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Hyperbolic Deep Learning in Computer Vision: A Survey
Pascal Mettes, Mina Ghadimi Atigh, Martin Keller-Ressel, Jeffrey Gu, Serena Yeung
TL;DR
Euclidean operators underpin vision, while the connections and shared themes across hyperbolic computer vision research have remained unclear. This survey categorizes supervised and unsupervised work and reviews its methods and applications. It synthesizes reported benefits and identifies open challenges for the field.
Problem
The survey addresses unclear connections across hyperbolic computer vision literature and a prior general survey that did not cover computer vision.
Source: §1 · p. 2Method
Results
The survey reports hyperbolic learning’s benefits for hierarchical learning, including reduced error severity, and generally small gains over Euclidean counterparts in contrastive learning.
Source: §5.1 · p. 19, §4.3.2 · p. 19Takeaways & Limitations
The survey identifies which problem types currently benefit most from hyperbolic learning and discusses open problems for future research.
Source: §5 · p. 19Hyperbolic networks can be numerically unstable and computationally heavier than Euclidean counterparts, making them less efficient depending on the model.
Source: §5.2 · p. 20
Abstract
from arXiv · showhide
Deep representation learning is a ubiquitous part of modern computer vision. While Euclidean space has been the de facto standard manifold for learning visual representations, hyperbolic space has recently gained rapid traction for learning in computer vision. Specifically, hyperbolic learning has shown a strong potential to embed hierarchical structures, learn from limited samples, quantify uncertainty, add robustness, limit error severity, and more. In this paper, we provide a categorization and in-depth overview of current literature on hyperbolic learning for computer vision. We research both supervised and unsupervised literature and identify three main research themes in each direction. We outline how hyperbolic learning is performed in all themes and discuss the main research problems that benefit from current advances in hyperbolic learning for computer vision. Moreover, we provide a high-level intuition behind hyperbolic geometry and outline open research questions to further advance research in this direction.
1 Introduction
Euclidean representations underpin modern computer vision, but their geometry is a poor fit for exponentially expanding hierarchies. This survey organizes hyperbolic computer vision research across supervised and unsupervised learning and reviews its motivations and open questions.
1 Introduction
- Modern vision architectures rely on Euclidean operators and implicitly or explicitly assume that data is best represented on regular grids.Source: §1 · p. 1
- Hierarchies grow exponentially with depth, whereas Euclidean ball volume grows polynomially, motivating a geometry better matched to hierarchical data.The survey presents hyperbolic geometry as a direct fit for hierarchies because of its hierarchical structure and exponential expansion.Source: §1 · pp. 1–2
- Hyperbolic visual embeddings have been applied to few-shot learning, zero-shot recognition, out-of-distribution generalization, uncertainty quantification, generative learning, and hierarchical representation learning.Prior hierarchical embeddings in hyperbolic space have represented deep, complex trees with minimal distortion.Source: §1 · p. 2
- The survey addresses unclear connections across the literature by categorizing supervised work into gyroplane, prototype, and sample-matching themes, and unsupervised work into generative, clustering, and self-supervised themes.It also reviews background and reported advantages, then outlines open challenges.Source: §1 · p. 2
2 Background on hyperbolic geometry
Hyperbolic geometry is characterized by constant negative curvature, exponential expansion, and tree-like structure. The section introduces common models and operations, then reviews mathematical tools and foundational work relevant to visual learning.
