# Knowledge Graph Completion via Complex Tensor Factorization

arXiv: https://arxiv.org/abs/1702.06879 · HTML: https://mlsanity.com/paper/1702.06879/

## Key facts

- Cited by 80 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2017.

## Built on by (most cited first, 30 of 80)

- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Knowledge Graph Reasoning with Relational Digraph](https://mlsanity.com/paper/2108.06040/) (2021)
- [Graph Attention Multi-Layer Perceptron](https://mlsanity.com/paper/2206.04355/) (2022)
- [A Review of Location Encoding for GeoAI: Methods and Applications](https://mlsanity.com/paper/2111.04006/) (2021)
- [Inductive Relation Prediction by BERT](https://mlsanity.com/paper/2103.07102/) (2021)
- [TempoQR: Temporal Question Reasoning over Knowledge Graphs](https://mlsanity.com/paper/2112.05785/) (2021)
- [Question Calibration and Multi-Hop Modeling for Temporal Question Answering](https://mlsanity.com/paper/2402.13188/) (2024)
- [Learning to Sample and Aggregate: Few-shot Reasoning over Temporal Knowledge Graphs](https://mlsanity.com/paper/2210.08654/) (2022)
- [Double Equivariance for Inductive Link Prediction for Both New Nodes and New Relation Types](https://mlsanity.com/paper/2302.01313/) (2023)
- [BiQUE: Biquaternionic Embeddings of Knowledge Graphs](https://mlsanity.com/paper/2109.14401/) (2021)
- [Knowledge Graph Completion with Pre-trained Multimodal Transformer and Twins Negative Sampling](https://mlsanity.com/paper/2209.07084/) (2022)
- [Convolutional Hypercomplex Embeddings for Link Prediction](https://mlsanity.com/paper/2106.15230/) (2021)
- [A Unified Framework for Rank-based Evaluation Metrics for Link Prediction in Knowledge Graphs](https://mlsanity.com/paper/2203.07544/) (2022)
- [Multimodal Learning on Graphs for Disease Relation Extraction](https://mlsanity.com/paper/2203.08893/) (2022)
- [Factor Graph Neural Networks](https://mlsanity.com/paper/2308.00887/) (2023)
- [Searching to Sparsify Tensor Decomposition for N-ary Relational Data](https://mlsanity.com/paper/2104.10625/) (2021)
- [Positive-Unlabeled Learning with Adversarial Data Augmentation for Knowledge Graph Completion](https://mlsanity.com/paper/2205.00904/) (2022)
- [Shrinking Embeddings for Hyper-Relational Knowledge Graphs](https://mlsanity.com/paper/2306.02199/) (2023)
- [Bilinear Scoring Function Search for Knowledge Graph Learning](https://mlsanity.com/paper/2107.00184/) (2021)
- [Neural Graph Databases](https://mlsanity.com/paper/2209.09732/) (2022)
- [Unleashing the Power of Imbalanced Modality Information for Multi-modal Knowledge Graph Completion](https://mlsanity.com/paper/2402.15444/) (2024)
- [Subgraph-aware Few-Shot Inductive Link Prediction via Meta-Learning](https://mlsanity.com/paper/2108.00954/) (2021)
- [Modeling Fine-grained Information via Knowledge-aware Hierarchical Graph for Zero-shot Entity Retrieval](https://mlsanity.com/paper/2211.10991/) (2022)
- [Exploring the Limits of Few-Shot Link Prediction in Knowledge Graphs](https://mlsanity.com/paper/2102.03419/) (2021)
- [Efficient Relation-aware Scoring Function Search for Knowledge Graph Embedding](https://mlsanity.com/paper/2104.10880/) (2021)
- [Message Intercommunication for Inductive Relation Reasoning](https://mlsanity.com/paper/2305.14074/) (2023)
- [Is Complex Query Answering Really Complex?](https://mlsanity.com/paper/2410.12537/) (2024)
- [KGTuner: Efficient Hyper-parameter Search for Knowledge Graph Learning](https://mlsanity.com/paper/2205.02460/) (2022)
- [R5: Rule Discovery with Reinforced and Recurrent Relational Reasoning](https://mlsanity.com/paper/2205.06454/) (2022)
- [Modality-Aware Negative Sampling for Multi-modal Knowledge Graph Embedding](https://mlsanity.com/paper/2304.11618/) (2023)

## Abstract (arXiv)

In statistical relational learning, knowledge graph completion deals with automatically understanding the structure of large knowledge graphs---labeled directed graphs---and predicting missing relationships---labeled edges. State-of-the-art embedding models propose different trade-offs between modeling expressiveness, and time and space complexity. We reconcile both expressiveness and complexity through the use of complex-valued embeddings and explore the link between such complex-valued embeddings and unitary diagonalization. We corroborate our approach theoretically and show that all real square matrices---thus all possible relation/adjacency matrices---are the real part of some unitarily diagonalizable matrix. This results opens the door to a lot of other applications of square matrices factorization. Our approach based on complex embeddings is arguably simple, as it only involves a Hermitian dot product, the complex counterpart of the standard dot product between real vectors, whereas other methods resort to more and more complicated composition functions to increase their expressiveness. The proposed complex embeddings are scalable to large data sets as it remains linear in both space and time, while consistently outperforming alternative approaches on standard link prediction benchmarks.