# Geometry-aware Distance Measure for Diverse Hierarchical Structures in Hyperbolic Spaces

arXiv: https://arxiv.org/abs/2506.18533 · HTML: https://mlsanity.com/paper/2506.18533/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 85 references that are in the map.
- First posted on arXiv 23 Jun 2025.

## Builds on (most cited first, 30 of 33)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [Meta-Learning for Semi-Supervised Few-Shot Classification](https://mlsanity.com/paper/1803.00676/) (2018)
- [TADAM: Task dependent adaptive metric for improved few-shot learning](https://mlsanity.com/paper/1805.10123/) (2018)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Adaptive Prototype Learning and Allocation for Few-Shot Segmentation](https://mlsanity.com/paper/2104.01893/) (2021)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Hyperbolic Vision Transformers: Combining Improvements in Metric Learning](https://mlsanity.com/paper/2203.10833/) (2022)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Finding Task-Relevant Features for Few-Shot Learning by Category Traversal](https://mlsanity.com/paper/1905.11116/) (2019)
- [Boosting Few-Shot Learning With Adaptive Margin Loss](https://mlsanity.com/paper/2005.13826/) (2020)
- [Clipped Hyperbolic Classifiers Are Super-Hyperbolic Classifiers](https://mlsanity.com/paper/2107.11472/) (2021)
- [Meta-Learning with Task-Adaptive Loss Function for Few-Shot Learning](https://mlsanity.com/paper/2110.03909/) (2021)
- [Fully Hyperbolic Convolutional Neural Networks for Computer Vision](https://mlsanity.com/paper/2303.15919/) (2023)
- [HyperML: A Boosting Metric Learning Approach in Hyperbolic Space for Recommender Systems](https://mlsanity.com/paper/1809.01703/) (2018)
- [HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections](https://mlsanity.com/paper/2106.03306/) (2021)
- [Hyperbolic Busemann Learning with Ideal Prototypes](https://mlsanity.com/paper/2106.14472/) (2021)
- [Learning to Forget for Meta-Learning](https://mlsanity.com/paper/1906.05895/) (2019)
- [Contrastive Multi-view Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2205.02618/) (2022)
- [Rethinking the compositionality of point clouds through regularization in the hyperbolic space](https://mlsanity.com/paper/2209.10318/) (2022)
- [Curved Geometric Networks for Visual Anomaly Recognition](https://mlsanity.com/paper/2208.01188/) (2022)
- [kHGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning](https://mlsanity.com/paper/2212.01793/) (2022)
- [Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/2302.06807/) (2023)
- [Dynamic Hyperbolic Attention Network for Fine Hand-object Reconstruction](https://mlsanity.com/paper/2309.02965/) (2023)
- [Hyperbolic Space with Hierarchical Margin Boosts Fine-Grained Learning from Coarse Labels](https://mlsanity.com/paper/2311.11019/) (2023)
- [Robust Geodesic Regression](https://mlsanity.com/paper/2007.04518/) (2020)

## Built on by (most cited first, 2 of 2)

- [Modality Alignment across Trees on Heterogeneous Hyperbolic Manifolds](https://mlsanity.com/paper/2510.27391/) (2025)
- [Learning Taxonomic Trees with Hierarchical Representation Regularization for Large Multimodal Models](https://mlsanity.com/paper/2607.02909/) (2026)

## Abstract (arXiv)

Learning in hyperbolic spaces has attracted increasing attention due to its superior ability to model hierarchical structures of data. Most existing hyperbolic learning methods use fixed distance measures for all data, assuming a uniform hierarchy across all data points. However, real-world hierarchical structures exhibit significant diversity, making this assumption overly restrictive. In this paper, we propose a geometry-aware distance measure in hyperbolic spaces, which dynamically adapts to varying hierarchical structures. Our approach derives the distance measure by generating tailored projections and curvatures for each pair of data points, effectively mapping them to an appropriate hyperbolic space. We introduce a revised low-rank decomposition scheme and a hard-pair mining mechanism to mitigate the computational cost of pair-wise distance computation without compromising accuracy. We present an upper bound on the low-rank approximation error using Talagrand's concentration inequality, ensuring theoretical robustness. Extensive experiments on standard image classification (MNIST, CIFAR-10 and CIFAR-100), hierarchical classification (5-level CIFAR-100), and few-shot learning tasks (mini-ImageNet, tiered-ImageNet) demonstrate the effectiveness of our method. Our approach consistently outperforms learning methods that use fixed distance measures, with notable improvements on few-shot learning tasks, where it achieves over 5% gains on mini-ImageNet. The results reveal that adaptive distance measures better capture diverse hierarchical structures, with visualization showing clearer class boundaries and improved prototype separation in hyperbolic spaces.