  Neighbor embedding methods $t$-SNE and UMAP are the de facto standard for
visualizing high-dimensional datasets. Motivated from entirely different
viewpoints, their loss functions appear to be unrelated. In practice, they
yield strongly differing embeddings and can suggest conflicting interpretations
of the same data. The fundamental reasons for this and, more generally, the
exact relationship between $t$-SNE and UMAP have remained unclear. In this
work, we uncover their conceptual connection via a new insight into contrastive
learning methods. Noise-contrastive estimation can be used to optimize $t$-SNE,
while UMAP relies on negative sampling, another contrastive method. We find the
precise relationship between these two contrastive methods and provide a
mathematical characterization of the distortion introduced by negative
sampling. Visually, this distortion results in UMAP generating more compact
embeddings with tighter clusters compared to $t$-SNE. We exploit this new
conceptual connection to propose and implement a generalization of negative
sampling, allowing us to interpolate between (and even extrapolate beyond)
$t$-SNE and UMAP and their respective embeddings. Moving along this spectrum of
embeddings leads to a trade-off between discrete / local and continuous /
global structures, mitigating the risk of over-interpreting ostensible features
of any single embedding. We provide a PyTorch implementation.
