  Topology based dimensionality reduction methods such as t-SNE and UMAP have
seen increasing success and popularity in high-dimensional data. These methods
have strong mathematical foundations and are based on the intuition that the
topology in low dimensions should be close to that of high dimensions. Given
that the initial topological structure is a precursor to the success of the
algorithm, this naturally raises the question: What makes a "good" topological
structure for dimensionality reduction? Insight into this will enable us to
design better algorithms which take into account both local and global
structure. In this paper which focuses on UMAP, we study the effects of node
connectivity (k-Nearest Neighbors vs mutual k-Nearest Neighbors) and relative
neighborhood (Adjacent via Path Neighbors) on dimensionality reduction. We
explore these concepts through extensive ablation studies on 4 standard image
and text datasets; MNIST, FMNIST, 20NG, AG, reducing to 2 and 64 dimensions.
Our findings indicate that a more refined notion of connectivity (mutual
k-Nearest Neighbors with minimum spanning tree) together with a flexible method
of constructing the local neighborhood (Path Neighbors), can achieve a much
better representation than default UMAP, as measured by downstream clustering
performance.
