Learning Structured Representations with Hyperbolic Embeddings (2412.01023v1)

Abstract: Most real-world datasets consist of a natural hierarchy between classes or an inherent label structure that is either already available or can be constructed cheaply. However, most existing representation learning methods ignore this hierarchy, treating labels as permutation invariant. Recent work [Zeng et al., 2022] proposes using this structured information explicitly, but the use of Euclidean distance may distort the underlying semantic context [Chen et al., 2013]. In this work, motivated by the advantage of hyperbolic spaces in modeling hierarchical relationships, we propose a novel approach HypStructure: a Hyperbolic Structured regularization approach to accurately embed the label hierarchy into the learned representations. HypStructure is a simple-yet-effective regularizer that consists of a hyperbolic tree-based representation loss along with a centering loss, and can be combined with any standard task loss to learn hierarchy-informed features. Extensive experiments on several large-scale vision benchmarks demonstrate the efficacy of HypStructure in reducing distortion and boosting generalization performance especially under low dimensional scenarios. For a better understanding of structured representation, we perform eigenvalue analysis that links the representation geometry to improved Out-of-Distribution (OOD) detection performance seen empirically. The code is available at \url{https://github.com/uiuctml/HypStructure}.

• The paper introduces HypStructure, a novel method that leverages hyperbolic geometry to embed label hierarchies and reduce distortion.
• The methodology combines hyperbolic centering and HypCPCC losses to boost both in-distribution classification and out-of-distribution detection.
• Empirical results on CIFAR10, CIFAR100, and a subset of ImageNet demonstrate improved embedding accuracy and interpretability over Euclidean approaches.

Learning Structured Representations with Hyperbolic Embeddings: An Overview

The paper addresses the need for learning structured representations that respect hierarchical relationships present in many real-world datasets. Traditional representation learning methods often disregard existing hierarchical structures, leading to potential semantic distortions. The authors propose HypStructure, a novel hyperbolic structured regularization approach, which leverages the properties of hyperbolic geometry to integrate label hierarchies into learned representations, thereby enhancing generalization and interpretability.

The paper begins by recognizing that datasets such as ImageNet and CIFAR exhibit natural hierarchical structures, yet common learning paradigms fail to utilize these structures effectively. This oversight can undermine the semantically meaningful organization of data classes. Recent approaches have attempted to incorporate hierarchies using Euclidean-based tree metrics, but suffer from inherent limitations due to Euclidean space's inability to accurately model tree-like data with low distortion.

Hyperbolic geometry, characterized by its negative curvature, provides a suitable framework for embedding hierarchical relationships. It allows for a continuous representation akin to a tree structure and supports exponential growth, making it well-suited for low-dimensional embeddings with minimal distortion. Thus, the paper introduces HypStructure, which combines a hyperbolic cophenetic correlation coefficient (HypCPCC) loss and a hyperbolic centering loss to embed hierarchical information explicitly.

The empirical evaluations performed on large-scale vision benchmarks, including CIFAR10, CIFAR100, and a subset of ImageNet, demonstrate HypStructure's effectiveness. Notably, HypStructure achieves substantial reductions in distortion, enhancing both in-distribution classification accuracy and out-of-distribution (OOD) detection performance. This is particularly evident under scenarios with resource constraints or low dimensions, where the geometry's representative capacity becomes critical.

Analyzing the performance, the research shows a significant improvement in embedding accuracy with the hyperbolic approach, compared to previous Euclidean-based methods. This is evidenced by improved CPCC scores and measured $\delta_{rel}$ (Gromov's hyperbolicity), showcasing more tree-like learned representations. Moreover, with the integration of centering constraints, the learned representations become less distorted and more interpretable, further bolstering generalization capabilities in downstream tasks.

A compelling theoretical aspect of the research is the eigenvalue analysis of learned representations, revealing insights into improved OOD detection. By linking eigenvalue behaviors to hierarchical feature structures, the paper provides a deeper understanding of representation dynamics in hierarchical settings, establishing a foundation for future exploration in structured representation learning.

In conclusion, this paper makes a significant contribution to the field by harnessing the geometric properties of hyperbolic spaces to improve hierarchical representation learning. The success of HypStructure in reducing distortion and enhancing performance on both ID and OOD tasks underscores the potential of hyperbolic embeddings in AI applications. Future research may focus on refining the integration of hierarchical data, exploring alternative hyperbolic models, and extending these insights to other domains, underpinning the evolution of semantically informed machine learning methodologies.