Source-linked AI summary
Hyperbolic Image Segmentation
Mina GhadimiAtigh, Julian Schoep, Erman Acar, Nanne van Noord, Pascal Mettes
TL;DR
Pixel-level segmentation is conventionally performed in Euclidean output spaces, but existing hyperbolic approaches do not scale directly to this setting. The paper introduces a tractable hierarchical hyperbolic classification formulation for segmentation and reports free uncertainty and boundary information, improved zero-label generalization, and benefits at low embedding dimensions.
Problem
Euclidean output spaces are the standard for pixel-level segmentation, while existing hyperbolic operations are memory-wise intractable for pixel-level optimization and inference.
Method
The paper reformulates hyperbolic multinomial logistic regression to enable tractable simultaneous pixel-level classification and incorporates hierarchical relations among class labels.
Results
Hyperbolic segmentation provides uncertainty and boundary information for free, improves zero-label generalization, and is preferred at fewer embedding dimensions.
Takeaways & Limitations
Hyperbolic embedding spaces offer practical and analytical opportunities for image segmentation, including potential explainability and on-device segmentation.
Takeaways & Limitations
The experiments use DeepLabv3+ and do not establish how hyperbolic embeddings behave in shallower or deeper architectures or beyond natural images such as medical imagery.
Abstract
from arXiv · showhide
For image segmentation, the current standard is to perform pixel-level optimization and inference in Euclidean output embedding spaces through linear hyperplanes. In this work, we show that hyperbolic manifolds provide a valuable alternative for image segmentation and propose a tractable formulation of hierarchical pixel-level classification in hyperbolic space. Hyperbolic Image Segmentation opens up new possibilities and practical benefits for segmentation, such as uncertainty estimation and boundary information for free, zero-label generalization, and increased performance in low-dimensional output embeddings.
1. Introduction
The paper challenges Euclidean pixel-level classification by extending hyperbolic learning to image segmentation, where prior hyperbolic vision methods mainly represented whole images or videos. It introduces a tractable pixel-level formulation and identifies practical benefits including free uncertainty and boundary information, zero-label generalization, and stronger low-dimensional performance.
- 1. Introduction: The proposed formulation makes hyperbolic classification tractable for simultaneous pixel-level segmentation optimization and inference.It bypasses explicit Möbius addition in hyperbolic multinomial logistic regression and supports hierarchical label knowledge.
- 1. Introduction: Hyperbolic embeddings extend prior vision uses of hyperbolic space from global image or video representations to pixel-level segmentation.The paper positions this as a nontrivial generalization because hyperbolic operations have different spatial complexity.
- 1. Introduction: Hyperbolic segmentation provides uncertainty estimates and semantic boundary information without additional parameters or multiple forward passes.The paper contrasts this with Bayesian uncertainty estimation, which requires additional computation or modeling choices.
- 1. Introduction: Hierarchical hyperbolic embeddings improve zero-label generalization, enabling reasoning over unseen categories compared with Euclidean counterparts.This benefit follows the paper’s use of hierarchical knowledge in the hyperbolic embedding space.
- 1. Introduction: Hyperbolic embeddings are preferred at fewer output dimensions, extending low-dimensional effectiveness to image segmentation.The paper connects this property to potential explainability and on-device segmentation.
2. Related work
Prior segmentation systems commonly classify pixels with Euclidean multinomial logistic regression within encoder-decoder architectures, while hyperbolic vision methods have largely focused on global representations. This work differs from prior instance-segmentation use by applying tractable hyperbolic classification directly within pixel-level segmentation.
- 2. Related work: Semantic segmentation commonly uses encoder-decoder architectures whose final pixel classification is performed through Euclidean multinomial logistic regression.Encoder-decoder designs reconstruct high-resolution maps using upsampling, multi-scale features, or context-enhancing modules.
- 2. Related work: Hyperbolic learning has been applied across taxonomies, text, graphs, and computer-vision tasks including hierarchical action search and few-shot learning.These applications motivate hyperbolic representations for data with latent hierarchical structure.
- 2. Related work: Earlier hierarchical segmentation methods used WordNet relations and joint image-feature–word-concept embeddings to predict hierarchical concepts.These approaches explicitly model hypernym–hyponym relations or hierarchy-specific concepts.
