# HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections

Source: [HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections](<https://www.sota2.com/research/paper/68fa6a5f1e209131cea27d0f>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T23:23:37.152Z

Authors: Ines Chami, Albert Gu, Dat Nguyen, Christopher R\\'e

Published: 2021-06-07T02:32:56.000Z

## Abstract

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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Reconstruction on phylo tree (test)](<https://www.sota2.com/research/sota/reconstruction-on-phylo-tree-test.md>): Reconstruction; phylo tree (test); Mean Reconstruction Error: 108.62.

- [Reconstruction on balancedtree synthetic (test)](<https://www.sota2.com/research/sota/reconstruction-on-balancedtree-synthetic-test.md>): Reconstruction; balancedtree synthetic (test); Mean Reconstruction Error: 7.8.

- [Reconstruction on unbalanced1 synthetic (test)](<https://www.sota2.com/research/sota/reconstruction-on-unbalanced1-synthetic-test.md>): Reconstruction; unbalanced1 synthetic (test); Mean Reconstruction Error: 6.51.

- [Reconstruction on unbalanced2 synthetic (test)](<https://www.sota2.com/research/sota/reconstruction-on-unbalanced2-synthetic-test.md>): Reconstruction; unbalanced2 synthetic (test); Mean Reconstruction Error: 7.35.

- [Reconstruction on diseasome (test)](<https://www.sota2.com/research/sota/reconstruction-on-diseasome-test.md>): Reconstruction; diseasome (test); Mean Reconstruction Error: 26.94.

- [Reconstruction on ca-CSphd (test)](<https://www.sota2.com/research/sota/reconstruction-on-ca-csphd-test.md>): Reconstruction; ca-CSphd (test); Mean Reconstruction Error: 87.99.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2106.03306>)

[PDF](<https://arxiv.org/pdf/2106.03306.pdf>)