Quaternion Knowledge Graph Embeddings (1904.10281v3)

Abstract: In this work, we move beyond the traditional complex-valued representations, introducing more expressive hypercomplex representations to model entities and relations for knowledge graph embeddings. More specifically, quaternion embeddings, hypercomplex-valued embeddings with three imaginary components, are utilized to represent entities. Relations are modelled as rotations in the quaternion space. The advantages of the proposed approach are: (1) Latent inter-dependencies (between all components) are aptly captured with Hamilton product, encouraging a more compact interaction between entities and relations; (2) Quaternions enable expressive rotation in four-dimensional space and have more degree of freedom than rotation in complex plane; (3) The proposed framework is a generalization of ComplEx on hypercomplex space while offering better geometrical interpretations, concurrently satisfying the key desiderata of relational representation learning (i.e., modeling symmetry, anti-symmetry and inversion). Experimental results demonstrate that our method achieves state-of-the-art performance on four well-established knowledge graph completion benchmarks.

• The paper introduces a novel quaternion embedding approach that models knowledge graphs using hypercomplex rotations and Hamilton products.
• It employs a scoring function based on the quaternion inner product to effectively capture latent interactions between entities and relations.
• Experimental results on benchmarks like WN18 and FB15K-237 demonstrate superior completion performance over complex-valued models.

Quaternion Knowledge Graph Embeddings: An Academic Overview

Knowledge graphs (KGs) are critical elements in the architecture of semantic applications, facilitating relational reasoning and structural representation learning. However, real-world knowledge graphs are often incomplete, prompting the need for effective methods for knowledge graph completion. The paper "Quaternion Knowledge Graph Embeddings" presents an innovative approach by introducing quaternion embeddings to model entities and relations within KGs, moving beyond traditional complex-valued representations.

The central tenet of this paper is the deployment of quaternion embeddings, marked by their hypercomplex nature with three imaginary components, for knowledge graph embeddings. The authors propose a novel mechanism in which relations are modeled as rotations in quaternion space, leveraging the Hamilton product to facilitate interaction between the entities and relations. This formulation harnesses the latent inter-dependencies in hypercomplex space, resulting in a more expressive model.

Methodology and Scoring Function

The paper delineates the employment of quaternion representations for entities in knowledge graphs, operationalized through a quaternion matrix for entities and another for relations. The relational rotation is achieved by normalizing the relational quaternion to a unit quaternion, eliminating scaling effects, and rotating the head entity quaternion using the Hamilton product. The quaternion inner product with the tail entity quaternion is then utilized as the scoring function for evaluating and learning knowledge graph embeddings.

This methodological framework offers multiple advantages:

1. Expressive Rotations: Quaternion embeddings in four-dimensional space allow for more expressive rotations compared to complex plane rotations, thereby offering increased degrees of freedom and robust modeling against spatial transformations and noise.
2. Hamilton Product: Ensures enhanced interdependency mapping among all components, thus capturing latent interactions effectively.
3. Desirable Properties: The framework generalizes current models like ComplEx, retaining the ability to model fundamental properties like symmetry, anti-symmetry, and inversion.

Experimental Evaluation

The authors underline the state-of-the-art performance of their model through extensive experiments on standard benchmarks such as WN18, FB15K, WN18RR, and FB15K-237. The results illustrate superior performance over existing models, highlighting the efficacy of hypercomplex space in representing KGs and completing missing links. The quaternion-based method outperforms complex-based counterparts like ComplEx and RotatE, particularly on datasets where trivial inverse relations are removed.

Implications and Future Directions

The implications of quaternion knowledge graph embeddings are profound, both practically and theoretically. Practically, this model demonstrates applicability across various knowledge-based AI tasks where incomplete information is a challenge. Theoretically, this work rekindles interest in hypercomplex numbers for machine learning tasks, suggesting further exploration of their properties and potential benefits.

Future research might explore extensions of this framework to more advanced hypercomplex systems like Octonions, as hinted within the framework's versatility. Additionally, while quaternion embeddings show promise, further refinement could address the potential complexities and challenges associated with expanding to systems with even more imaginary components.

In conclusion, this paper contributes a significant step forward in knowledge graph embeddings, offering a methodologically sound and highly effective approach through quaternion mathematics, thereby laying the groundwork for expanded use of hypercomplex systems in artificial intelligence tasks.