[Paper Review] Group Representation Theory for Knowledge Graph Embedding

Chen Cai|arXiv (Cornell University)|Sep 11, 2019

Advanced Graph Neural Networks27 references4 citations

TL;DR

This paper introduces a group representation theory framework for knowledge graph embedding, unifying existing methods through group actions. By applying Schur's lemma, it proves RotatE can model relations from any finite Abelian group, establishing a theoretical foundation for its expressiveness and generalization across group structures.

ABSTRACT

Knowledge graph embedding has recently become a popular way to model relations and infer missing links. In this paper, we present a group theoretical perspective of knowledge graph embedding, connecting previous methods with different group actions. Furthermore, by utilizing Schur's lemma from group representation theory, we show that the state of the art embedding method RotatE can model relations from any finite Abelian group.

Motivation & Objective

• To establish a theoretical framework linking knowledge graph embedding methods to group representation theory.
• To explain the expressiveness of existing methods like RotatE through the lens of group actions.
• To demonstrate that RotatE can represent relations from any finite Abelian group using Schur's lemma.
• To unify diverse knowledge graph embedding techniques under a common algebraic structure.

Proposed method

• Modeling knowledge graph entities and relations as group actions within a representation space.
• Applying Schur's lemma to analyze the structure of invariant subspaces in the embedding space.
• Formalizing RotatE as a unitary representation of a finite Abelian group.
• Using group homomorphisms to map relations to rotations in complex vector space.
• Deriving conditions under which a representation is irreducible and invariant under group operations.
• Proving that any finite Abelian group relation can be embedded using RotatE's rotation mechanism.

Experimental results

Research questions

• RQ1 How can knowledge graph embedding methods be systematically unified under a group-theoretic framework?
• RQ2 What is the theoretical basis for RotatE's ability to model diverse relations?
• RQ3 Can RotatE represent relations from any finite Abelian group?
• RQ4 What role do group representations and Schur's lemma play in ensuring model expressiveness?
• RQ5 How do different group actions correspond to existing embedding architectures?

Key findings

• RotatE can model relations from any finite Abelian group, as proven using Schur's lemma.
• The theoretical framework unifies existing methods by interpreting them as different group actions on the embedding space.
• Schur's lemma ensures that the embedding space respects the algebraic structure of the underlying group.
• The framework explains why RotatE achieves strong performance on link prediction tasks.
• The representation of relations as group elements provides a principled way to generalize beyond observed relations.
• The results validate that RotatE's rotation mechanism is not arbitrary but grounded in group representation theory.