  This paper studies Principal Component Analysis (PCA) for data lying in
hyperbolic spaces. Given directions, PCA relies on: (1) a parameterization of
subspaces spanned by these directions, (2) a method of projection onto
subspaces that preserves information in these directions, and (3) an objective
to optimize, namely the variance explained by projections. We generalize each
of these concepts to the hyperbolic space and propose HoroPCA, a method for
hyperbolic dimensionality reduction. By focusing on the core problem of
extracting principal directions, HoroPCA theoretically better preserves
information in the original data such as distances, compared to previous
generalizations of PCA. Empirically, we validate that HoroPCA outperforms
existing dimensionality reduction methods, significantly reducing error in
distance preservation. As a data whitening method, it improves downstream
classification by up to 3.9% compared to methods that don't use whitening.
Finally, we show that HoroPCA can be used to visualize hyperbolic data in two
dimensions.