2.1 What is hyperbolic geometry?
- Hyperbolic space has constant negative curvature, exponential volume growth, and a tree-like structure, unlike Euclidean space’s polynomial volume growth.Gromov hyperbolicity is finite but non-zero for hyperbolic space, zero for tree graphs, and infinite for Euclidean space.Source: §2.1 · p. 2
2.2 Models of hyperbolic geometry
- The Poincaré, hyperboloid, Klein, and upper half-space models represent the same geometry with different coordinates and trade-offs in visualization, numerical efficiency, and calculations.The Poincaré model is conformal; the Klein model has straight geodesics, while the hyperboloid model retains some linear structure.Source: §2.2 · pp. 2–3
2.3 Five core hyperbolic operations
- Five core operations cover much of hyperbolic learning: distances, geodesics, geodesic extension, exponential maps, and distance-preserving translations.The exponential map is commonly used to map visual representations from Euclidean into hyperbolic space.Source: §2.3 · pp. 3–4
2.4 Gyrovectorspace calculus
- Gyrovectorspace calculus supplies operations for the Poincaré ball, including addition and scalar products; curvature rescaling makes Euclidean space a continuous limit as curvature approaches zero.Source: §2.4 · p. 4
2.5 Non-visual hyperbolic learning
- Non-visual foundations include hyperbolic embeddings of hierarchies, hyperbolic neural-network layers, and graph and text methods; graph networks have shown classification improvements especially on graphs with latent hierarchies.Other surveyed applications include molecular and cellular structures, recommendation, skeletal action recognition, LiDAR, point clouds, and 3D shapes.Source: §2.5 · pp. 4–6
3 Supervised hyperbolic visual learning
Supervised hyperbolic vision learning mostly maps standard-network features into hyperbolic space and optimizes them through three strategies: sample-to-gyroplane, sample-to-prototype, and sample-to-sample learning. The survey organizes current work around these strategies and the tasks they address.
3.1 Sample-to-gyroplane learning
- Clipping Euclidean embeddings before the exponential map addresses vanishing gradients and improves hyperbolic classification, out-of-distribution detection, and adversarial robustness.The gradient problem arises as embeddings are pushed toward the Poincaré-ball boundary; clipping is particularly useful with many classes, including ImageNet.Source: §3.1 · p. 7
- Hyperbolic logistic regression supports image segmentation with per-pixel hierarchical classification, where lower embedding norms correlate with greater uncertainty near semantic boundaries.The approach rewrites Möbius addition to enable per-pixel evaluation across image batches and replaces one-hot softmax with hierarchical softmax.Source: §3.1 · p. 7, Figure 3 · p. 8
3.2 Sample-to-prototype learning
- Hyperbolic ProtoNet is competitive with Euclidean prototypical learning for few-shot tasks and can achieve better accuracy with a 4-layer ConvNet backbone.It uses the Einstein midpoint to average samples in the Klein model, avoiding expensive Fréchet-mean calculations in the Poincaré ball.Source: §3.2 · p. 8
- For limited-data few-shot learning, hyperbolic feature augmentation and neural-ODE distribution estimation improve performance, while hyperbolic clipping outperforms ProtoNet and Hyperbolic ProtoNet on CUB Birds and miniImageNet.Taylor-series reformulations of hyperbolic neural networks are also reported to improve scalability and compatibility and outperform Hyperbolic ProtoNet.Source: §3.2 · p. 9
- Hierarchical class prototypes support classification, zero-shot recognition, action search, and open-set recognition, with methods embedding or jointly learning taxonomies and visual representations.Other work uses hierarchical prototypes for multi-granularity action explanations and continual learning, where angle-regularization and neighbor-robustness losses preserve old-data geometry.Source: §3.2 · pp. 9–10
3.3 Sample-to-sample learning
- Hyperbolic metric learning with vision transformers consistently outperforms Euclidean alternatives and achieves state-of-the-art results on fine-grained datasets, while embeddings show latent hierarchical neighborhoods.The approach uses a hyperbolic pairwise cross-entropy loss with supervised or self-supervised vision-transformer encoders.Source: §3.3 · pp. 10–11, Figure 5 · p. 11
- Hyperbolic metric-learning variants address few-shot class-incremental learning, noisy labels, unannotated hierarchical relations, semantic hashing, face recognition, and multimodal alignment.Hyperbolic image-text representations also outperform Euclidean CLIP in capturing hierarchical multimodal relations.Source: §3.3 · p. 11
- Sample-to-set distances retain support examples to improve few-shot robustness to outliers, while grouplet-based matching offers a hybrid between costly sample-to-sample and less-accurate sample-to-prototype learning.Grouplets are random sample subsets matched with prototypes through differentiable optimal transport; their hyperbolic embeddings improve fine-grained metric learning.Source: §3.3 · p. 11
4 Unsupervised hyperbolic visual learning
Unsupervised hyperbolic visual learning has three dominant directions: generative learning, clustering, and self-supervised learning.
