  TSNE and UMAP are two of the most popular dimensionality reduction algorithms
due to their speed and interpretable low-dimensional embeddings. However, while
attempts have been made to improve on TSNE's computational complexity, no
existing method can obtain TSNE embeddings at the speed of UMAP. In this work,
we show that this is indeed possible by combining the two approaches into a
single method. We theoretically and experimentally evaluate the full space of
parameters in the TSNE and UMAP algorithms and observe that a single parameter,
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 underpinning the UMAP framework, as well as how to reconcile
them with existing TSNE interpretations. Based on our analysis, we propose a
new dimensionality reduction algorithm, GDR, that combines previously
incompatible techniques from TSNE and UMAP and can replicate the results of
either algorithm by changing the normalization. As a further advantage, GDR
performs the optimization faster than available UMAP methods and thus an order
of magnitude faster than available TSNE methods. Our implementation is
plug-and-play with the traditional UMAP and TSNE libraries and can be found at
github.com/Andrew-Draganov/GiDR-DUN.
