# A Set-to-Set Distance Measure in Hyperbolic Space

arXiv: https://arxiv.org/abs/2506.18529 · HTML: https://mlsanity.com/paper/2506.18529/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 68 references that are in the map.
- First posted on arXiv 23 Jun 2025.

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

- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [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)
- [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 Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Few-Shot Learning via Embedding Adaptation with Set-to-Set Functions](https://mlsanity.com/paper/1812.03664/) (2018)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Hyperbolic Attention Networks](https://mlsanity.com/paper/1805.09786/) (2018)
- [Hyperbolic Vision Transformers: Combining Improvements in Metric Learning](https://mlsanity.com/paper/2203.10833/) (2022)
- [Hyperbolic Image Segmentation](https://mlsanity.com/paper/2203.05898/) (2022)
- [Negative Sample Matters: A Renaissance of Metric Learning for Temporal Grounding](https://mlsanity.com/paper/2109.04872/) (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)
- [Fully Hyperbolic Convolutional Neural Networks for Computer Vision](https://mlsanity.com/paper/2303.15919/) (2023)
- [Hyperbolic Busemann Learning with Ideal Prototypes](https://mlsanity.com/paper/2106.14472/) (2021)
- [ZeroER: Entity Resolution using Zero Labeled Examples](https://mlsanity.com/paper/1908.06049/) (2019)
- [Poincar'e ResNet](https://mlsanity.com/paper/2303.14027/) (2023)
- [Capturing implicit hierarchical structure in 3D biomedical images with self-supervised hyperbolic representations](https://mlsanity.com/paper/2012.01644/) (2020)
- [Hyperbolic Uncertainty Aware Semantic Segmentation](https://mlsanity.com/paper/2203.08881/) (2022)
- [Auto-FuzzyJoin: Auto-Program Fuzzy Similarity Joins Without Labeled Examples](https://mlsanity.com/paper/2103.04489/) (2021)
- [Adaptive Poincar'e Point to Set Distance for Few-Shot Classification](https://mlsanity.com/paper/2112.01719/) (2021)
- [GPr-Net: Geometric Prototypical Network for Point Cloud Few-Shot Learning](https://mlsanity.com/paper/2304.06007/) (2023)
- [Hyp-UML: Hyperbolic Image Retrieval with Uncertainty-aware Metric Learning](https://mlsanity.com/paper/2310.08390/) (2023)

## Abstract (arXiv)

We propose a hyperbolic set-to-set distance measure for computing dissimilarity between sets in hyperbolic space. While point-to-point distances in hyperbolic space effectively capture hierarchical relationships between data points, many real-world applications require comparing sets of hyperbolic data points, where the local structure and the global structure of the sets carry crucial semantic information. The proposed the underlinehyperbolic underlineset-underlineto-underlineset underlinedistance measure (HS2SD) integrates both global and local structural information: global structure through geodesic distances between Einstein midpoints of hyperbolic sets, and local structure through topological characteristics of the two sets. To efficiently compute topological differences, we prove that using a finite Thue-Morse sequence of degree and adjacency matrices can serve as a robust approximation to capture the topological structure of a set. In this case, by considering the topological differences, HS2SD provides a more nuanced understanding of the relationships between two hyperbolic sets. Empirical evaluation on entity matching, standard image classification, and few-shot image classification demonstrates that our distance measure outperforms existing methods by effectively modeling the hierarchical and complex relationships inherent in hyperbolic sets.