing categorical constructions for deep learning called πππ«π and πππ§π¬ 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 ππππ_n βπππ«π(π’ππͺπ (β^n Γ nΓ -)). 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.
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