Relations Are Channels: Knowledge Graph Embedding via Kraus Decompositions
2026-05-11 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors study how knowledge graph embedding (KGE) models represent relationships between entities as mathematical operations. They identify three key properties that these operations should have and show these correspond to a known mathematical structure called a Kraus channel. Using this insight, they create a new KGE model, KrausKGE, which can naturally handle complex relations involving many entities and supports multi-step reasoning without extra components. Their model also introduces a way to measure how complex each relation is, backed by theory. Experiments show KrausKGE performs better than existing methods, especially when dealing with relations connecting many entities.
Knowledge Graph EmbeddingRelation OperatorKraus ChannelKraus Representation TheoremLinearityTrace PreservationComplete PositivityMulti-hop ReasoningRelation ComplexityEntity Embeddings
Authors
Sayan Kumar Chaki
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 $κ= 1$ 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.