  Dimension reduction (DR) techniques such as t-SNE, UMAP, and TriMAP have
demonstrated impressive visualization performance on many real world datasets.
One tension that has always faced these methods is the trade-off between
preservation of global structure and preservation of local structure: these
methods can either handle one or the other, but not both. In this work, our
main goal is to understand what aspects of DR methods are important for
preserving both local and global structure: it is difficult to design a better
method without a true understanding of the choices we make in our algorithms
and their empirical impact on the lower-dimensional embeddings they produce.
Towards the goal of local structure preservation, we provide several useful
design principles for DR loss functions based on our new understanding of the
mechanisms behind successful DR methods. Towards the goal of global structure
preservation, our analysis illuminates that the choice of which components to
preserve is important. We leverage these insights to design a new algorithm for
DR, called Pairwise Controlled Manifold Approximation Projection (PaCMAP),
which preserves both local and global structure. Our work provides several
unexpected insights into what design choices both to make and avoid when
constructing DR algorithms.
