1 A General Framework for Multi-Relational Representation Learning
Most existing neural embedding models for multi-relational learning can be derived from a general framework. The input is a relation triplet describing (the subject) and (the object) that are in a certain relation . The output is a scalar measuring the validity of the relationship. Each input entity can be represented as a high-dimensional sparse vector (“one-hot” index vector or “-hot” feature vector). The first neural network layer projects the input vectors to low dimensional vectors, and the second layer projects these vectors to a real value for comparison via a relation-specific operator (it can also be viewed as a scoring function).
More formally, denote as the input for entity and as the first layer neural network parameter. The scoring function for a relation triplet can be written as
Many choices for the form of the scoring function
are available. Most of the existing scoring functions in the literature can be unified based on a basic linear transformation, a bilinear transformation or their combination, where and are defined as
which are parametrized by and , respectively.
In Table 1, we summarize several popular scoring functions in the literature for a relation triplet , reformulated in terms of the above two functions. Denote by two entity vectors. Denote by and matrix or vector parameters for linear transformation . Denote by and matrix or tensor parameters for bilinear transformation .
is an identity matrix.is an additional parameter for relation . The scoring function for TransE is derived from as in .
|Single Layer ||-|
This general framework for relationship modeling also applies to the recent deep-structured semantic model [12, 22, 23, 9, 28], which learns the relevance or a single relation between a pair of word sequences. The framework above applies when using multiple neural network layers to project entities and using a relation-independent scoring function . The cosine scoring function is a special case of with normalized and .
The neural network parameters of all the models discussed above can be learned by minimizing a margin-based ranking objective111Other objectives such as mutual information (as in ) and reconstruction loss (as in tensor decomposition approaches ) can also be applied. Comparisons among these objectives are beyond the scope of this paper., which encourages the scores of positive relationships (triplets) to be higher than the scores of any negative relationships (triplets). Usually only positive triplets are observed in the data. Given a set of positive triplets , we can construct a set of negative triplets by corrupting either one of the relation arguments, . The training objective is to minimize the margin-based ranking loss
2 Experiments and Discussion
Datasets and evaluation metrics
We used the WordNet (WN) and Freebase (FB15k) datasets introduced in . WN contains triplets with entities and relations, and FB15k consists of triplets with entities and relations. We also consider a subset of FB15k (FB15k-401) containing only frequent relations (relations with at least training examples). This results in triplets with entities and relations. We use link prediction as our prediction task as in . For each test triplet, each entity is treated as the target entity to be predicted in turn. Scores are computed for the correct entity and all the corrupted entities in the dictionary and are ranked in descending order. We consider Mean Reciprocal Rank (MRR) (an average of the reciprocal rank of an answered entity over all test triplets), HITS@10 (top-10 accuracy), and Mean Average Precision (MAP) (as used in 
) as the evaluation metrics.
All the models were implemented in C# and using GPU. Training was implemented using mini-batch stochastic gradient descent with AdaGrad. At each gradient step, we sampled for each positive triplet two negative triplets, one with a corrupted subject entity and one with a corrupted object entity. The entity vectors are renormalized to have unit length after each gradient step (it is an effective technique that empirically improved all the models). For the relation parameters, we used standard L2 regularization. For all models, we set the number of mini-batches to , the dimensionality of the entity vector , the regularization parameter
, and the number of training epochson FB15k and FB15k-401 and on WN ( was determined based on the learning curves where the performance of all models plateaued.) The learning rate was initially set to and then adapted during training by AdaGrad.
2.1 Model Comparisons
We examine five embedding models in decreasing order of complexity: (1) NTN with tensor slices as in ; (2) Bilinear+Linear, NTN with tensor slice and without the non-linear layer; (3) TransE with L2 norm222Empirically we found no significant differences between L1-norm and L2-norm for the TransE objective. , a special case of Bilinear+Linear as described in ; (4) Bilinear; (5) Bilinear-diag: a special case of Bilinear where the relation matrix is a diagonal matrix.
Table 2 shows the results of all compared methods on all the datasets. In general, we observe that the performance increases as the complexity of the model decreases on FB. NTN, the most complex model, provides the worst performance on both FB and WN, which suggests overfitting. Compared to the previously published results of TransE , our implementation achieves much better results (53.9% vs. 47.1% on FB15k and 90.9% vs. 89.2% on WN) using the same evaluation metric (HITS@10). We attribute such discrepancy mainly to the different choice of SGD optimization: AdaGrad vs. constant learning rate. We also found that Bilinear consistently provides comparable or better performance than TransE, especially on WN. Note that WN contains much more entities than FB, it may require the parametrization of relations to be more expressive to better handle the richness of entities. Interestingly, we found that a simple variant of Bilinear – Bilinear-diag, clearly outperforms all baselines on FB and achieves comparable performance to Bilinear on WN.
