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

2025-06-23

AI Summary

Existing hyperbolic learning methods employ fixed curvature and uniform distance metrics, limiting their ability to model heterogeneous hierarchical structures prevalent in real-world data. To address this, we propose a geometry-aware dynamic hyperbolic distance metric that adaptively generates personalized projections and local curvatures for each data pair—enabling the first input-dependent parameterization of hyperbolic distance functions. Our method incorporates low-rank parameterization to reduce computational overhead and integrates hard-negative mining to enhance discriminability. We further establish a theoretical upper bound on estimation error via Talagrand’s concentration inequality. Extensive experiments demonstrate state-of-the-art performance across image classification, hierarchical classification, and few-shot learning tasks: on mini-ImageNet, our approach achieves over 5% absolute accuracy improvement. Visualizations confirm sharper class boundaries and higher prototype separation, validating improved geometric expressivity.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Adapting distance measures for diverse hierarchical structures in hyperbolic spaces
• Mitigating computational cost of pair-wise distance without accuracy loss
• Improving performance on few-shot learning and hierarchical classification tasks

Innovation

• Dynamic hyperbolic distance measure adapts to hierarchies
• Tailored projections and curvatures for each data pair
• Low-rank decomposition reduces computational cost effectively