HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections
This addresses dimensionality reduction for data in hyperbolic spaces, which is incremental as it generalizes PCA concepts to a specific geometric setting.
The paper tackles the problem of principal component analysis (PCA) for data in hyperbolic spaces by proposing HoroPCA, a method for hyperbolic dimensionality reduction that better preserves distances and reduces error in distance preservation, improving downstream classification by up to 3.9%.
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.