# Hyperbolic VAE via Latent Gaussian Distributions

arXiv: https://arxiv.org/abs/2209.15217 · HTML: https://mlsanity.com/paper/2209.15217/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 43 references that are in the map.
- First posted on arXiv 30 Sep 2022.

## Builds on (most cited first, 24 of 24)

- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [World Models](https://mlsanity.com/paper/1803.10122/) (2018)
- [Learning Latent Dynamics for Planning from Pixels](https://mlsanity.com/paper/1811.04551/) (2018)
- [Mastering Atari with Discrete World Models](https://mlsanity.com/paper/2010.02193/) (2020)
- [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)
- [Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Images](https://mlsanity.com/paper/2011.10650/) (2020)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [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)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Universal Statistics of Fisher Information in Deep Neural Networks: Mean Field Approach](https://mlsanity.com/paper/1806.01316/) (2018)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Spherical Latent Spaces for Stable Variational Autoencoders](https://mlsanity.com/paper/1808.10805/) (2018)
- [Skill-based Model-based Reinforcement Learning](https://mlsanity.com/paper/2207.07560/) (2022)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Igeood: An Information Geometry Approach to Out-of-Distribution Detection](https://mlsanity.com/paper/2203.07798/) (2022)
- [A Rotated Hyperbolic Wrapped Normal Distribution for Hierarchical Representation Learning](https://mlsanity.com/paper/2205.13371/) (2022)
- [Zero-Shot Fine-Grained Classification by Deep Feature Learning with Semantics](https://mlsanity.com/paper/1707.00785/) (2017)
- [GeoSeq2Seq: Information Geometric Sequence-to-Sequence Networks](https://mlsanity.com/paper/1710.09363/) (2017)

## Built on by (most cited first, 14 of 14)

- [Manifold Integrated Gradients: Riemannian Geometry for Feature Attribution](https://mlsanity.com/paper/2405.09800/) (2024)
- [Hyperbolic Dataset Distillation](https://mlsanity.com/paper/2505.24623/) (2025)
- [Continual Hyperbolic Learning of Instances and Classes](https://mlsanity.com/paper/2506.10710/) (2025)
- [Curvature Learning for Generalization of Hyperbolic Neural Networks](https://mlsanity.com/paper/2508.17232/) (2025)
- [LangVAE and LangSpace: Building and Probing for Language Model VAEs](https://mlsanity.com/paper/2505.00004/) (2025)
- [Modality Alignment across Trees on Heterogeneous Hyperbolic Manifolds](https://mlsanity.com/paper/2510.27391/) (2025)
- [On Probabilistic Pullback Metrics for Latent Hyperbolic Manifolds](https://mlsanity.com/paper/2410.20850/) (2024)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [HLoOP -- Hyperbolic 2-space Local Outlier Probabilities](https://mlsanity.com/paper/2312.03895/) (2023)
- [Bridging Compositional and Distributional Semantics: A Survey on Latent Semantic Geometry via AutoEncoder](https://mlsanity.com/paper/2506.20083/) (2025)
- [Complex variational autoencoders admit K"ahler structure](https://mlsanity.com/paper/2511.15172/) (2025)
- [Mutual Information Collapse Explains Disentanglement Failure in β-VAEs](https://mlsanity.com/paper/2602.09277/) (2026)
- [Distilling Latent Manifolds: Resolution Extrapolation by Variational Autoencoders](https://mlsanity.com/paper/2603.14536/) (2026)
- [Understanding Latent Diffusability via Fisher Geometry](https://mlsanity.com/paper/2604.02751/) (2026)

## Abstract (arXiv)

We propose a Gaussian manifold variational auto-encoder (GM-VAE) whose latent space consists of a set of Gaussian distributions. It is known that the set of the univariate Gaussian distributions with the Fisher information metric form a hyperbolic space, which we call a Gaussian manifold. To learn the VAE endowed with the Gaussian manifolds, we propose a pseudo-Gaussian manifold normal distribution based on the Kullback-Leibler divergence, a local approximation of the squared Fisher-Rao distance, to define a density over the latent space. In experiments, we demonstrate the efficacy of GM-VAE on two different tasks: density estimation of image datasets and environment modeling in model-based reinforcement learning. GM-VAE outperforms the other variants of hyperbolic- and Euclidean-VAEs on density estimation tasks and shows competitive performance in model-based reinforcement learning. We observe that our model provides strong numerical stability, addressing a common limitation reported in previous hyperbolic-VAEs.