Recommender systems are increasingly spreading to different areas like e-commerce or video streaming to alleviate information overload. One of the most fundamental methods for recommendation is Collaborative Filtering (CF), which leverages historical user-item interactions to infer user preferences. In recent years, Graph Neural Networks (GNNs) have been extensively studied to capture graph structures in CF tasks. Despite this remarkable progress, local structure modeling and embedding distortion still remain two notable limitations in the majority of GNN-based CF methods. Therefore, in this paper, we propose a novel Hyperbolic Graph Transformer architecture, to tackle the long-tail problems in CF tasks. Specifically, the proposed framework is comprised of two essential modules: 1) Local Hyperbolic Graph Convolutional Network (LHGCN), which performs graph convolution entirely in the hyperbolic manifold and captures the local structure of each node; 2) Hyperbolic Transformer, which is comprised of hyperbolic cross-attention mechanisms to capture global information. Furthermore, to enable its feasibility on large-scale data, we introduce an unbiased approximation of the cross-attention for linear computational complexity, with a theoretical guarantee in approximation errors. Empirical experiments demonstrate that our proposed model outperforms the leading collaborative filtering methods and significantly mitigates the long-tail issue in CF tasks. Our implementations are available in https://github.com/EnkiXin/Hgformer.

Cross-Domain Recommendation (CDR) seeks to utilize knowledge from different domains to alleviate the problem of data sparsity in the target recommendation domain, and has been gaining more attention in recent years. Although there have been notable advances in this area, most current methods represent users and items in Euclidean space, which is not ideal for handling long-tail distributed data in recommendation systems. Additionally, adding data from other domains can worsen the long-tail characteristics of the entire dataset, making it harder to train CDR models effectively. Recent studies have shown that hyperbolic methods are particularly suitable for modeling long-tail distributions, which has led us to explore hyperbolic representations for users and items in CDR scenarios. However, due to the distinct characteristics of the different domains, applying hyperbolic representation learning to CDR tasks is quite challenging. In this paper, we introduce a new framework called Hyperbolic Contrastive Learning (HCTS), designed to capture the unique features of each domain while enabling efficient knowledge transfer between domains. We achieve this by embedding users and items from each domain separately and mapping them onto distinct hyperbolic manifolds with adjustable curvatures for prediction. To improve the representations of users and items in the target domain, we develop a hyperbolic contrastive learning module for knowledge transfer. Extensive experiments on real-world datasets demonstrate that hyperbolic manifolds are a promising alternative to Euclidean space for CDR tasks. The codes are available at https://github.com/EnkiXin/hcts.