Laplacian Features for Learning with Hyperbolic Space

Due to its geometric properties, hyperbolic space can support high-fidelity embeddings of tree- and graph-structured data. As a result, various hyperbolic networks have been developed which outperform Euclidean networks on many tasks: e.g. hyperbolic graph convolutional networks (GCN) can outperform vanilla GCN on some graph learning tasks. However, most existing hyperbolic networks are complicated, computationally expensive, and numerically unstable -- and they cannot scale to large graphs due to these shortcomings. With more and more hyperbolic networks proposed, it is becoming less and less clear what key component is necessary to make the model behave. In this paper, we propose HyLa, a simple and minimal approach to using hyperbolic space in networks: HyLa maps once from a hyperbolic-space embedding to Euclidean space via the eigenfunctions of the Laplacian operator in the hyperbolic space. We evaluate HyLa on graph learning tasks including node classification and text classification, where HyLa can be used together with any graph neural networks. When used with a linear model, HyLa shows significant improvements over hyperbolic networks and other baselines

Hyperbolic Interaction Model For Hierarchical Multi-Label Classification

Different from the traditional classification tasks which assume mutual exclusion of labels, hierarchical multi-label classification (HMLC) aims to assign multiple labels to every instance with the labels organized under hierarchical relations. Besides the labels, since linguistic ontologies are intrinsic hierarchies, the conceptual relations between words can also form hierarchical structures. Thus it can be a challenge to learn mappings from word hierarchies to label hierarchies. We propose to model the word and label hierarchies by embedding them jointly in the hyperbolic space. The main reason is that the tree-likeness of the hyperbolic space matches the complexity of symbolic data with hierarchical structures. A new Hyperbolic Interaction Model (HyperIM) is designed to learn the label-aware document representations and make predictions for HMLC. Extensive experiments are conducted on three benchmark datasets. The results have demonstrated that the new model can realistically capture the complex data structures and further improve the performance for HMLC comparing with the state-of-the-art methods. To facilitate future research, our code is publicly available