# CO-SNE: Dimensionality Reduction and Visualization for Hyperbolic Data

arXiv: https://arxiv.org/abs/2111.15037 · HTML: https://mlsanity.com/paper/2111.15037/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 34 references that are in the map.
- First posted on arXiv 30 Nov 2021.

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

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [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 Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Hyperbolic Attention Networks](https://mlsanity.com/paper/1805.09786/) (2018)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [An Analysis of the t-SNE Algorithm for Data Visualization](https://mlsanity.com/paper/1803.01768/) (2018)
- [HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections](https://mlsanity.com/paper/2106.03306/) (2021)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Unsupervised Discovery of the Long-Tail in Instance Segmentation Using Hierarchical Self-Supervision](https://mlsanity.com/paper/2104.01257/) (2021)
- [Robust Large-Margin Learning in Hyperbolic Space](https://mlsanity.com/paper/2004.05465/) (2020)

## Built on by (most cited first, 28 of 28)

- [Compositional Entailment Learning for Hyperbolic Vision-Language Models](https://mlsanity.com/paper/2410.06912/) (2024)
- [Graph-level Representation Learning with Joint-Embedding Predictive Architectures](https://mlsanity.com/paper/2309.16014/) (2023)
- [Hyperbolic Audio-visual Zero-shot Learning](https://mlsanity.com/paper/2308.12558/) (2023)
- [Curved Geometric Networks for Visual Anomaly Recognition](https://mlsanity.com/paper/2208.01188/) (2022)
- [Learning Weakly Supervised Audio-Visual Violence Detection in Hyperbolic Space](https://mlsanity.com/paper/2305.18797/) (2023)
- [Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein](https://mlsanity.com/paper/2402.02239/) (2024)
- [Hyperbolic-constraint Point Cloud Reconstruction from Single RGB-D Images](https://mlsanity.com/paper/2412.09055/) (2024)
- [SHMamba: Structured Hyperbolic State Space Model for Audio-Visual Question Answering](https://mlsanity.com/paper/2406.09833/) (2024)
- [OpenHype: Hyperbolic Embeddings for Hierarchical Open-Vocabulary Radiance Fields](https://mlsanity.com/paper/2510.21441/) (2025)
- [Interpretable Dimensionality Reduction by Feature Preserving Manifold Approximation and Projection](https://mlsanity.com/paper/2211.09321/) (2022)
- [Hyp-UML: Hyperbolic Image Retrieval with Uncertainty-aware Metric Learning](https://mlsanity.com/paper/2310.08390/) (2023)
- [Accelerating hyperbolic t-SNE](https://mlsanity.com/paper/2401.13708/) (2024)
- [Unsupervised Feature Learning with Emergent Data-Driven Prototypicality](https://mlsanity.com/paper/2307.01421/) (2023)
- [Hyperbolic Delaunay Geometric Alignment](https://mlsanity.com/paper/2404.08608/) (2024)
- [Improving Hyperbolic Representations via Gromov-Wasserstein Regularization](https://mlsanity.com/paper/2407.10495/) (2024)
- [HyperPath: Knowledge-Guided Hyperbolic Semantic Hierarchy Modeling for WSI Analysis](https://mlsanity.com/paper/2506.16398/) (2025)
- [Learning Protein-Ligand Binding in Hyperbolic Space](https://mlsanity.com/paper/2508.15480/) (2025)
- [HyHTM: Hyperbolic Geometry based Hierarchical Topic Models](https://mlsanity.com/paper/2305.09258/) (2023)
- [Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Forms](https://mlsanity.com/paper/2411.10889/) (2024)
- [Machine Unlearning in Hyperbolic vs. Euclidean Multimodal Contrastive Learning: Adapting Alignment Calibration to MERU](https://mlsanity.com/paper/2503.15166/) (2025)
- [Normalized Maximum Likelihood Code-Length on Riemannian Data Spaces](https://mlsanity.com/paper/2508.21466/) (2025)
- [Randomized HyperSteiner: A Stochastic Delaunay Triangulation Heuristic for the Hyperbolic Steiner Minimal Tree](https://mlsanity.com/paper/2510.09328/) (2025)
- [HyperAlign: Hyperbolic Entailment Cones for Adaptive Text-to-Image Alignment Assessment](https://mlsanity.com/paper/2601.04614/) (2026)
- [Attend to Anything: Foundation Model for Unified Human Attention Modeling](https://mlsanity.com/paper/2606.03540/) (2026)
- [hyperbolix: Hyperbolic Deep Learning in JAX](https://mlsanity.com/paper/2609.28248/) (2026)
- [Density-based Isometric Mapping](https://mlsanity.com/paper/2403.02531/) (2024)
- [HyNeuralMap: Hyperbolic Mapping of Visual Semantics to Neural Hierarchies](https://mlsanity.com/paper/2605.09392/) (2026)
- [HypOProto: Hyperbolic Ordinal Prototypes for Left Ventricular Filling Pressure Classification](https://mlsanity.com/paper/2606.19804/) (2026)

## Abstract (arXiv)

Hyperbolic space can naturally embed hierarchies that often exist in real-world data and semantics. While high-dimensional hyperbolic embeddings lead to better representations, most hyperbolic models utilize low-dimensional embeddings, due to non-trivial optimization and visualization of high-dimensional hyperbolic data. We propose CO-SNE, which extends the Euclidean space visualization tool, t-SNE, to hyperbolic space. Like t-SNE, it converts distances between data points to joint probabilities and tries to minimize the Kullback-Leibler divergence between the joint probabilities of high-dimensional data X and low-dimensional embedding Y. However, unlike Euclidean space, hyperbolic space is inhomogeneous: A volume could contain a lot more points at a location far from the origin. CO-SNE thus uses hyperbolic normal distributions for X and hyperbolic underlineCauchy instead of t-SNE's Student's t-distribution for Y, and it additionally seeks to preserve X's individual distances to the underlineOrigin in Y. We apply CO-SNE to naturally hyperbolic data and supervisedly learned hyperbolic features. Our results demonstrate that CO-SNE deflates high-dimensional hyperbolic data into a low-dimensional space without losing their hyperbolic characteristics, significantly outperforming popular visualization tools such as PCA, t-SNE, UMAP, and HoroPCA which is also designed for hyperbolic data.