# Hyperbolic Busemann Learning with Ideal Prototypes

arXiv: https://arxiv.org/abs/2106.14472 · HTML: https://mlsanity.com/paper/2106.14472/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 53 references that are in the map.
- First posted on arXiv 28 Jun 2021.

## Builds on (most cited first, 30 of 35)

- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Rethinking Spatiotemporal Feature Learning: Speed-Accuracy Trade-offs in Video Classification](https://mlsanity.com/paper/1712.04851/) (2017)
- [Meta-Learning for Semi-Supervised Few-Shot Classification](https://mlsanity.com/paper/1803.00676/) (2018)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [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)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Attribute Prototype Network for Zero-Shot Learning](https://mlsanity.com/paper/2008.08290/) (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)
- [Transferrable Prototypical Networks for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1904.11227/) (2019)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Graph Prototypical Networks for Few-shot Learning on Attributed Networks](https://mlsanity.com/paper/2006.12739/) (2020)
- [Hyperspherical Prototype Networks](https://mlsanity.com/paper/1901.10514/) (2019)
- [Lorentzian Graph Convolutional Networks](https://mlsanity.com/paper/2104.07477/) (2021)
- [Episode-based Prototype Generating Network for Zero-Shot Learning](https://mlsanity.com/paper/1909.03360/) (2019)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [Graph Geometry Interaction Learning](https://mlsanity.com/paper/2010.12135/) (2020)
- [Differentiating through the Fr'echet Mean](https://mlsanity.com/paper/2003.00335/) (2020)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [Learning the Predictability of the Future](https://mlsanity.com/paper/2101.01600/) (2021)
- [Multimodal Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/2011.08899/) (2020)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Deep Learning on Small Datasets without Pre-Training using Cosine Loss](https://mlsanity.com/paper/1901.09054/) (2019)

## Built on by (most cited first, 30 of 41)

- [Hyperbolic Image Segmentation](https://mlsanity.com/paper/2203.05898/) (2022)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Poincar'e ResNet](https://mlsanity.com/paper/2303.14027/) (2023)
- [Maximum Class Separation as Inductive Bias in One Matrix](https://mlsanity.com/paper/2206.08704/) (2022)
- [Lorentzian Residual Neural Networks](https://mlsanity.com/paper/2412.14695/) (2024)
- [Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections](https://mlsanity.com/paper/2211.10066/) (2022)
- [Exploring Data Geometry for Continual Learning](https://mlsanity.com/paper/2304.03931/) (2023)
- [Hyperbolic Dataset Distillation](https://mlsanity.com/paper/2505.24623/) (2025)
- [Hyperbolic Random Forests](https://mlsanity.com/paper/2308.13279/) (2023)
- [HyperFed: Hyperbolic Prototypes Exploration with Consistent Aggregation for Non-IID Data in Federated Learning](https://mlsanity.com/paper/2307.14384/) (2023)
- [HMSN: Hyperbolic Self-Supervised Learning by Clustering with Ideal Prototypes](https://mlsanity.com/paper/2305.10926/) (2023)
- [Hyperbolic vs Euclidean Embeddings in Few-Shot Learning: Two Sides of the Same Coin](https://mlsanity.com/paper/2309.10013/) (2023)
- [Low-distortion and GPU-compatible Tree Embeddings in Hyperbolic Space](https://mlsanity.com/paper/2502.17130/) (2025)
- [Hierarchical Explanations for Video Action Recognition](https://mlsanity.com/paper/2301.00436/) (2023)
- [Continual Hyperbolic Learning of Instances and Classes](https://mlsanity.com/paper/2506.10710/) (2025)
- [Hyperbolic Busemann Neural Networks](https://mlsanity.com/paper/2602.18858/) (2026)
- [Hyperbolic Geometry in Computer Vision: A Survey](https://mlsanity.com/paper/2304.10764/) (2023)
- [Hyperspherical Classification with Dynamic Label-to-Prototype Assignment](https://mlsanity.com/paper/2403.16937/) (2024)
- [Learning Multi-Manifold Embedding for Out-Of-Distribution Detection](https://mlsanity.com/paper/2409.12479/) (2024)
- [Adversarial Attacks on Hyperbolic Networks](https://mlsanity.com/paper/2412.01495/) (2024)
- [A group-theoretic framework for machine learning in hyperbolic spaces](https://mlsanity.com/paper/2501.06934/) (2025)
- [Few-shot Classification with Hypersphere Modeling of Prototypes](https://mlsanity.com/paper/2211.05319/) (2022)
- [Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Experts](https://mlsanity.com/paper/2402.03460/) (2024)
- [Geometry-aware Distance Measure for Diverse Hierarchical Structures in Hyperbolic Spaces](https://mlsanity.com/paper/2506.18533/) (2025)
- [HIDISC: A Hyperbolic Framework for Domain Generalization with Generalized Category Discovery](https://mlsanity.com/paper/2510.17188/) (2025)
- [A survey of manifold learning and its applications for multimedia](https://mlsanity.com/paper/2310.12986/) (2023)
- [Hyp2Nav: Hyperbolic Planning and Curiosity for Crowd Navigation](https://mlsanity.com/paper/2407.13567/) (2024)
- [Balanced Hyperbolic Embeddings Are Natural Out-of-Distribution Detectors](https://mlsanity.com/paper/2506.10146/) (2025)
- [Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing](https://mlsanity.com/paper/2510.04579/) (2025)
- [Learning Visual-Semantic Subspace Representations](https://mlsanity.com/paper/2405.16213/) (2024)

## Abstract (arXiv)

Hyperbolic space has become a popular choice of manifold for representation learning of various datatypes from tree-like structures and text to graphs. Building on the success of deep learning with prototypes in Euclidean and hyperspherical spaces, a few recent works have proposed hyperbolic prototypes for classification. Such approaches enable effective learning in low-dimensional output spaces and can exploit hierarchical relations amongst classes, but require privileged information about class labels to position the hyperbolic prototypes. In this work, we propose Hyperbolic Busemann Learning. The main idea behind our approach is to position prototypes on the ideal boundary of the Poincar'e ball, which does not require prior label knowledge. To be able to compute proximities to ideal prototypes, we introduce the penalised Busemann loss. We provide theory supporting the use of ideal prototypes and the proposed loss by proving its equivalence to logistic regression in the one-dimensional case. Empirically, we show that our approach provides a natural interpretation of classification confidence, while outperforming recent hyperspherical and hyperbolic prototype approaches.