  tSNE and UMAP are popular dimensionality reduction algorithms due to their
speed and interpretable low-dimensional embeddings. Despite their popularity,
however, little work has been done to study their full span of differences. We
theoretically and experimentally evaluate the space of parameters in both tSNE
and UMAP and observe that a single one -- the normalization -- is responsible
for switching between them. This, in turn, implies that a majority of the
algorithmic differences can be toggled without affecting the embeddings. We
discuss the implications this has on several theoretic claims behind UMAP, as
well as how to reconcile them with existing tSNE interpretations.
  Based on our analysis, we provide a method (\ourmethod) that combines
previously incompatible techniques from tSNE and UMAP and can replicate the
results of either algorithm. This allows our method to incorporate further
improvements, such as an acceleration that obtains either method's outputs
faster than UMAP. We release improved versions of tSNE, UMAP, and \ourmethod
that are fully plug-and-play with the traditional libraries at
https://github.com/Andrew-Draganov/GiDR-DUN
