A Probabilistic Generative Model of Free Categories
This work addresses a niche problem in applied category theory and machine learning, offering an incremental approach by combining existing concepts from both fields.
The paper tackles the problem of generating morphisms in free monoidal categories by defining a probabilistic generative model that uses acyclic directed wiring diagrams, and it achieves competitive reconstruction performance on the Omniglot dataset.
Applied category theory has recently developed libraries for computing with morphisms in interesting categories, while machine learning has developed ways of learning programs in interesting languages. Taking the analogy between categories and languages seriously, this paper defines a probabilistic generative model of morphisms in free monoidal categories over domain-specific generating objects and morphisms. The paper shows how acyclic directed wiring diagrams can model specifications for morphisms, which the model can use to generate morphisms. Amortized variational inference in the generative model then enables learning of parameters (by maximum likelihood) and inference of latent variables (by Bayesian inversion). A concrete experiment shows that the free category prior achieves competitive reconstruction performance on the Omniglot dataset.