## 2 Related Work
### Machine-Learning-Based Model  
Machine-learning-based models typically adopt a geometric approach, embedding entities into a specific space and using relation vectors to represent the transformation between entities, including translation and rotation. TransE [19] is the most classical model that represents relations as translation patterns. However, TransE has limitations in that translations can only express inverse and symmetric relations, which causes some triples to not be correctly represented. TransH [24], TransR [25], and TransD [26] are all improvements built upon the score function of TransE.  
Following translation models, bilinear models such as DisMult [27] and ComplEx [28] were developed, which employ similarity-based scoring functions to gauge the likelihood of triples by matching potential entity and relation semantics in the embedding vector space. DisMult builds on the framework of RESCAL [29], modeling entities as vectors and relations as matrices, with the relation matrix being defined as a diagonal matrix. HoIE [30] combines the advantages of RESCAL and DisMult by representing both entities and relations as vectors in space and introducing a loop mechanism for interactions between head and tail entities. This results in a powerful but computationally efficient model. ComplEx, on the other hand, introduces complex embeddings, which represent the embeddings of entities and relations in complex space instead of real space. Additionally, ComplEx modifies the scoring function of the model to an asymmetric form, thereby better capturing asymmetric relations.  
Drawing inspiration from complex space bilinear models such as DisMult and ComplEx, RotatE treats relations as rotation operations between entities. This enables relations to express a wide range of attributes, including symmetric, anti-symmetric, inverse, and synthetic attributes. Additionally, TorusE [31], which appeared around the same time as RotatE, defines embeddings as translations on tight Lie groups, embedding entities and relations in spatial torus, thereby eliminating the need for regularization steps. QuatE [23] further developed the concept of complex space and extended it into a quaternion space with two rotation surfaces, allowing for more flexible representation compared to that achievable in a complex space. Additionally, QuatE utilizes Hamilton products to capture internal dependencies among entities and relations, resulting in a simplified and more efficient process for relation modeling. Continuing in this vein, DualE [32] further extended the space to an octonion space, which offers a means for representing both translation and rotation patterns. In recent years, researchers have developed models that embed KGs into more complex spaces. MuRP [33] and MuRE [33] incorporate multi-relation graphs into Poincare balls in hyperbolic space and transform entity embeddings through the learning of relation-specific parameters via Mobius matrix vector multiplication and Mobius addition. RoTH [34] combines hyperbolic reflections and rotations with attention to simultaneously capture hierarchical and logical patterns in a hyperbolic space. ReTIN [35] introduces real-time influence information on historical facts in hyperbolic geometric space to capture potential hierarchical structures and other rich semantic patterns.  
Although some of machine-learning-based models note the difficulty of embedding complex relations, they address the problem posed by single rotation patterns, which cannot adequately model hierarchical structures and relations between entities.