4 Unsupervised hyperbolic visual learning
- The survey identifies generative learning, clustering, and self-supervised learning as the three dominant directions in unsupervised hyperbolic computer vision.It discusses each direction separately.Source: §4 · p. 12
4.1 Generative approaches
Generative approaches use hyperbolic latent spaces and layers in VAEs, GANs, and normalizing flows. Reported comparisons show advantages in some settings, especially at low latent dimensions, with performance depending on the model and setting.
4.1.1 Hyperbolic VAEs
- Hyperbolic representations outperform Euclidean ones on MNIST at low latent dimensions, but Euclidean representations overtake them starting at dimension 10.Nagano et al. use wrapped-normal sampling and density estimation in a hyperbolic VAE; Mathieu et al. report improved test log-likelihoods from the Riemannian normal and gyroplane layer, especially at low dimensions.Source: §4.1.1 · pp. 12–13
- RoWN fixes the wrapped normal’s principal-axis misalignment and outperforms HWN in several settings, including Atari 2600 Breakout image generation.Cho et al. describe the rotated hyperbolic wrapped normal as a new sampling process for diagonal covariance in the Lorentz model.Source: §4.1.1 · p. 13, Figure 7 · p. 13
4.1.2 Hyperbolic GANs
- Hyperbolic GAN variants generally improve Inception Score and Frechet Inception Distance on MNIST; their best learned curvatures are close to zero.The reported best improvements are in the GAN architecture, and good results are observed at large latent dimensions.Source: §4.1.2 · p. 13
- HAEGAN generalizes WGAN-GP to the Lorentz model using an encoder, generator, critic, decoder, and hyperbolic operations.Compared with Lazcano et al. (2021), it reports improved MNIST image-generation results; its HCat layer has bounded gradients compared with Shimizu et al. (2021).Source: §4.1.2 · pp. 13–14
- Li et al. encode category hierarchy in hyperbolic space for few-shot image generation, reporting competitive results and promise for image-to-image transfer.Their method maps image representations into hyperbolic space, rescales them to a desired radius, and samples along a geodesic to a point from seen categories.Source: §4.1.2 · p. 14
4.1.3 Hyperbolic Normalizing Flows
- Wrapped hyperbolic coupling uses parallel transport to access manifold regions beyond the origin, while hyperbolic flows outperform hyperbolic VAEs at low latent dimensions.On MNIST, hyperbolic models exceed equivalent Euclidean models at low dimensions, but Euclidean models overtake them as early as latent dimension 6.Source: §4.1.3 · pp. 14–15, Figure 9 · p. 15
4.2 Clustering
Hyperbolic clustering methods represent hierarchical structure through embeddings, parent-child objectives, and last common ancestors. The surveyed approaches include fixed-leaf tree construction, direct tree optimization, and multi-view clustering.
4.2 Clustering
- Monath et al. (2019) construct a hierarchical clustering by optimizing hyperbolic representations for data points and a fixed number of internal nodes.Their parent-child margin encourages children to have larger norms than parents, and they apply the method to ImageNet representations from a pretrained Inception network.Source: §4.2 · pp. 15–16
- The clustering objective uses Dasgupta’s cost and its continuous last-common-ancestor reformulation, alongside a parent-child margin objective.The margin objective encourages parent nodes to have smaller norm than their children.Source: §4.2 · pp. 15–16
- Chami et al. (2020a) optimize hyperbolic embeddings of the whole tree and iteratively merge pairs using their hyperbolic LCA distance from the origin.Unlike Monath et al. (2019), their method does not require pre-existing hyperbolic embeddings.Source: §4.2 · pp. 15–16
- Lin et al. (2022) combine reconstruction, cross-view contrastive learning, and weighted triplet loss with hyperbolic hierarchical clustering for multi-view data.Positive examples are views of the same object; negatives are views of different objects, with examples mined using k-nearest-neighbor graphs.Source: §4.2 · p. 16
4.3 Self-supervised learning
Self-supervised visual learning in hyperbolic space uses either triplet-loss-based self-supervision or contrastive losses. The surveyed approaches apply these methods to biomedical segmentation, long-tail instance segmentation, video action prediction, and scene-object representation learning.
