Aspects of Artificial Intelligence: Transforming Machine Learning Systems Naturally
Xiuzhan Guo
TL;DR
The paper reframes ML systems as categories $({\mathbf M},R)$ of elements and relations and argues that transformations between systems should be structure-preserving functors. It develops a categorical toolkit—quotients, Yoneda embeddings, presheaves, and representable functors—to analyze and compare ML elements and their interactions, and to model the data-model loop. Central contributions include formalizing ML system transformations as functors and natural transformations, introducing adjunctions as optimal problem-solving loops, and showing how monads and universal properties organize and extend ML objects. This provides a principled, compositional framework for reasoning about dynamic, data-driven ML pipelines and motivates algebraic extensions to ML systems for robust, scalable reasoning.
Abstract
In this paper, we study the machine learning elements which we are interested in together as a machine learning system, consisting of a collection of machine learning elements and a collection of relations between the elements. The relations we concern are algebraic operations, binary relations, and binary relations with composition that can be reasoned categorically. A machine learning system transformation between two systems is a map between the systems, which preserves the relations we concern. The system transformations given by quotient or clustering, representable functor, and Yoneda embedding are highlighted and discussed by machine learning examples. An adjunction between machine learning systems, a special machine learning system transformation loop, provides the optimal way of solving problems. Machine learning system transformations are linked and compared by their maps at 2-cell, natural transformations. New insights and structures can be obtained from universal properties and algebraic structures given by monads, which are generated from adjunctions.