# Robust Large-Margin Learning in Hyperbolic Space

arXiv: https://arxiv.org/abs/2004.05465 · HTML: https://mlsanity.com/paper/2004.05465/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Apr 2020.

## Built on by (most cited first, 21 of 21)

- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Clipped Hyperbolic Classifiers Are Super-Hyperbolic Classifiers](https://mlsanity.com/paper/2107.11472/) (2021)
- [A Hyperbolic-to-Hyperbolic Graph Convolutional Network](https://mlsanity.com/paper/2104.06942/) (2021)
- [CO-SNE: Dimensionality Reduction and Visualization for Hyperbolic Data](https://mlsanity.com/paper/2111.15037/) (2021)
- [Hardness of Learning Neural Networks under the Manifold Hypothesis](https://mlsanity.com/paper/2406.01461/) (2024)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Linear Classifiers in Product Space Forms](https://mlsanity.com/paper/2102.10204/) (2021)
- [Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/2302.06807/) (2023)
- [It Is All About Data: A Survey on the Effects of Data on Adversarial Robustness](https://mlsanity.com/paper/2303.09767/) (2023)
- [Hyperbolic Random Forests](https://mlsanity.com/paper/2308.13279/) (2023)
- [Highly Scalable and Provably Accurate Classification in Poincare Balls](https://mlsanity.com/paper/2109.03781/) (2021)
- [HyperFed: Hyperbolic Prototypes Exploration with Consistent Aggregation for Non-IID Data in Federated Learning](https://mlsanity.com/paper/2307.14384/) (2023)
- [Hyperbolic Space with Hierarchical Margin Boosts Fine-Grained Learning from Coarse Labels](https://mlsanity.com/paper/2311.11019/) (2023)
- [Unitary convolutions for learning on graphs and groups](https://mlsanity.com/paper/2410.05499/) (2024)
- [Learning Fine-grained Domain Generalization via Hyperbolic State Space Hallucination](https://mlsanity.com/paper/2504.08020/) (2025)
- [Adversarial for Good? How the Adversarial ML Community's Values Impede Socially Beneficial Uses of Attacks](https://mlsanity.com/paper/2107.10302/) (2021)
- [Provably Accurate and Scalable Linear Classifiers in Hyperbolic Spaces](https://mlsanity.com/paper/2203.03730/) (2022)
- [A Geometry-Aware Algorithm to Learn Hierarchical Embeddings in Hyperbolic Space](https://mlsanity.com/paper/2407.16641/) (2024)
- [Why are hyperbolic neural networks effective? A study on hierarchical representation capability](https://mlsanity.com/paper/2402.02478/) (2024)
- [Towards Non-Euclidean Foundation Models: Advancing AI Beyond Euclidean Frameworks](https://mlsanity.com/paper/2505.14417/) (2025)
- [Convex Relaxation for Solving Large-Margin Classifiers in Hyperbolic Space](https://mlsanity.com/paper/2405.17198/) (2024)

## Abstract (arXiv)

Recently, there has been a surge of interest in representation learning in hyperbolic spaces, driven by their ability to represent hierarchical data with significantly fewer dimensions than standard Euclidean spaces. However, the viability and benefits of hyperbolic spaces for downstream machine learning tasks have received less attention. In this paper, we present, to our knowledge, the first theoretical guarantees for learning a classifier in hyperbolic rather than Euclidean space. Specifically, we consider the problem of learning a large-margin classifier for data possessing a hierarchical structure. We provide an algorithm to efficiently learn a large-margin hyperplane, relying on the careful injection of adversarial examples. Finally, we prove that for hierarchical data that embeds well into hyperbolic space, the low embedding dimension ensures superior guarantees when learning the classifier directly in hyperbolic space.