---
title: "Complex Embeddings for Simple Link Prediction — Trouillon 2016"
description: "The paper addresses link prediction, that is, automatically completing missing facts in large… 26 references, cited by 420 papers here."
authors: ["Théo Trouillon", "Johannes Welbl", "Sebastian Riedel", "Éric Gaussier", "Guillaume Bouchard"]
year: 2016
venue: "arXiv:1606.06357"
source: "https://arxiv.org/abs/1606.06357"
doi: "10.48550/arxiv.1606.06357"
arxiv: "1606.06357"
licence: "arXiv — non-exclusive licence to distribute"
licence_url: "http://arxiv.org/licenses/nonexclusive-distrib/1.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/trouillon-2016-complex-embeddings-simple-link-prediction/"
---
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# Complex Embeddings for Simple Link Prediction

**Authors:** Théo Trouillon, [Johannes Welbl](https://md.research.elamur.ai/free/en/authors/johannes-welbl/), [Sebastian Riedel](https://md.research.elamur.ai/free/en/authors/sebastian-riedel/), [Éric Gaussier](https://md.research.elamur.ai/free/en/authors/eric-gaussier/) and [Guillaume Bouchard](https://md.research.elamur.ai/free/en/authors/guillaume-bouchard/) · **Year:** 2016 · **Venue:** arXiv:1606.06357 · **Licence:** [arXiv — non-exclusive licence to distribute](http://arxiv.org/licenses/nonexclusive-distrib/1.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/1606.06357) · [PDF](https://arxiv.org/pdf/1606.06357) · [DOI 10.48550/arxiv.1606.06357](https://doi.org/10.48550/arxiv.1606.06357) · [arXiv 1606.06357](https://arxiv.org/abs/1606.06357) · [OpenAlex](https://openalex.org/W2432356473)

**Citations:** 1,069 citations (OpenAlex, 2026-10-14) · [420](https://md.research.elamur.ai/free/en/papers/trouillon-2016-complex-embeddings-simple-link-prediction/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
The paper addresses link prediction, that is, automatically completing missing facts in large knowledge bases such as Freebase or DBPedia. The authors represent each fact as a "subject–relation–object" triple and complete a three-dimensional tensor via latent factorization. The novelty of the ComplEx model is that entity and relation vectors take complex values, and the score of a fact is computed as the real part of the Hermitian product. Because one of the vectors is conjugated, the score depends on the order of the arguments. The model therefore captures both symmetric and antisymmetric relations, while memory and computation time grow linearly with the dimensionality. Theoretically, the approach rests on the spectral decomposition of normal matrices, and for practical implementation an equivalent formulation using only real numbers is given. On standard benchmarks, ComplEx consistently outperforms stronger and more complex models such as NTN and HolE. Thus, making a model capture more types of relations does not necessarily require inflating the number of parameters: it is enough to change the numerical space of the embeddings.

## Key points
- The ComplEx scoring function is Re(\<w_r, e_s, ē_o>) with e_s, e_o, w_r ∈ C^K. The complexity is O(K) in both time and memory. For comparison, RESCAL requires O(K²), and HolE requires O(K log K) in time due to circular correlation.
- DistMult is symmetric in subject and object by construction. ComplEx removes this limitation without additional parameters.
- For FB15K, the authors separately studied the effect of the number of negative examples per positive one (from 1 to 200). The best result was obtained with 100 negatives: filtered MRR 0.737 and Hits@1 64.8%. With 200 negatives, accuracy drops again. As a trade-off between accuracy and training time, the authors recommend 50.
- The code is open on GitHub: `ttrouill/complex`. The paper was published at ICML 2016.
- Among future directions, the authors name smarter generation of negative examples and complex embeddings in other dot-product-based models, in particular in deep neural networks.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

## Institutions

- [University College London](https://md.research.elamur.ai/free/en/universities/university-college-london/) — University, London, United Kingdom · 88 works · 19,432 citations
- [Université Grenoble Alpes](https://md.research.elamur.ai/free/en/universities/universite-grenoble-alpes/) — University, Saint-Martin-d'Hères, France · 20 works · 8,059 citations

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## Cite

Cite as: Théo Trouillon, Johannes Welbl, Sebastian Riedel, Éric Gaussier and Guillaume Bouchard. 2016. Complex Embeddings for Simple Link Prediction. arXiv:1606.06357.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{trouillon2016complex,
  title = {{Complex Embeddings for Simple Link Prediction}},
  author = {Théo Trouillon, Johannes Welbl, Sebastian Riedel, Éric Gaussier and Guillaume Bouchard},
  year = {2016},
  doi = {10.48550/arxiv.1606.06357},
  eprint = {1606.06357},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.1606.06357}
}
```

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*Updated 31 October 2026*