# A Rotated Hyperbolic Wrapped Normal Distribution for Hierarchical Representation Learning

arXiv: https://arxiv.org/abs/2205.13371 · HTML: https://mlsanity.com/paper/2205.13371/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 39 references that are in the map.
- First posted on arXiv 25 May 2022.

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

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (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)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Word Representations via Gaussian Embedding](https://mlsanity.com/paper/1412.6623/) (2014)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [Knowledge Association with Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2010.02162/) (2020)
- [Hierarchical Embeddings for Hypernymy Detection and Directionality](https://mlsanity.com/paper/1707.07273/) (2017)

## Built on by (most cited first, 10 of 10)

- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Hyperbolic VAE via Latent Gaussian Distributions](https://mlsanity.com/paper/2209.15217/) (2022)
- [Geometric Relational Embeddings: A Survey](https://mlsanity.com/paper/2304.11949/) (2023)
- [Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2502.01512/) (2025)
- [Stable and Consistent Prediction of 3D Characteristic Orientation via Invariant Residual Learning](https://mlsanity.com/paper/2306.11406/) (2023)
- [Hyperbolic Metric Learning for Visual Outlier Detection](https://mlsanity.com/paper/2403.15260/) (2024)
- [A group-theoretic framework for machine learning in hyperbolic spaces](https://mlsanity.com/paper/2501.06934/) (2025)
- [HLoOP -- Hyperbolic 2-space Local Outlier Probabilities](https://mlsanity.com/paper/2312.03895/) (2023)
- [Hyperboloid GPLVM for Discovering Continuous Hierarchies via Nonparametric Estimation](https://mlsanity.com/paper/2410.16698/) (2024)
- [HyperST: Hierarchical Hyperbolic Learning for Spatial Transcriptomics Prediction](https://mlsanity.com/paper/2511.22107/) (2025)

## Abstract (arXiv)

We present a rotated hyperbolic wrapped normal distribution (RoWN), a simple yet effective alteration of a hyperbolic wrapped normal distribution (HWN). The HWN expands the domain of probabilistic modeling from Euclidean to hyperbolic space, where a tree can be embedded with arbitrary low distortion in theory. In this work, we analyze the geometric properties of the diagonal HWN, a standard choice of distribution in probabilistic modeling. The analysis shows that the distribution is inappropriate to represent the data points at the same hierarchy level through their angular distance with the same norm in the Poincar'e disk model. We then empirically verify the presence of limitations of HWN, and show how RoWN, the proposed distribution, can alleviate the limitations on various hierarchical datasets, including noisy synthetic binary tree, WordNet, and Atari 2600 Breakout. The code is available at https://github.com/ml-postech/RoWN.