Hyperbolic Bernstein Neural Networks: Enhancing graph convolutions in non-Euclidean spaces

Area of Science:

• Machine Learning
• Graph Neural Networks
• Hyperbolic Geometry

Background:

• Graph Convolutional Neural Networks (GCNs) embed graph data, but Euclidean embeddings distort features in power-law graphs.
• Hyperbolic embeddings offer reduced distortion but existing hyperbolic GCNs struggle with adequate convolution approximation due to simple filters.

Purpose of the Study:

• To propose Hyperbolic Bernstein Neural Networks (HBNN) for improved node classification and link prediction.
• To extend Bernstein polynomials to hyperbolic space using Möbius operations for enhanced filter approximation.
• To enable learning of high-order complex filters in hyperbolic space for effective convolution.

Main Methods:

• HBNN estimates filters using an order-K Bernstein polynomial approximation in hyperbolic space.
• Coefficients for each polynomial order are set as learnable parameters.
• Möbius operations are utilized to extend Bernstein polynomials to the hyperbolic manifold.

Main Results:

• HBNN effectively approximates complex filters in hyperbolic space, enabling better convolution.
• The proposed method demonstrates improved ability to learn the hierarchical structure of graph nodes.
• Experiments show superior performance of HBNN over mainstream methods in node classification and link prediction tasks.

Conclusions:

• HBNN offers a significant advancement in hyperbolic graph neural networks.
• The approach effectively addresses limitations in approximating convolutions in hyperbolic space.
• HBNN provides a powerful new tool for analyzing complex graph-structured data.