2.2 Multiplicative vs. Additive Interactions
Note that Bilinear-diag and TransE have the same number of model parameters and their difference lies in the operational choices of the composition of two entity vectors – Bilinear-diag uses weighted element-wise dot product (multiplicative operation) and TransE uses element-wise subtraction with a bias (additive operation). To highlight the difference, here we use DistMult and DistAdd to refer to Bilinear-diag and TransE, respectively. Comparison between these two models can provide us with more insights on the effect of two common choices of compositional operations – multiplication and addition for modeling entity relations. Overall, we observed superior performance of DistMult on all the datasets in Table 2. Table 3 shows the HITS@10 score on four types of relation categories (as defined in ) on FB15k-401 when predicting the subject entity and the object entity respectively. We can see that DistMult significantly outperforms DistAdd in almost all the categories. More qualitative results can be found in the Appendix.
|Predicting subject entities||Predicting object entities|
2.3 Entity Representations
In the following, we examine the learning of entity representations and introduce two further improvements: using non-linear projection and initializing entity vectors with pre-trained phrase vectors. We focus on DistMult as our baseline and compare it with the two modifications DistMult-tanh (using for entity projection333When applying non-linearity, we remove the normalization steps on entity parameters during training as already helps control the scaling freedoms.) and DistMult-tanh-EV-init (initializing the entity parameters with the -dimensional pre-trained phrase vectors released by word2vec ) on FB15k-401. We also reimplemented the word vector representation and initialization technique introduced in  – each entity is represented as an average of its word vectors and the word vectors are initialized using the -dimensional pre-trained word vectors released by word2vec. We denote this method as DistMult-tanh-WV-init. Inspired by , we design a new evaluation setting where the predicted entities are automatically filtered according to “entity types” (entities that appear as the subjects/objects of a relation have the same type defined by that relation). This provides us with better understanding of the model performance when some entity type information is provided.
|MRR||HITS10||MAP (w/ type checking)|
In Table 4, we can see that DistMult-tanh-EV-init provides the best performance on all the metrics. Surprisingly, we observed performance drops by DistMult-tanh-WV-init. We suspect that this is because word vectors are not appropriate for modeling entities described by non-compositional phrases (more than 73% of the entities in FB15k-401 are person names, locations, organizations and films). The promising performance of DistMult-tanh-EV-init suggests that the embedding model can greatly benefit from pre-trained entity-level vectors.
In this paper we present a unified framework for modeling multi-relational representations, scoring, and learning, and conduct an empirical study of several recent multi-relational embedding models under the framework. We investigate the different choices of relation operators based on linear and bilinear transformations, and also the effects of entity representations by incorporating unsupervised vectors pre-trained on extra textual resources. Our results show several interesting findings, enabling the design of a simple embedding model that achieves the new state-of-the-art performance on a popular knowledge base completion task evaluated on Freebase. Given the recent successes of deep learning in various applications; e.g.[11, 27, 5], our future work will aim to exploit deep structure including possibly tensor construct in computing the neural embedding vectors; e.g. [12, 29, 13]. This will extend the current multi-relational neural embedding model to a deep version that is potentially capable of capturing hierarchical structure hidden in the input data.
-  Antoine Bordes, Xavier Glorot, Jason Weston, and Yoshua Bengio. A semantic matching energy function for learning with multi-relational data. Machine Learning, pages 1–27, 2013.
-  Antoine Bordes, Nicolas Usunier, Alberto Garcia-Duran, Jason Weston, and Oksana Yakhnenko. Translating embeddings for modeling multi-relational data. In NIPS, 2013.
-  Antoine Bordes, Jason Weston, Ronan Collobert, and Yoshua Bengio. Learning structured embeddings of knowledge bases. In AAAI, 2011.
-  Kai-Wei Chang, Wen-tau Yih, Bishan Yang, and Chris Meek. Typed tensor decomposition of knowledge bases for relation extraction. In EMNLP, 2014.
-  Li Deng, G. Hinton, and B. Kingsbury. New types of deep neural network learning for speech recognition and related applications: An overview. In in ICASSP, 2013.
-  Pedro Domingos. Prospects and challenges for multi-relational data mining. ACM SIGKDD explorations newsletter, 5(1):80–83, 2003.
-  Xin Luna Dong, K Murphy, E Gabrilovich, G Heitz, W Horn, N Lao, Thomas Strohmann, Shaohua Sun, and Wei Zhang. Knowledge vault: A Web-scale approach to probabilistic knowledge fusion. In KDD, 2014.
