1 Introduction
Largescale knowledge graphs Nickel et al. (2016a) are indispensable resources for knowledgeintensive applications such as question answering, dialog systems, and distantly supervised relation extraction. A knowledge graph is a collection of triplets representing the fact that (binary) relation holds between subject entity and object entity . Although efforts continue to enrich existing knowledge graphs with more facts, many facts are still missing Nickel et al. (2016a). Knowledge graph completion (KGC) aims to automatically detect missing facts in an incomplete knowledge graph, and has become an active field of research in recent years.
Knowledge graph embedding (KGE) is a promising approach to KGC. It embeds entities and relations in vector space, and defines a
scoring function to evaluate the degree of factuality of a given triplet in terms of vector operations.Bilinear KGE models are a popular choice for a scoring function, along with those based on translation and neural networks. RESCAL
Nickel et al. (2011) adopts a generic bilinear form as the scoring function, given by . In this formula, are the dimensional vector embeddings of entities and , respectively, and is the matrix embedding of relation . Some of the more recent models have constrained the relation matrices to be diagonal. DistMult Yang et al. (2015) and ComplEx Trouillon et al. (2016) are two such diagonal models. HolE Nickel et al. (2016b) does not use diagonal relation matrices, but has been shown Hayashi and Shimbo (2017) to be isomorphic to ComplEx. These models have a smaller number of parameters than RESCAL, making them less prone to overfitting, and the performance is usually better.While all these models were designed with a specific task of KGC in mind, i.e., computing the factuality of triplets, another important task on knowledge graphs was pursued by Guu et al. (2015) and Lin et al. (2015). This latter task, called path query answering (path QA), is to answer composite queries that consist of a cascade of relations, as opposed to an atomic relation. See Figure 1 for instance. A query “Is Beatrice a child of a paternal uncle of William?” can be answered by predicting the truth value of the triplet (William, fatherOf/brotherOf/fatherOf, Beatrice) where fatherOf/brotherOf/fatherOf is a binary relation not present in the knowledge graph as a relation (edge) label but is composed of a cascade of three atomic relations.^{1}^{1}1We regard inverse relations (e.g., ) also as atomic relations. Composite queries are also called path queries, as they can be represented as paths in a knowledge graph; see, e.g., the blue line in Figure 1(a). Notice however that some of the edges in the path may be missing due to the incompleteness of the knowledge graph; even in such circumstances, the model must ideally be able to answer path queries correctly.
Guu et al. (2015) extended the existing KGE approaches to path QA. For example, to answer a general path query with RESCAL, a composite relation is modeled by matrix product , and the score for the given query is modeled by . This formulation is also applicable to DistMult and ComplEx, which use diagonal relation matrices. In diagonalized models, however, relation matrices are commutative, in the sense that for any pair of relations .
Commutativity of relation matrices was not recognized as an issue in the past research because the main focus was on predicting the truth value of atomic triplets. However, when path queries are concerned, commutativity poses a problem. Consider, for example, a relation sequence
fatherOf/brotherOf/fatherOf 
and its permutation
Although these are two distinct paths (cf. Figure 1(b, c)), in bilinear models with commutative relation matrices, they are represented by the same product of relation matrices, which thereby makes the truth values of these permutated queries indistinguishable by their scores.
Drawing on the observation above, this paper proposes a new KGE model called BlockHolE, wherein relations are represented by block circulant matrices. This makes relation matrices noncommutative, and thus it does not suffer from the issues arising from commutativity, yet in general manages to reduce the number of parameters compared with RESCAL. It can be interpreted as a generalization of HolE and ComplEx, and also subsumes RESCAL as an extreme case. We report experimental results in both path and atomic QA tasks.
2 Notation and preliminaries
Symbol  Description 

sets of real/complex numbers  
th component of vector  
component of matrix  
transpose of  
conjugate of  
componentwise (Hadamard) product  
circular convolution  
circular correlation  
real part of complex number  
diagonal matrix with main diagonal  
circulant matrix determined by  
sum of the componentwise products of  
discrete Fourier matrix  
set of entities  
set of relations  
set of observed facts (triplets)  
set of ground truth facts  
Knowledge graph induced by facts 
We first introduce symbols and notation used in this paper, followed by some preliminaries on circulant matrices, circular convolution, correlation, and Fourier transform. The summary of symbols can be found in Table 1.
Let be the set of reals, and be the set of complex numbers. Let denote the th component of vector , and let the element of matrix . For a complex number , vector , and matrix , let , , and denote their complex conjugate, respectively.
Let , , and be dimensional (real or complex) vectors. Let denote an diagonal matrix with the main diagonal components given by . We write to denote the componentwise product of and ; i.e., , or , . We also write .
