  Data visualizations summarize high-dimensional distributions in two or three
dimensions. Dimensionality reduction entails a loss of information, and what is
preserved differs between methods. Existing methods preserve the local or the
global geometry of the points, and most techniques do not consider labels. Here
we introduce "hypersphere2sphere" (H2S), a new method that aims to visualize
not the points, but the relationships between the labeled distributions. H2S
fits a hypersphere to each labeled set of points in a high-dimensional space
and visualizes each hypersphere as a sphere in 3D (or circle in 2D). H2S
perfectly captures the geometry of up to 4 hyperspheres in 3D (or 3 in 2D), and
approximates the geometry for larger numbers of distributions, matching the
sizes (radii), and the pairwise separations (between-center distances) and
overlaps (along the center-connection line). The resulting visualizations are
robust to sampling imbalances. Leveraging labels and the sphere as the simplest
geometrical primitive, H2S provides an important addition to the toolbox of
visualization techniques.
