# On Probabilistic Pullback Metrics for Latent Hyperbolic Manifolds

arXiv: https://arxiv.org/abs/2410.20850 · HTML: https://mlsanity.com/paper/2410.20850/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 49 references that are in the map.
- First posted on arXiv 28 Oct 2024.

## Builds on (most cited first, 25 of 25)

- [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)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [Geometrically Enriched Latent Spaces](https://mlsanity.com/paper/2008.00565/) (2020)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [Metrics for Probabilistic Geometries](https://mlsanity.com/paper/1411.7432/) (2014)
- [Data Augmentation in High Dimensional Low Sample Size Setting Using a Geometry-Based Variational Autoencoder](https://mlsanity.com/paper/2105.00026/) (2021)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Learning Riemannian Manifolds for Geodesic Motion Skills](https://mlsanity.com/paper/2106.04315/) (2021)
- [What is a meaningful representation of protein sequences?](https://mlsanity.com/paper/2012.02679/) (2020)
- [Hyperbolic Representation Learning: Revisiting and Advancing](https://mlsanity.com/paper/2306.09118/) (2023)
- [Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning](https://mlsanity.com/paper/2305.18962/) (2023)
- [Hyperbolic VAE via Latent Gaussian Distributions](https://mlsanity.com/paper/2209.15217/) (2022)
- [Bringing motion taxonomies to continuous domains via GPLVM on hyperbolic manifolds](https://mlsanity.com/paper/2210.01672/) (2022)
- [Hyperbolic Random Forests](https://mlsanity.com/paper/2308.13279/) (2023)
- [Leveraging Hyperbolic Embeddings for Coarse-to-Fine Robot Design](https://mlsanity.com/paper/2311.00462/) (2023)
- [Isometric Gaussian Process Latent Variable Model for Dissimilarity Data](https://mlsanity.com/paper/2006.11741/) (2020)

## Built on by (most cited first, 1 of 1)

- [Taxonomy-aware Dynamic Motion Generation on Hyperbolic Manifolds](https://mlsanity.com/paper/2509.21281/) (2025)

## Abstract (arXiv)

Probabilistic Latent Variable Models (LVMs) excel at modeling complex, high-dimensional data through lower-dimensional representations. Recent advances show that equipping these latent representations with a Riemannian metric unlocks geometry-aware distances and shortest paths that comply with the underlying data structure. This paper focuses on hyperbolic embeddings, a particularly suitable choice for modeling hierarchical relationships. Previous approaches relying on hyperbolic geodesics for interpolating the latent space often generate paths crossing low-data regions, leading to highly uncertain predictions. Instead, we propose augmenting the hyperbolic manifold with a pullback metric to account for distortions introduced by the LVM's nonlinear mapping and provide a complete development for pullback metrics of Gaussian Process LVMs (GPLVMs). Our experiments demonstrate that geodesics on the pullback metric not only respect the geometry of the hyperbolic latent space but also align with the underlying data distribution, significantly reducing uncertainty in predictions.