A deeper dive into UMAP theory.
In its simplest sense, the UMAP algorithm consists of two steps: construction of a graph in high dimensions followed by an optimization step to find the most similar graph in lower dimensions. In order to achieve this goal, the algorithm relies on a number of insights from algebraic topology and Riemannian geometry. Despite the intimidating mathematics, the intuitions behind the core principles are actually quite simple: UMAP essentially constructs a weighted graph from the high dimensional data, with edge strength representing how “close” a given point is to another, then projects this graph down to a lower dimensionality. The advanced mathematics gives UMAP a solid footing with which to handle the challenges of doing this in high dimensions with real data.