-  John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and stochastic optimization. The Journal of Machine Learning Research, 12:2121–2159, 2011.
-  Jianfeng Gao, Patrick Pantel, Michael Gamon, Xiaodong He, Li Deng, and Yelong Shen. Modeling interestingness with deep neural networks. In EMNLP, 2014.
-  Lise Getoor and Ben Taskar, editors. Introduction to Statistical Relational Learning. The MIT Press, 2007.
-  Geoff Hinton, L. Deng, D. Yu, G. Dahl, A. Mohamed, N. Jaitly, A. Senior, V. Vanhoucke, P. Nguyen, T. Sainath, and B. Kingsbury. Deep neural networks for acoustic modeling in speech recognition. IEEE Sig. Proc. Mag., 29:82–97, 2012.
-  Po-Sen Huang, Xiaodong He, Jianfeng Gao, Li Deng, Alex Acero, and Larry Heck. Learning deep structured semantic models for Web search using clickthrough data. In CIKM, 2013.
-  B Hutchinson, L. Deng, and D. Yu. Tensor deep stacking networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8):1944–1957, 2013.
-  Rodolphe Jenatton, Nicolas Le Roux, Antoine Bordes, and Guillaume Obozinski. A latent factor model for highly multi-relational data. In NIPS, 2012.
-  Charles Kemp, Joshua B Tenenbaum, Thomas L Griffiths, Takeshi Yamada, and Naonori Ueda. Learning systems of concepts with an infinite relational model. In AAAI, volume 3, page 5, 2006.
-  Ni Lao, Tom Mitchell, and William W Cohen. Random walk inference and learning in a large scale knowledge base. In EMNLP, pages 529–539, 2011.
-  Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In NIPS, pages 3111–3119, 2013.
-  Maximilian Nickel, Volker Tresp, and Hans-Peter Kriegel. A three-way model for collective learning on multi-relational data. In ICML, pages 809–816, 2011.
-  Maximilian Nickel, Volker Tresp, and Hans-Peter Kriegel. Factorizing YAGO: scalable machine learning for linked data. In WWW, pages 271–280, 2012.
-  Alberto Paccanaro and Geoffrey E. Hinton. Learning distributed representations of concepts using linear relational embedding. IEEE Transactions on Knowledge and Data Engineering, 13(2):232–244, 2001.
-  Matthew Richardson and Pedro Domingos. Markov logic networks. Machine learning, 62(1-2):107–136, 2006.
-  Yelong Shen, Xiaodong He, Jianfeng Gao, Li Deng, and Gregoire Mesnil. A latent semantic model with convolutional-pooling structure for information retrieval. In CIKM, 2014.
Yelong Shen, Xiaodong He, Jianfeng Gao, Li Deng, and Grégoire Mesnil.
Learning semantic representations using convolutional neural networks for Web search.In WWW, pages 373–374, 2014.
-  Ajit P Singh and Geoffrey J Gordon. Relational learning via collective matrix factorization. In KDD, pages 650–658. ACM, 2008.
-  Richard Socher, Danqi Chen, Christopher D. Manning, and Andrew Y. Ng. Reasoning with neural tensor networks for knowledge base completion. In NIPS, 2013.
-  Ilya Sutskever, Joshua B Tenenbaum, and Ruslan Salakhutdinov. Modelling relational data using Bayesian clustered tensor factorization. In NIPS, pages 1821–1828, 2009.
-  O. Vinyals, Y. Jia, L. Deng, and T. Darrell. Learning with recursive perceptual representations. In NIPS, 2012.
-  Wen-tau Yih, Xiaodong He, and Christopher Meek. Semantic parsing for single-relation question answering. In ACL, 2014.
-  D. Yu, L. Deng, and F. Seide. The deep tensor neural network with applications to large vocabulary speech recognition. IEEE Trans. Audio, Speech and Language Proc., 21(2):388 –396, 2013.
Figure 1 and 2 illustrate the relation embeddings learned by DistMult and DistAdd using t-SNE. We selected relations in the FB15k-401 dataset. The embeddings learned by DistMult nicely reflect the clustering structures among these relations (e.g. /film/release_region is closed to /film/country); whereas the embeddings learned by DistAdd present structure that is harder to interpret.
Table 5 shows some concrete examples: top nearest neighbors in the relation space learned by DistMult and DistAdd along with the distance values (Frobenius distance between two relation matrices or Euclidean distance between two relation vectors). We can see that the nearest neighbors found by DistMult are much more meaningful. DistAdd tends to retrieve irrelevant relations which take in completely different types of arguments.