# Relations Are Channels: Knowledge Graph Embedding via Kraus Decompositions

arXiv: https://arxiv.org/abs/2605.10317 · HTML: https://mlsanity.com/paper/2605.10317/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 51 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 29 of 29)

- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [Embedding Entities and Relations for Learning and Inference in Knowledge Bases](https://mlsanity.com/paper/1412.6575/) (2014)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space](https://mlsanity.com/paper/1902.10197/) (2019)
- [Convolutional 2D Knowledge Graph Embeddings](https://mlsanity.com/paper/1707.01476/) (2017)
- [KG-BERT: BERT for Knowledge Graph Completion](https://mlsanity.com/paper/1909.03193/) (2019)
- [TuckER: Tensor Factorization for Knowledge Graph Completion](https://mlsanity.com/paper/1901.09590/) (2019)
- [Neural Bellman-Ford Networks: A General Graph Neural Network Framework for Link Prediction](https://mlsanity.com/paper/2106.06935/) (2021)
- [DeepPath: A Reinforcement Learning Method for Knowledge Graph Reasoning](https://mlsanity.com/paper/1707.06690/) (2017)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Quaternion Knowledge Graph Embeddings](https://mlsanity.com/paper/1904.10281/) (2019)
- [A Survey on Knowledge Graph-Based Recommender Systems](https://mlsanity.com/paper/2003.00911/) (2020)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Multi-Hop Knowledge Graph Reasoning with Reward Shaping](https://mlsanity.com/paper/1808.10568/) (2018)
- [Orthogonal Recurrent Neural Networks with Scaled Cayley Transform](https://mlsanity.com/paper/1707.09520/) (2017)
- [Learning to Exploit Long-term Relational Dependencies in Knowledge Graphs](https://mlsanity.com/paper/1905.04914/) (2019)
- [Modeling Relation Paths for Representation Learning of Knowledge Bases](https://mlsanity.com/paper/1506.00379/) (2015)
- [A*Net: A Scalable Path-based Reasoning Approach for Knowledge Graphs](https://mlsanity.com/paper/2206.04798/) (2022)
- [Knowledge Graph Embedding: A Survey from the Perspective of Representation Spaces](https://mlsanity.com/paper/2211.03536/) (2022)
- [HousE: Knowledge Graph Embedding with Householder Parameterization](https://mlsanity.com/paper/2202.07919/) (2022)
- [TransG : A Generative Mixture Model for Knowledge Graph Embedding](https://mlsanity.com/paper/1509.05488/) (2015)
- [Ultrahyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2206.00449/) (2022)
- [Geometry Interaction Knowledge Graph Embeddings](https://mlsanity.com/paper/2206.12418/) (2022)
- [End-to-End Entity Classification on Multimodal Knowledge Graphs](https://mlsanity.com/paper/2003.12383/) (2020)
- [Generalizing Knowledge Graph Embedding with Universal Orthogonal Parameterization](https://mlsanity.com/paper/2405.08540/) (2024)
- [Knowledge Graph Embedding using Graph Convolutional Networks with Relation-Aware Attention](https://mlsanity.com/paper/2102.07200/) (2021)
- [SQUIRE: A Sequence-to-sequence Framework for Multi-hop Knowledge Graph Reasoning](https://mlsanity.com/paper/2201.06206/) (2022)
- [MEIM: Multi-partition Embedding Interaction Beyond Block Term Format for Efficient and Expressive Link Prediction](https://mlsanity.com/paper/2209.15597/) (2022)
- [Block-Diagonal Orthogonal Relation and Matrix Entity for Knowledge Graph Embedding](https://mlsanity.com/paper/2401.05967/) (2024)

## Abstract (arXiv)

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 mboxw-Kraus channels, which satisfy completeness by construction within their respective spaces. Building on this theory, we propose textscKrausKGE, 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 textscKrausKGE consistently outperforms strong baselines on N-to-N relations, with performance gains that increase monotonically with relation fan-out, in alignment with theoretical predictions.