4.3.1 Hyperbolic self-supervision
- Hsu et al. use a hierarchical triplet loss to bring a child patch toward its parent and separate non-overlapping negative patches.This encourages subpatch representations to become descendants of the main patch and distant patches to occupy other branches.Source: §4.3.1 · p. 16
- Hsu et al. cluster hyperbolic patch representations using the Fréchet mean, which lacks a closed-form solution on the Poincaré ball and is computed iteratively.The strategy is reported effective for synthetic biological data and 3D brain tumor MRI scans.Source: §4.3.1 · pp. 16–17
- Weng et al. combine three self-supervised losses with hyperbolic k-means clustering and report improvement over other partially supervised methods on LVIS long-tail segmentation.Their mean maps representations to the hyperboloid and uses a weighted midpoint with a closed-form formula, making it more computationally efficient than the Fréchet mean.Source: §4.3.1 · p. 17
4.3.2 Hyperbolic contrastive learning
- Surís et al. pretrain hyperbolic video representations by predicting future-frame embeddings contrastively, then freeze them for action prediction.When actions are uncertain, the geodesic midpoint can represent their more general parent action.Source: §4.3.2 · pp. 17–18
- Ge et al.’s HCL adds a hyperbolic branch to MoCo that contrasts scene regions with contained objects and non-contained negative objects.The resulting hierarchy places scenes near the Poincaré-ball edge and objects nearer the origin; out-of-context objects generally have higher distance to scene images.Source: §4.3.2 · p. 18, Figure 11 · p. 18
- Yue et al. adapt SimCLR, SupCon, and adversarially robust RoCL to hyperbolic contrastive learning by replacing dot-product similarity or Euclidean contrastive losses with hyperbolic distance.Their approach uses a hyperbolic branch without retaining a Euclidean branch.Source: §4.3.2 · p. 18
- Hyperbolic contrastive learning generally achieves small gains over Euclidean counterparts, while HCL reports small gains over MoCo and box-augmented MoCo.HCL also finds out-of-context objects generally have higher distance to scene images.Source: §4.3.2 · pp. 18–19
5 Conclusions and future outlook
The survey reviews supervised and unsupervised hyperbolic vision learning, identifying when it appears most effective and outlining open research challenges. It highlights gains in hierarchical, few-sample, robust, and low-dimensional learning, while noting unresolved technical and infrastructure needs.
5.1 When is hyperbolic learning most effective?
- Hyperbolic hierarchical learning can reduce error severity, improve retrieval consistency, and support zero-shot generalization.The survey describes embedding class hierarchies and mapping examples from seen classes to their embeddings as a route to generalizing to unseen classes.Source: §5.1 · p. 19
- In few-shot learning, hyperbolic methods consistently improve accuracy, and hyperbolic generative learning performs better with constrained data sources.The survey says these results indicate generalization from fewer examples.Source: §5.1 · p. 19
- Hyperbolic learning has improved out-of-distribution detection, uncertainty quantification, out-of-context sample identification, and adversarial robustness.The survey also reports improved performance with smaller unsupervised embedding spaces and reduced supervised embedding dimensions while maintaining downstream performance.Source: §5.1 · p. 19
5.2 Open research questions
- Fully hyperbolic learning remains open: it requires rethinking network layers, and questions remain about models, curvature, scalability, and performance across tasks.Two cited works describe hyperbolic convolutional networks and Poincaré ResNet components, providing a foundation but leaving many questions unresolved.Source: §5.2 · pp. 19–20
- Hyperbolic networks can be numerically unstable and computationally heavier than Euclidean counterparts, reducing efficiency.Reported challenges include issues at the ball boundary and vanishing gradients.Source: §5.2 · p. 20
- The field lacks a shared open-source code base, so researchers must implement advances themselves or import repositories from other papers.The survey identifies continued open-source development as important for attracting practitioners and advancing the field.Source: §5.2 · p. 20
- Hyperbolic foundation models are a longer-term prospect because they require progress on stable fully hyperbolic learning and open-source development.Early work has shown large-scale and/or multimodal learning to be viable with hyperbolic embeddings.Source: §5.2 · p. 20