- 2. Related work: Prior hyperbolic instance segmentation applied hyperbolic representations after predicted instance masks, whereas this work integrates hyperbolic classification into pixel-level segmentation.The distinction is between post hoc processing and classification as part of the segmentation model itself.
3. Image segmentation on the hyperbole
The paper reformulates pixel-level hyperbolic classification for image segmentation so it can operate efficiently, then incorporates hierarchical class structure through hyperbolic embeddings and hierarchical softmax.
- 3.1. Background: The Poincaré ball model: The model maps Euclidean network outputs into the Poincaré ball and classifies each pixel using distances to hyperbolic gyroplanes.The output at pixel location (i, j) is represented as zij, and class logits are derived from the distance to each class gyroplane.
- 3.2. Tractable pixel-level hyperbolic classification: Pixel-wise segmentation starts from an image-to-image representation and normally applies a linear layer followed by softmax to produce per-pixel class probabilities.The hyperbolic formulation replaces this Euclidean classification stage while retaining cross-entropy optimization.
- 3.2. Tractable pixel-level hyperbolic classification: 132 GB to 1.1 GB: factoring out explicit Möbius addition makes hyperbolic pixel-level classification tractable in a standard segmentation setting.The reformulation computes only the inner product and squared norm required by the class likelihood.
- 3.2. Tractable pixel-level hyperbolic classification: The efficient reformulation reduces the required computation from a tensor in R^(W×H×C×n) to operations on tensors in R^(W×H×C).The squared norm is computed as a summation of three tensors after precomputing shared terms.
- 3.3. Hierarchical hyperbolic class embedding: Hierarchical softmax assigns class probability through the root-to-leaf path, normalizing each node over sibling classes in the same subtree.The resulting class gyroplanes are visualized as hierarchically consistent embeddings, with curvature affecting their placement in the Poincaré disk.
4. Analyses
Analyses show that hyperbolic segmentation provides interpretable uncertainty, improves zero-label generalization with hierarchical knowledge, and remains effective in low-dimensional embeddings.
- 4.2. Uncertainty and boundary information for free: A 1,000-fold inference speed-up is obtained because hyperbolic uncertainty maps require no additional passes, unlike Bayesian Euclidean uncertainty.Both uncertainty maps focus on semantic boundaries and occluded areas, but Bayesian uncertainty uses 1,000 inference passes.
- 4.2. Uncertainty and boundary information for free: Hyperbolic confidence correlates with distance to semantic boundaries and is highest for foreground pixels, followed by background and boundary pixels.The boundary experiment classifies pixels as boundary, background, or foreground using proximity to another class.
- 4.3. Zero-label generalization: Combining hierarchical knowledge with hyperbolic embeddings raises zero-label mIOU from 18.53 to 20.76 on COCO-Stuff-10k and from 16.15 to 34.87 on Pascal VOC.The comparison uses 256-dimensional outputs and shows larger gains on Pascal VOC.
- 4.4. Low-dimensional embedding effectiveness: In 2- and 3-dimensional embeddings, hyperbolic segmentation outperforms DeepLabV3+, while remaining comparable in 10- and 256-dimensional settings.At three dimensions, mIOU improves by 4.32 percentage points on COCO-Stuff-10k and 4.99 on ADE20K.
- 4.5. Further ablations: Curvature changes have little effect at 256 dimensions, whereas curvature 10 reduces performance in the three-dimensional setting.The authors use validation to select curvature values between 0.1 and 2.
- 4.5. Further ablations: Hierarchical softmax generally outperforms flat softmax, especially on hierarchical metrics, although flat softmax slightly leads on a standard metric at 256 dimensions.The advantage of hierarchical softmax decreases as embedding dimensionality increases.
5. Conclusions
Hyperbolic Image Segmentation makes pixel-level hyperbolic classification tractable and demonstrates practical benefits for semantic segmentation. Its evaluation is limited to a DeepLabv3+ backbone, natural images, and related settings.
- Hyperbolic Image Segmentation provides a tractable formulation of hyperbolic multinomial logistic regression for pixel-level image segmentation.The paper presents this as the first approach for image segmentation in hyperbolic embedding spaces.
- The analyses show free uncertainty and boundary information, improved zero-label generalization, and better performance in low-dimensional embedding spaces.
- The experiments use DeepLabv3+ and do not establish effects in shallower or deeper architectures or beyond natural images such as medical data.The authors also flag possible negative societal applications in surveillance and military settings.