The Theory behind UMAP?

TL;DR

This paper critically examines the theoretical foundations of UMAP, correcting errors in prior unpublished drafts, and provides a comprehensive derivation of the underlying functors and their relation to the algorithm.

Contribution

It offers a corrected, self-contained derivation of the metric realization and related functors underlying UMAP, clarifying its theoretical basis.

Findings

Provides an explicit description of the metric realization.

Corrects errors in the original unpublished draft by Spivak.

Discusses the theoretical properties and correspondence of UMAP.

Abstract

In 2018, McInnes et al. introduced a dimensionality reduction algorithm called UMAP, which enjoys wide popularity among data scientists. Their work introduces a finite variant of a functor called the metric realization, based on an unpublished draft by Spivak. This draft contains many errors, most of which are reproduced by McInnes et al. and subsequent publications. This article aims to repair these errors and provide a self-contained document with the full derivation of Spivak's functors and McInnes et al.'s finite variant. We contribute an explicit description of the metric realization and related functors. At the end, we discuss the UMAP algorithm, as well as claims about properties of the algorithm and the correspondence of McInnes et al.'s finite variant to the UMAP algorithm.