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HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections

arXiv 2106.03306 ↗First posted 7 Jun 2021cs.LGcited by 47

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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.

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

Builds on

14 of 37 references.

References · 14

  1. 2017Poincar'e Embeddings for Learning Hierarchical Representationscited by 674
  2. 2019Hyperbolic Graph Convolutional Neural Networkscited by 400
  3. 2018Hyperbolic Neural Networkscited by 393
  4. 2019Hyperbolic Graph Neural Networkscited by 244
  5. 2018Representation Tradeoffs for Hyperbolic Embeddingscited by 204
  6. 2020Low-Dimensional Hyperbolic Knowledge Graph Embeddingscited by 195
  7. 2018Poincar'e GloVe: Hyperbolic Word Embeddingscited by 155
  8. 2019Multi-relational Poincar'e Graph Embeddingscited by 126
  9. 2020From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clusteringcited by 65
  10. 2017Neural Embeddings of Graphs in Hyperbolic Spacecited by 58
  11. 2018Large-Margin Classification in Hyperbolic Spacecited by 41
  12. 2017Hyperbolic Representation Learning for Fast and Efficient Neural Question Answeringcited by 23
2 more
  1. 2020A Theory of Hyperbolic Prototype Learningcited by 3
  2. 2020Nearest Neighbor Search for Hyperbolic Embeddingscited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 5; 2023: 12; 2024: 6; 2025: 11; 2026: 11.

47 papers.

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  1. 2024Compositional Entailment Learning for Hyperbolic Vision-Language Modelscited by 72
  2. 2021CO-SNE: Dimensionality Reduction and Visualization for Hyperbolic Datacited by 28
  3. 2023Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signalscited by 19
  4. 2023Hyperbolic Audio-visual Zero-shot Learningcited by 19
  5. 2025HLFormer: Enhancing Partially Relevant Video Retrieval with Hyperbolic Learningcited by 17
  6. 2022Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projectionscited by 16
  7. 2022Curved Geometric Networks for Visual Anomaly Recognitioncited by 15
  8. 2021Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Designcited by 14
  9. 2022Small Transformers Compute Universal Metric Embeddingscited by 12
  10. 2024Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wassersteincited by 12
  11. 2023Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Spacecited by 11
  12. 2023Hyperbolic Random Forestscited by 9
35 more
  1. 2022Autoencoding Hyperbolic Representation for Adversarial Generationcited by 7
  2. 2022Weisfeiler and Leman go Hyperbolic: Learning Distance Preserving Node Representationscited by 7
  3. 2023Fast hyperboloid decision tree algorithmscited by 7
  4. 2024SHMamba: Structured Hyperbolic State Space Model for Audio-Visual Question Answeringcited by 7
  5. 2025PHyCLIP: ℓ1-Product of Hyperbolic Factors Unifies Hierarchy and Compositionality in Vision-Language Representation Learningcited by 7
  6. 2023Principal Component Analysis in Space Formscited by 5
  7. 2026Neural Networks on Symmetric Spaces of Noncompact Typecited by 5
  8. 2023Leveraging Hyperbolic Embeddings for Coarse-to-Fine Robot Designcited by 4
  9. 2026HyFI: Hyperbolic Feature Interpolation for Brain-Vision Alignmentcited by 4
  10. 2023Hyp-UML: Hyperbolic Image Retrieval with Uncertainty-aware Metric Learningcited by 3
  11. 2023Agri-GNN: A Novel Genotypic-Topological Graph Neural Network Framework Built on GraphSAGE for Optimized Yield Predictioncited by 3
  12. 2023Zoo Guide to Network Embeddingcited by 2
  13. 2023Directed Scattering for Knowledge Graph-based Cellular Signaling Analysiscited by 2
  14. 2024Hyperbolic Delaunay Geometric Alignmentcited by 2
  15. 2025A group-theoretic framework for machine learning in hyperbolic spacescited by 2
  16. 2025HyperPath: Knowledge-Guided Hyperbolic Semantic Hierarchy Modeling for WSI Analysiscited by 2
  17. 2025Geometry-aware Distance Measure for Diverse Hierarchical Structures in Hyperbolic Spacescited by 2
  18. 2025Learning Protein-Ligand Binding in Hyperbolic Spacecited by 2
  19. 2026Hyperbolic Busemann Neural Networkscited by 2
  20. 2026Uncertainty-guided Compositional Alignment with Part-to-Whole Semantic Representativeness in Hyperbolic Vision-Language Modelscited by 2
  21. 2023O(k)-Equivariant Dimensionality Reduction on Stiefel Manifoldscited by 1
  22. 2024Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Formscited by 1
  23. 2025Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicingcited by 1
  24. 2025Randomized HyperSteiner: A Stochastic Delaunay Triangulation Heuristic for the Hyperbolic Steiner Minimal Treecited by 1
  25. 2025Understanding and Improving Hyperbolic Deep Reinforcement Learningcited by 1
  26. 2025H2em: Learning Hierarchical Hyperbolic Embeddings for Compositional Zero-Shot Learningcited by 1
  27. 2026ARGENT: Adaptive Hierarchical Image-Text Representationscited by 1
  28. 2024Tempered Calculus for ML: Application to Hyperbolic Model Embeddingcited by 0
  29. 2025Cartan Networks: Group theoretical Hyperbolic Deep Learningcited by 0
  30. 2026Intrinsic Concept Extraction Based on Compositional Interpretabilitycited by 0
  31. 2026HAC: Parameter-Efficient Hyperbolic Adaptation of CLIP for Zero-Shot VQAcited by 0
  32. 2026HyFL-CLIP: Hyperbolic Fine-Tuning of CLIP for Robust Long-Context Understandingcited by 0
  33. 20261-Lipschitz Neural Networks on Hadamard Manifoldscited by 0
  34. 2026hyperbolix: Hyperbolic Deep Learning in JAXcited by 0
  35. 2026Rethinking the Fully Hyperbolic Vision Transformer in Polar Coordinatescited by 0