Coalgebras for categorical deep learning: Representability and universal approximation

TL;DR

This paper develops a coalgebraic framework for categorical deep learning, establishing a universal approximation theorem for equivariant maps and providing a categorical foundation for invariant representations in neural networks.

Contribution

It introduces a coalgebraic formalism for equivariant representations and proves a universal approximation theorem within this categorical setting.

Findings

Coalgebraic formalism generalizes group actions and equivariant maps.

A functorial embedding relates data sets to vector spaces preserving invariance.

Continuous equivariant functions can be universally approximated in this framework.

Abstract

Categorical deep learning (CDL) has recently emerged as a framework that leverages category theory to unify diverse neural architectures. While geometric deep learning (GDL) is grounded in the specific context of invariants of group actions, CDL aims to provide domain-independent abstractions for reasoning about models and their properties. In this paper, we contribute to this program by developing a coalgebraic foundation for equivariant representation in deep learning, as classical notions of group actions and equivariant maps are naturally generalized by the coalgebraic formalism. Our first main result demonstrates that, given an embedding of data sets formalized as a functor from SET to VECT, and given a notion of invariant behavior on data sets modeled by an endofunctor on SET, there is a corresponding endofunctor on VECT that is compatible with the embedding in the sense that this lifted functor recovers the analogous notion of invariant behavior on the embedded data. Building on this foundation, we then establish a universal approximation theorem for equivariant maps in this generalized setting. We show that continuous equivariant functions can be approximated within our coalgebraic framework for a broad class of symmetries. This work thus provides a categorical bridge between the abstract specification of invariant behavior and its concrete realization in neural architectures.