Hyperbolic space can naturally embed hierarchies that often exist in
real-world data and semantics. While high-dimensional hyperbolic embeddings
lead to better representations, most hyperbolic models utilize low-dimensional
embeddings, due to non-trivial optimization and visualization of
high-dimensional hyperbolic data.
We propose CO-SNE, which extends the Euclidean space visualization tool,
t-SNE, to hyperbolic space. Like t-SNE, it converts distances between data
points to joint probabilities and tries to minimize the Kullback-Leibler
divergence between the joint probabilities of high-dimensional data $X$ and
low-dimensional embedding $Y$. However, unlike Euclidean space, hyperbolic
space is inhomogeneous: A volume could contain a lot more points at a location
far from the origin. CO-SNE thus uses hyperbolic normal distributions for $X$
and hyperbolic \underline{C}auchy instead of t-SNE's Student's t-distribution
for $Y$, and it additionally seeks to preserve $X$'s individual distances to
the \underline{O}rigin in $Y$.
We apply CO-SNE to naturally hyperbolic data and supervisedly learned
hyperbolic features. Our results demonstrate that CO-SNE deflates
high-dimensional hyperbolic data into a low-dimensional space without losing
their hyperbolic characteristics, significantly outperforming popular
visualization tools such as PCA, t-SNE, UMAP, and HoroPCA which is also
designed for hyperbolic data.

本文提出了 CO-SNE 将 Euclidean 空间降维工具 t-SNE 扩展到 hyperbolic 空间，采用超几何正态分布和超几何 Cauchy 分布，旨在保留数据点之间的距离和数据的超几何特性，经实验证明效果优于 PCA，t-SNE，UMAP 和 HoroPCA 等流行的可视化工具。