A Study of LFT Embeddings in the Second Order Clifford Algebra <italic>Cl</italic>(R&#x00B2;,&#x00BA;)

Abstract

Knowledge graph embedding models represent entities as vectors in continuous spaces and their relations by geometric transformations, mainly translation, and rotation. However, multi-relational knowledge graphs contain complex sub-graph structures, for which these two families of models fail to solely preserve. The complexities of these sub-graph structures often require embedding vector spaces with additional structures and composite transformations. A typical intrinsic additional algebraic structure on these embedding vector spaces has the potential to allow models to preserve complex relational patterns, such as non-commutativity, which are intrinsically related to the vector space. In this paper, we use linear fractional transformations ( $\mathrm {LFT_{}}$ ) in the noncommutative Clifford Algebra of order 2, $Cl(\mathbb {R}^{2,0})$ , to embed relations. $\mathrm {LFT_{}}$ can be understood as transformations that generalize Möbius transformations to the framework of Clifford algebras, in a non-commutative settings. Thus, the $\mathrm {LFT_{}}$ relation embeddings in our proposed models can exhibit geometric transformations namely translation, rotation, reflection, homothety, inversion, and their composites. By examining the preservation of complex structures using eleven benchmark knowledge graph datasets, our models showed results which are superior or competitive to existing state-of-the-art models relative to learning relational, many-to-many structural, and hierarchical patterns. This literature does not only advance the performance of knowledge graph embeddings but also emphasizes the importance of robust mathematical foundations and particularly the potential of Clifford algebras in embedding models.

Keywords

• Representation learning
• knowledge graph embeddings
• Clifford algebra
• geometric algebra
• linear fractional and Möbius transformations