# Ultrahyperbolic Representation Learning

arXiv: https://arxiv.org/abs/2007.00211 · HTML: https://mlsanity.com/paper/2007.00211/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2020.

## Built on by (most cited first, 17 of 17)

- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Pseudo-Riemannian Graph Convolutional Networks](https://mlsanity.com/paper/2106.03134/) (2021)
- [Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach](https://mlsanity.com/paper/2106.04941/) (2021)
- [Ultrahyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2206.00449/) (2022)
- [SpaceMesh: A Continuous Representation for Learning Manifold Surface Meshes](https://mlsanity.com/paper/2409.20562/) (2024)
- [Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learning](https://mlsanity.com/paper/2309.04082/) (2023)
- [kHGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning](https://mlsanity.com/paper/2212.01793/) (2022)
- [Directed Graph Embeddings in Pseudo-Riemannian Manifolds](https://mlsanity.com/paper/2106.08678/) (2021)
- [Feature Generation for Long-tail Classification](https://mlsanity.com/paper/2111.05956/) (2021)
- [D-HYPR: Harnessing Neighborhood Modeling and Asymmetry Preservation for Digraph Representation Learning](https://mlsanity.com/paper/2112.11734/) (2021)
- [Neural Spacetimes for DAG Representation Learning](https://mlsanity.com/paper/2408.13885/) (2024)
- [Capacity Bounds for Hyperbolic Neural Network Representations of Latent Tree Structures](https://mlsanity.com/paper/2308.09250/) (2023)
- [Normed Spaces for Graph Embedding](https://mlsanity.com/paper/2312.01502/) (2023)
- [Towards Non-Euclidean Foundation Models: Advancing AI Beyond Euclidean Frameworks](https://mlsanity.com/paper/2505.14417/) (2025)
- [Pseudo-Riemannian Embedding Models for Multi-Relational Graph Representations](https://mlsanity.com/paper/2212.03720/) (2022)
- [Variational autoencoders with latent high-dimensional steady geometric flows for dynamics](https://mlsanity.com/paper/2410.10137/) (2024)
- [PR-CapsNet: Pseudo-Riemannian Capsule Network with Adaptive Curvature Routing for Graph Learning](https://mlsanity.com/paper/2512.08218/) (2025)

## Abstract (arXiv)

In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic space which is well suited for tree-like data. In this paper, we propose a representation living on a pseudo-Riemannian manifold of constant nonzero curvature. It is a generalization of hyperbolic and spherical geometries where the nondegenerate metric tensor need not be positive definite. We provide the necessary learning tools in this geometry and extend gradient-based optimization techniques. More specifically, we provide closed-form expressions for distances via geodesics and define a descent direction to minimize some objective function. Our novel framework is applied to graph representations.