For dimensional real vectors^{2}^{2}2Generally, circular convolution, circular correlation, and circulant matrices are defined over . However, in this paper, it suffices to define them over . , and denote circular convolution and circular correlation, respectively defined by
where vector indices that do not fall in the range must be interpreted by .
For dimensional real vector , let
be an operation that converts a vector to a circulant matrix of size .
A circulant matrix can be diagonalized as , where is the discrete Fourier matrix of order . Also, circular convolution and correlation can be written in terms of : , and . It follows that
(1)  
(2) 
These equations imply that circular convolution and correlation can be computed in time using the fast Fourier transform (FFT).
3 Knowledge graph embedding using bilinear maps
A knowledge graph is a labeled multigraph , where is the set of entities (or vertices), is the set of relation labels (or edge labels), and defines the observed instances of binary relations over entities (or labeled edges). An item is called a triplet, with and called its subject and object, respectively. For every entity in , it is assumed that contains at least one triplet with or ; likewise, for every relation in , is assumed to contain at least one triplet . Because determines the sets and of entities and relations, we write to denote the knowledge graph determined by .
Aside from observed triplets , we also assume the presence of a set of (ground truth) facts, which is a strict superset of , i.e., . Thus, is not fully observable.
3.1 Knowledge graph completion
Knowledge graph completion (KGC) is the task of identifying the set of ground truth facts from observed facts (or equivalently, from ).
A popular approach to KGC is to design a scoring function quantifying how likely a triplet is true. This scoring function is learned from the observed triplets , in a way that it generalizes well to unobserved triplets ; i.e., the score must be high for both observed and unobserved facts, and it must be low for nonfactual triplets.
In knowledge graph embedding (KGE)–based approaches to KGC, the scoring function is defined in terms of the embeddings of entities and relations; i.e., , , and are embedded as objects in a vector space, and is defined in terms of some operations over these objects.
3.2 Bilinear models for knowledge graph embedding
Below, we describe some of the popular KGE models that use bilinear maps to define scoring functions.
3.2.1 Rescal
RESCAL Nickel et al. (2011) provides the most general form of bilinear scoring function.
(3) 
where are the vector embeddings of entities and , respectively, and is the matrix representing relation . Thus, parameters are required per relation, which is not only a computational burden but also the cause of overfitting during training Kazemi and Poole (2018).
3.2.2 DistMult
DistMult Yang et al. (2015) is a model obtained by restricting the relation matrices of RESCAL to diagonal; i.e., , . The scoring function is thus
(4) 
Although the number of parameters is reduced considerably, the scoring function (4) is symmetric with respect to the entities, i.e., . This is a severe limitation because most realworld relations are nonsymmetric.
3.2.3 ComplEx: Complex embedding
The complex embedding (ComplEx) Trouillon et al. (2016) represents entities and relations as dimensional vectors as in DistMult, but their components are complexvalued.
The scoring function of ComplEx is given by
where are the embeddings of , , and , respectively. The number of parameters in ComplEx is , and the score is computable in time linear in the dimension of vector space. Unlike DistMult, ComplEx can model nonsymmetric relations, since in general.
3.2.4 HolE: Holographic embedding
The holographic embedding (HolE) Nickel et al. (2016b) uses circular correlation to define a scoring function
(5) 
where are dimensional real vectors representing relation , and entities and , respectively. HolE has only parameters per relation, and it can model nonsymmetric relations since in general. Computing circular correlation requires time if FFT is employed. Eq. (5) is not a bilinear form, but it has been shown Hayashi and Shimbo (2017) that HolE is isomorphic to ComplEx, and thus any model in HolE can be converted to an equivalent model in ComplEx, and vice versa.
4 Path question answering over a knowledge graph
4.1 Path query answering
Let be the set of ground truth facts, and let be its induced knowledge graph. For relations , we call a relation path of length . When , the relation path is atomic; otherwise, it is composite. Let . We say a path query holds (or “is true”) in (or with respect to ) if
where and . Path query answering (path QA) is the task of predicting the truth value of path queries with respect to the unobserved set of ground truth facts, when its incomplete subset is only available. In other words, we want to predict that is true if a path from to exists in , although some of the edges that constitute the path may be missing in the observed graph .
For atomic path queries (i.e., those with length ), path QA reduces to that of knowledge graph completion introduced in Section 3.1. Thus, it is natural to address general path QA by extending the scoring function of KGC methods so that composite relation is allowed in place of atomic relation ; i.e., by defining . Previous work Guu et al. (2015) explored this direction, which is also pursued in the rest of this paper.
