Relations Are Channels: Knowledge Graph Embedding via Kraus Decompositions
Authors: Sayan Kumar Chaki
Organizations: Inria, Laboratoire Hubert Curien, Université Jean Monnet
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.
Knowledge graph embedding (KGE) demonstrates its effectiveness for predicting missing links in knowledge graphs (KGs) by projecting entities and relations into a low-dimensional vector space. It is crucial for KGE models to effectively capture inference patterns (patterns) inherent in KGs, such as symmetry/antisymmetry, inversion and composition. Although recent KGE models exhibit strong capabilities in modeling such diverse patterns, they suffer from inherent limitations stemming from pattern over-generalization, where embeddings learned from only a single pattern instance inevitably generalize that pattern to all related instances, i.e., generalize the pattern universally. To address this issue, we propose PogRE (Pattern Over-Generalization Robust Embedding), a simple but effective method that utilizes dense linear transformations and compound operations for relation representation. Our theoretical analysis demonstrates that a dense linear transformation allows a pattern to become progressively universal as more triples are observed in the pattern. Furthermore, after observing d+1 linearly independent entities (d+1 denotes the dimension of entity), the linear transformation guarantees universal generalization of the pattern across all related instances. Experimental results on three standard benchmark datasets show that PogRE outperforms existing state-of-the-art KGE models in link prediction. Moreover, our empirical results indicate that PogRE effectively addresses the negative impact of over-generalization.
Large language models extracting knowledge graphs from text capture only explicitly stated facts, often leaving semantically related entities disconnected across documents. We present an additive, engine-neutral second pass that discovers these latent ties without altering extracted facts. Each document is chunked and embedded once; top-k nearest- neighbor queries across existing chunks yield candidate node pairs via entity membership maps. Candidate pairs are scored using Shepard inverse-distance weighting with a rescaled chord distance metric, avoiding the threshold-collapsing flaw of affine cosine scoring behind a k-NN gate. Un-gated per-pair accumulators form a commutative monoid, ensuring the pipeline is strictly order-independent and scales incrementally without recomputing prior documents. Implemented across FalkorDB, Kinetica, ArangoDB, and Neo4j, our method shows that 768- and 240-dimensional embeddings retain 92% and 72% edge fidelity against a 3072-D baseline while achieving a 25x faster top-k formulation.
Knowledge Graph Embeddings (KGEs) support a wide range of downstream tasks over Knowledge Graphs (KGs). In practice, KGs evolve as new entities and facts are added, motivating Continual Knowledge Graph Embedding (CKGE) methods that update embeddings over time. Current CKGE approaches address catastrophic forgetting (i.e., the performance degradation on previously learned tasks) primarily by limiting changes to existing embeddings. However, we show that this view is incomplete. When new entities are introduced, their embeddings can interfere with previously learned ones, causing the model to predict them in place of previously correct answers. This phenomenon, which we call entity interference, has been largely overlooked and is not accounted for in current CKGE evaluation protocols. As a result, the assessment of catastrophic forgetting becomes misleading, and CKGE methods performance is systematically overestimated. To address this issue, we introduce a corrected CKGE evaluation protocol that accounts for entity interference. Through experiments on multiple benchmarks, we show that ignoring this effect can lead to performance overestimation of up to 25%, particularly in scenarios with significant entity growth. We further analyze how different CKGE methods and KGE models are affected by the different sources of forgetting, and introduce a catastrophic forgetting metric tailored to CKGE.