# HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections

arXiv: https://arxiv.org/abs/2106.03306 · HTML: https://mlsanity.com/paper/2106.03306/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 37 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 14 of 14)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Poincar'e GloVe: Hyperbolic Word Embeddings](https://mlsanity.com/paper/1810.06546/) (2018)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Neural Embeddings of Graphs in Hyperbolic Space](https://mlsanity.com/paper/1705.10359/) (2017)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Hyperbolic Representation Learning for Fast and Efficient Neural Question Answering](https://mlsanity.com/paper/1707.07847/) (2017)
- [A Theory of Hyperbolic Prototype Learning](https://mlsanity.com/paper/2010.07744/) (2020)
- [Nearest Neighbor Search for Hyperbolic Embeddings](https://mlsanity.com/paper/2009.00836/) (2020)

## Built on by (most cited first, 30 of 47)

- [Compositional Entailment Learning for Hyperbolic Vision-Language Models](https://mlsanity.com/paper/2410.06912/) (2024)
- [CO-SNE: Dimensionality Reduction and Visualization for Hyperbolic Data](https://mlsanity.com/paper/2111.15037/) (2021)
- [Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals](https://mlsanity.com/paper/2303.05798/) (2023)
- [Hyperbolic Audio-visual Zero-shot Learning](https://mlsanity.com/paper/2308.12558/) (2023)
- [HLFormer: Enhancing Partially Relevant Video Retrieval with Hyperbolic Learning](https://mlsanity.com/paper/2507.17402/) (2025)
- [Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections](https://mlsanity.com/paper/2211.10066/) (2022)
- [Curved Geometric Networks for Visual Anomaly Recognition](https://mlsanity.com/paper/2208.01188/) (2022)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [Small Transformers Compute Universal Metric Embeddings](https://mlsanity.com/paper/2209.06788/) (2022)
- [Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein](https://mlsanity.com/paper/2402.02239/) (2024)
- [Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/2302.06807/) (2023)
- [Hyperbolic Random Forests](https://mlsanity.com/paper/2308.13279/) (2023)
- [Autoencoding Hyperbolic Representation for Adversarial Generation](https://mlsanity.com/paper/2201.12825/) (2022)
- [Weisfeiler and Leman go Hyperbolic: Learning Distance Preserving Node Representations](https://mlsanity.com/paper/2211.02501/) (2022)
- [Fast hyperboloid decision tree algorithms](https://mlsanity.com/paper/2310.13841/) (2023)
- [SHMamba: Structured Hyperbolic State Space Model for Audio-Visual Question Answering](https://mlsanity.com/paper/2406.09833/) (2024)
- [PHyCLIP: ℓ1-Product of Hyperbolic Factors Unifies Hierarchy and Compositionality in Vision-Language Representation Learning](https://mlsanity.com/paper/2510.08919/) (2025)
- [Neural Networks on Symmetric Spaces of Noncompact Type](https://mlsanity.com/paper/2601.01097/) (2026)
- [Principal Component Analysis in Space Forms](https://mlsanity.com/paper/2301.02750/) (2023)
- [Leveraging Hyperbolic Embeddings for Coarse-to-Fine Robot Design](https://mlsanity.com/paper/2311.00462/) (2023)
- [Hyperbolic Busemann Neural Networks](https://mlsanity.com/paper/2602.18858/) (2026)
- [HyFI: Hyperbolic Feature Interpolation for Brain-Vision Alignment](https://mlsanity.com/paper/2603.22721/) (2026)
- [Hyp-UML: Hyperbolic Image Retrieval with Uncertainty-aware Metric Learning](https://mlsanity.com/paper/2310.08390/) (2023)
- [Agri-GNN: A Novel Genotypic-Topological Graph Neural Network Framework Built on GraphSAGE for Optimized Yield Prediction](https://mlsanity.com/paper/2310.13037/) (2023)
- [A group-theoretic framework for machine learning in hyperbolic spaces](https://mlsanity.com/paper/2501.06934/) (2025)
- [Understanding and Improving Hyperbolic Deep Reinforcement Learning](https://mlsanity.com/paper/2512.14202/) (2025)
- [Zoo Guide to Network Embedding](https://mlsanity.com/paper/2305.03474/) (2023)
- [Directed Scattering for Knowledge Graph-based Cellular Signaling Analysis](https://mlsanity.com/paper/2309.07813/) (2023)
- [Hyperbolic Delaunay Geometric Alignment](https://mlsanity.com/paper/2404.08608/) (2024)
- [HyperPath: Knowledge-Guided Hyperbolic Semantic Hierarchy Modeling for WSI Analysis](https://mlsanity.com/paper/2506.16398/) (2025)

## Abstract (arXiv)

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.