Using Enriched Category Theory to Construct the Nearest Neighbour Classification Algorithm (2312.16529v2)

Abstract: This paper is the first to construct and motivate a Machine Learning algorithm solely with Enriched Category Theory, supplementing evidence that Category Theory can provide valuable insights into the construction and explainability of Machine Learning algorithms. It is shown that a series of reasonable assumptions about a dataset lead to the construction of the Nearest Neighbours Algorithm. This construction is produced as an extension of the original dataset using profunctors in the category of Lawvere metric spaces, leading to a definition of an Enriched Nearest Neighbours Algorithm, which, consequently, also produces an enriched form of the Voronoi diagram. Further investigation of the generalisations this construction induces demonstrates how the $k$ Nearest Neighbours Algorithm may also be produced. Moreover, how the new construction allows metrics on the classification labels to inform the outputs of the Enriched Nearest Neighbour Algorithm: Enabling soft classification boundaries and dependent classifications. This paper is intended to be accessible without any knowledge of Category Theory.

• The paper demonstrates that Enriched Category Theory systematically constructs the traditional NNA via profunctors in Lawvere metric spaces.
• It introduces an enriched Voronoi diagram that enhances transparency and interpretability in classification tasks.
• The approach provides a theoretical basis to reframe other machine learning algorithms, paving the way for future research in categorical ML.

Abstract

The paper presents a novel use of Enriched Category Theory to construct and explain a machine learning algorithm known as the Nearest Neighbour Algorithm (NNA). The authors demonstrate that Enriched Category Theory can serve as a theoretical framework for developing robust and interpretable algorithms by systematically constructing the NNA through the application of this advanced mathematical concept. Specifically, they rely on profunctors in the category of Lawvere metric spaces to create an algorithm that not only performs classification tasks but also generates an enriched version of the Voronoi diagram.

Introduction

In the machine learning domain, there's a recognized tension between the intuitive, heuristic-based algorithm design and the need for clear, explainable models. As machine learning algorithms often work as "black boxes," their inner workings can be opaque, making diagnosis, improvement, and behavioral guarantees challenging. This paper addresses this challenge by leveraging the comparative nature inherent to learning tasks with the comprehensive formalism of Enriched Category Theory, allowing algorithms' data interactions to be encoded explicitly rather than implicitly.

Background and Theory

Enriched Category Theory provides a systematic way to define and compare complex structures, making explicit the relationships that, in machine learning, typically remain implicit. The theory emphasizes comparisons within data, using profunctors to measure relationships, rather than direct representation of data points. The researchers apply this approach to Lawvere metric spaces, which in essence can be viewed as a generalized space where the notion of distance between data points is defined but does not strictly adhere to all traditional metric axioms. This enriched structure provides a more nuanced view of data and their relationships, potentially leading to more transparent machine learning models and algorithms.

Algorithm Construction

The Nearest Neighbour Algorithm typically classifies data points by considering their proximity to the known data points in a metric space. Utilizing Enriched Category Theory, the authors reconstruct NNA in a more abstract setting, replacing the standard datasets with enriched categorical constructs. They demonstrate that under certain conditions, their enriched theoretical framework leads to an algorithm that behaves identical to the traditional NNA but brings the advantages of categorical insight, potentially facilitating a better understanding of the algorithm's behavior and efficacy. This categorical NNA operates through intertwined functors and profunctors that encode distances and classes within the metric space, yielding an enriched classifier that is both intellectually satisfying and practically useful.

Future Directions and Conclusion

Looking forward, the potential of Enriched Category Theory extends beyond the specifics of the NNA, as many machine learning algorithms could potentially be reframed within this categorical language. The authors suggest exploring k Nearest Neighbours Algorithm within this context as well as examining how the enriched structures handle various types of data and learning problems. Moreover, they speculate that certain machine learning challenges could be addressed by selecting the most suitable "base of enrichment," rather than continually inventing new algorithms. This paper underscores the significance of the emerging field of categorical machine learning, proposing that Enriched Category Theory—often seen as a highly abstract mathematical theory—has practical applications in the field of machine learning and warrants further attention from both mathematicians and computer scientists.