Hyperbolic Graph Convolutional Neural Networks (1910.12933v1)

Abstract: Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unclear how to transform the features into hyperbolic embeddings with the right amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural Network (HGCN), the first inductive hyperbolic GCN that leverages both the expressiveness of GCNs and hyperbolic geometry to learn inductive node representations for hierarchical and scale-free graphs. We derive GCN operations in the hyperboloid model of hyperbolic space and map Euclidean input features to embeddings in hyperbolic spaces with different trainable curvature at each layer. Experiments demonstrate that HGCN learns embeddings that preserve hierarchical structure, and leads to improved performance when compared to Euclidean analogs, even with very low dimensional embeddings: compared to state-of-the-art GCNs, HGCN achieves an error reduction of up to 63.1% in ROC AUC for link prediction and of up to 47.5% in F1 score for node classification, also improving state-of-the art on the Pubmed dataset.

• The paper introduces hyperbolic GCNNs that employ both the Poincaré ball and hyperboloid models to better capture tree-like data structures.
• It utilizes concepts from differential and hyperbolic geometry to derive geodesics, parallel transports, and projections for robust network optimization.
• The study demonstrates improved performance in hierarchical clustering and link prediction, indicating strong potential for advanced geometric deep learning.

Hyperbolic Graph Convolutional Neural Networks

Overview

The paper addresses the use of hyperbolic geometry in graph convolutional neural networks (GCNNs), focusing specifically on hyperbolic spaces for their potential in representing complex structures such as hierarchies and tree-like data. The paper leverages the unique properties of hyperbolic geometry to enhance the capabilities of conventional GCNNs.

Key Concepts in Differential and Hyperbolic Geometry

The foundation of this research is grounded in differential and hyperbolic geometry:

• Differential Geometry: Central concepts such as manifolds, tangent spaces, and Riemannian metrics are utilized to define and manipulate curved spaces. The Riemannian manifold's ability to measure distances on surfaces is critical for understanding the spatial relationships within the data.
• Hyperbolic Geometry: The research utilizes models like the Poincaré ball and the hyperboloid model, each offering distinct advantages in terms of optimization stability and interpretability. This involves defining distances using complex logarithmic and exponential mappings in hyperbolic space.

Geometric Models and Their Use

The paper provides a detailed comparison between the Poincaré ball model and the hyperboloid model:

• Poincaré Ball Model: Defined with negative curvature, it provides an interpretable framework wherein embeddings can be visualized directly.
• Hyperboloid Model: This model is shown to be more stable for optimization. The connection between these models, through isomorphic transformations, ensures flexibility in applications, allowing mappings from one model to the other.

Mathematical Foundations and Results

Several mathematical results underpin the work, including:

• Geodesics: The derivation of unit-speed geodesics in hyperbolic space, essential for path calculations and distance measures within the neural network.
• Parallel Transport and Projections: Techniques for translating vectors across the manifold and projecting points into tangent spaces, which are crucial for optimization within the manifold constraints.
• Curvature: A lemma demonstrating how embeddings in hyperbolic spaces with varying curvatures can be transformed while preserving specific mathematical properties. This contributes to the flexibility and adaptability of the model across different dataset characteristics.

Implications and Future Directions

The findings of this paper have significant implications for the use of hyperbolic spaces in neural network design:

• Practical Applications: By effectively capturing the hierarchical nature of data, these models promise improvements in tasks such as link prediction in knowledge graphs and hierarchical clustering.
• Theoretical Insights: The paper contributes to the growing body of work on non-Euclidean neural networks, suggesting pathways for future research on scalable and robust geometric deep learning models.

Speculation on Future Advances

Future developments may explore the integration of hyperbolic neural networks with other geometric frameworks, optimizing computational efficiency and exploring higher-dimensional manifolds to capture richer datasets. The paper's results encourage further exploration into adaptive curvatures and the dynamic nature of embedding spaces to improve learning outcomes in diverse applications.

This paper adds a valuable perspective to the use of advanced geometric techniques in modern machine learning frameworks, aligning with ongoing research towards more effective and efficient data representations.