Zhihao Li

Menglin Yang, Zhihao Li, Min Zhou et al.

Considering the prevalence of the power-law distribution in user-item networks, hyperbolic space has attracted considerable attention and achieved impressive performance in the recommender system recently. The advantage of hyperbolic recommendation lies in that its exponentially increasing capacity is well-suited to describe the power-law distributed user-item network whereas the Euclidean equivalent is deficient. Nonetheless, it remains unclear which kinds of items can be effectively recommended by the hyperbolic model and which cannot. To address the above concerns, we take the most basic recommendation technique, collaborative filtering, as a medium, to investigate the behaviors of hyperbolic and Euclidean recommendation models. The results reveal that (1) tail items get more emphasis in hyperbolic space than that in Euclidean space, but there is still ample room for improvement; (2) head items receive modest attention in hyperbolic space, which could be considerably improved; (3) and nonetheless, the hyperbolic models show more competitive performance than Euclidean models. Driven by the above observations, we design a novel learning method, named hyperbolic informative collaborative filtering (HICF), aiming to compensate for the recommendation effectiveness of the head item while at the same time improving the performance of the tail item. The main idea is to adapt the hyperbolic margin ranking learning, making its pull and push procedure geometric-aware, and providing informative guidance for the learning of both head and tail items. Extensive experiments back up the analytic findings and also show the effectiveness of the proposed method. The work is valuable for personalized recommendations since it reveals that the hyperbolic space facilitates modeling the tail item, which often represents user-customized preferences or new products.

Menglin Yang, Min Zhou, Tong Zhang et al.

Graph representation learning in Euclidean space, despite its widespread adoption and proven utility in many domains, often struggles to effectively capture the inherent hierarchical and complex relational structures prevalent in real-world data, particularly for datasets exhibiting a highly non-Euclidean latent anatomy or power-law distributions. Hyperbolic geometry, with its constant negative curvature and exponential growth property, naturally accommodates such structures, offering a promising alternative for learning rich graph representations. This survey paper provides a comprehensive review of the rapidly evolving field of Hyperbolic Graph Learning (HGL). We systematically categorize and analyze existing methods broadly dividing them into (1) hyperbolic graph embedding-based techniques, (2) graph neural network-based hyperbolic models, and (3) emerging paradigms. Beyond methodologies, we extensively discuss diverse applications of HGL across multiple domains, including recommender systems, knowledge graphs, bioinformatics, and other relevant scenarios, demonstrating the broad applicability and effectiveness of hyperbolic geometry in real-world graph learning tasks. Most importantly, we identify several key challenges that serve as directions for advancing HGL, including handling complex data structures, developing geometry-aware learning objectives, ensuring trustworthy and scalable implementations, and integrating with foundation models, e.g., large language models. We highlight promising research opportunities in this exciting interdisciplinary area. A comprehensive repository can be found at https://github.com/digailab/awesome-hyperbolic-graph-learning.

Haoze Song, Zhihao Li, Xiaobo Zhang et al.

Neural Operators have emerged as powerful tools for learning mappings between function spaces. Among them, the kernel integral operator has been widely validated on universally approximating various operators. Although many advancements following this definition have developed effective modules to better approximate the kernel function defined on the original domain (with $d$ dimensions, $d=1, 2, 3\dots$), the unclarified evolving mechanism in the embedding spaces blocks researchers' view to design neural operators that can fully capture the target system evolution. Drawing on the Schrödingerisation method in quantum simulations of partial differential equations (PDEs), we elucidate the linear evolution mechanism in neural operators. Based on that, we redefine neural operators on a new $d+1$ dimensional domain. Within this framework, we implement a Schrödingerised Kernel Neural Operator (SKNO) aligning better with the $d+1$ dimensional evolution. In experiments, the $d+1$ dimensional evolving designs in our SKNO consistently outperform other baselines across ten benchmarks of increasing difficulty, ranging from the simple 1D heat equation to the highly nonlinear 3D Rayleigh-Taylor instability. We also validate the resolution-invariance of SKNO on mixing-resolution training and zero-shot super-resolution tasks. In addition, we show the impact of different lifting and recovering operators on the prediction within the redefined NO framework, reflecting the alignment between our model and the underlying $d+1$ dimensional evolution.