Relations Are Channels: Knowledge Graph Embedding via Kraus Decompositions

TL;DR

This paper introduces Kraus channel theory to knowledge graph embedding, providing a principled foundation for relation operators, and proposes KrausKGE, a new model that improves handling complex relations and reasoning tasks.

Contribution

It establishes a Kraus channel framework for KGE, generalizes existing models, and introduces KrausKGE, which outperforms baselines on complex relation tasks.

Findings

KrausKGE outperforms strong baselines on N-to-N relations.

Theoretical lower bound relates to relation matrix rank.

Performance increases with relation fan-out.

Abstract

Knowledge graph embedding (KGE) models typically represent each relation as an operator on entity embeddings. In this work, we identify three structural axioms that any principled relation operator must satisfy, linearity, trace preservation, and complete positivity, and show that they characterize a Kraus channel structure via the Kraus representation theorem. The completeness constraint defining this family is equivalent to these axioms, providing a principled foundation rather than an externally imposed condition. Under this formulation, most existing operator-based KGE models are recoverable as special cases with Kraus rank $\kappa = 1$ under specific embedding choices. We further generalize this characterization to arbitrary metric geometries by introducing \mbox{w-Kraus} channels, which satisfy completeness by construction within their respective spaces. Building on this theory, we propose \textsc{KrausKGE}, a principled KGE model that naturally handles $1$-to-$N$ and $N$-to-$N$ relations, supports $k$-hop reasoning without requiring explicit path encoders, and eliminates the need for norm constraints on entity embeddings. Additionally, our framework yields the first theoretically grounded per-relation complexity measure in the KGE literature, with a provable lower bound in terms of the empirical relation matrix rank. Empirical evaluation demonstrates that \textsc{KrausKGE} consistently outperforms strong baselines on $N$-to-$N$ relations, with performance gains that increase monotonically with relation fan-out, in alignment with theoretical predictions.