The Theory behind UMAP?
This work clarifies the theoretical basis of UMAP for researchers and practitioners, but it is incremental as it focuses on fixing existing errors rather than introducing new methods.
The paper addresses errors in the theoretical foundation of UMAP, a popular dimensionality reduction algorithm, by repairing mistakes from an unpublished draft and providing a corrected, self-contained derivation of the underlying functors and their finite variant.
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 Their work introduces a finite variant of a functor called the metric realization, based on an unpublished draft by Spivak.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.