[论文解读] Complex Embeddings for Simple Link Prediction

Théo Trouillon, Johannes Welbl+3 more|arXiv (Cornell University)|Jun 20, 2016

Advanced Graph Neural Networks参考文献 25被引用 1,127

一句话总结

本文介绍 ComplEx，一种使用 Hermitian 点积的复值嵌入方法，能够建模对称和反对称关系以用于链接预测，在线性空间/时间复杂度下达到最先进的结果。

ABSTRACT

In statistical relational learning, the link prediction problem is key to automatically understand the structure of large knowledge bases. As in previous studies, we propose to solve this problem through latent factorization. However, here we make use of complex valued embeddings. The composition of complex embeddings can handle a large variety of binary relations, among them symmetric and antisymmetric relations. Compared to state-of-the-art models such as Neural Tensor Network and Holographic Embeddings, our approach based on complex embeddings is arguably simpler, as it only uses the Hermitian dot product, the complex counterpart of the standard dot product between real vectors. Our approach is scalable to large datasets as it remains linear in both space and time, while consistently outperforming alternative approaches on standard link prediction benchmarks.

研究动机与目标

• Motivate link prediction in large knowledge bases and address antisymmetric relations without excessive parameters.
• Propose a complex-valued embedding model that uses the Hermitian dot product for relation scoring.
• Extend to multi-relational data with shared entity embeddings and relation-specific complex weights.
• Demonstrate scalability and empirical performance on standard benchmarks (FB15K, WN18).
• Provide an equivalent real-valued reformulation to aid practical implementation.

提出的方法

• Model relations as low-rank, complex-valued matrices X = (E W Ÿer^T) and predict scores via X_{so} = e_s^T W Ÿr o (complex conjugate of e_o).
• Use a shared entity embedding E  C^{n x K} and a relation embedding w_r  C^K per relation.
• Represent multi-relational data with a logit link: P(Y_{rso}=1) = sigmoid(<w_r, e_s, Ÿr e_o>).
• Explain that the real part of the complex decomposition suffices to approximate real-valued scores while preserving antisymmetry.
• Argue that the model handles symmetric and antisymmetric relations via the Hermitian product, with linear space/time complexity.
• Provide an equivalent real-valued formulation to facilitate implementation.

实验结果

研究问题

• RQ1 Can complex-valued embeddings with the Hermitian dot product jointly model symmetric and antisymmetric relations in knowledge graphs?
• RQ2 Do ComplEx embeddings outperform real-valued baselines (e.g., DistMult, TransE, HolE) on standard link prediction benchmarks?
• RQ3 Is the approach scalable to web-scale knowledge bases with linear time and space complexity?
• RQ4 How does the model perform on datasets with antisymmetric relations (e.g., WN18) versus symmetric relations?
• RQ5 Can a real-valued reformulation retain performance while simplifying implementation?

主要发现

• ComplEx outperforms state-of-the-art baselines on FB15K and WN18 in filtered MRR and Hits@k metrics.
• The model accurately captures antisymmetric relations, improving on datasets with such patterns (e.g., WN18).
• Complex embeddings enable a simple, scalable dot-product-based composition that remains linear in space and time.
• An equivalent real-valued reformulation is provided, aiding practical adoption without requiring complex arithmetic.
• Training with negative sampling and logistic loss yields strong empirical performance across datasets.