A Probabilistic Generative Model of Free Categories

TL;DR

This paper introduces a probabilistic generative model for morphisms in free monoidal categories, leveraging wiring diagrams and variational inference to learn and generate morphisms, with promising results on the Omniglot dataset.

Contribution

It presents the first probabilistic generative model for free categories, integrating category theory with machine learning techniques for morphism generation and inference.

Findings

Achieves competitive reconstruction performance on Omniglot

Demonstrates effective learning of category-theoretic structures

Shows potential for applications in program synthesis and category-based modeling

Abstract

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.