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

Abstract

Graph Convolutional Neural Networks (GCNs) embed graph data into either Euclidean or non-Euclidean spaces. For power-law distributed graphs, Euclidean embeddings distort input features because they prevent the recovery of distances between nodes, while hyperbolic embeddings show smaller distortion in embedding input features. However, previous hyperbolic graph neural networks cannot approximate convolutions adequately as they only apply simple filters. Hyperbolic Graph Convolutional Neural Networks (HGCN) is a good illustration, constructed based on the approximation of first-order Chebyshev polynomial filters. Here, Hyperbolic Bernstein Neural Networks (HBNN) are proposed, extending Bernstein polynomials to hyperbolic space through Möbius operations for node classification and link prediction tasks. HBNN estimates filters by an order-K Bernstein polynomials approximation in hyperbolic space while setting the coefficients of each polynomial order as learnable parameters. This approach enables HBNN to learn high-order complex filters in hyperbolic space, therefore approximating convolution effectively. Experiments demonstrate that HBNN can better learn the hierarchical structure of nodes and leads to improved performance compared to other mainstream methods in node classification and link prediction tasks.