4.2 Issues in existing KGE models applied to path QA
We now discuss the extension of existing bilinear KGE models to path QA. We begin with RESCAL, which is the most general among existing bilinear models. In RESCAL, if we assume for true triplets , we can model path QA as computing
(6) 
As seen in this formula, a composite relation is represented by the product of the matrices for atomic relations Guu et al. (2015).
Likewise, DistMult and ComplEx can also be used for path QA, by computing
and
respectively. However, because diagonal matrices are commutative, the score of is equal to any path query in which are permutated, such as . That is, because , their truth values cannot be distinguished by the magnitude of scores. More recent bilinear models such as ANALOGY^{3}^{3}3 We categorize ANALOGY as a diagonal model because each block diagonal element of its relation matrices can be substituted by a single equivalent complexvalued component. Liu et al. (2017) and SimplE Kazemi and Poole (2018) also represent relations by diagonal matrices, and thus they can only model commutative relation paths. Moreover, for SimplE, which represents subject and object entities in different vector spaces, it is not clear how it can be applied to path QA.
In the translationbased model TransE Bordes et al. (2013), the scoring function is given by^{4}^{4}4The original TransE defines a penalty function, which gives a smaller value if a triplet is more likely to be true. We thus changed the sign to make it a scoring function in Eq. (7).
(7) 
Guu et al. (2015) extended this function for a path query by
(8) 
Thus, a composite relation is represented as the sum of the embedding vectors for its constituent atomic relations. Unfortunately, Eq. (8) is also invariant with the permutation of relations , and their order is not respected.
5 Knowledge graph embedding with block circulant matrices
5.1 BlockHolE
In this section, we propose a bilinear KGE model suitable for path QA. In this model, the relation matrices are noncommutative. It thus respects the order of relations in a path query. Further, it has a smaller number of parameters than RESCAL in general. To be specific, our model constrains the relation matrices to be block circulant.
A matrix is block circulant if it can be written in the form
(9) 
where each , , is a circulant matrix determined by . Thus, if the dimension of the matrix in Eq. (9) is , we have . A block circulant matrix is noncommutative when ; i.e., for two block circulant matrices , , in general.
Substituting a block circulant matrix of Eq. (9) for matrix in the bilinear scoring function (Eq. (3)) yields
(10) 
where , and , . Recall that , and thus , . Using equalities Nickel et al. (2016b) and to rewrite Eq. (10), we have
(11) 
We call this model BlockHolE, after the fact that it reduces to HolE when ; cf. Eq. (5). Also, BlockHolE is identical to dimensional RESCAL when (or equivalently ).
The number of parameters in BlockHolE is (or ), and naive computation of Eq. (11) takes time using FFT. However, we can make this computation faster by exploiting the duality of the Fourier transform, as shown below.
5.2 Fast computation in complex space
Using a similar technique used by Hayashi and Shimbo Hayashi and Shimbo (2017) to show the equivalence of ComplEx and HolE, we can eliminate Fourier transform to speed up the computation of BlockHolE scores. We first rewrite Eq. (11) as follows:
where is the discrete Fourier matrix. Here we used Eq. (1) to derive the second equation, and to derive the third. Defining complex vectors , , and yields
(12) 
On the basis of Eq. (12), we train directly in complex space (i.e., the Fourier domain) instead of and use it as the vector embedding of entity , for all ; similarly, is directly trained in complex space to represent relation . The number of parameters in this model is , and Eq. (12) can be computed in time. Typically, we set . For instance, in the experiment of Section 6, we set and , and thus . In this case, factor is negligible and the computational complexity is linear in .
5.3 Modeling path QA
BlockHolE can be used in path QA as follows. First, for any and , let
Then, Eq. (12) can be rewritten as
and we can compute the score of relation paths by
Since for , this scoring function respects the order of relations in .
6 Experiments
In this section, we report the results of empirical evaluation investigating the commutativity property of bilinear KGE models on the path QA task. As expected, the proposed BlockHolE model, which uses noncommutative relation matrices, outperformed commutative bilinear KGE models.
6.1 Dataset and evaluation protocol
WN11  FB13  
Train  112,581  316,232  
Base  Valid  2,609  5,908 
Test  10,544  23,733  
Train  2,129,539  6,266,058  
Path  Valid  11,277  27,163 
TestDeduction  24,749  77,883  
TestInduction  21,828  31,674 
WN11  FB13  
Base  Deduction  Induction  Base  Deduction  Induction  
P@10  MQ  P@10  MQ  P@10  MQ  P@10  MQ  P@10  MQ  P@10  MQ  
DistMult  45.6  83.0  33.5  97.7  29.6  79.8  62.7  91.6  63.6  86.4  59.3  86.5 
ComplEx  60.9  83.1  68.7  99.2  46.1  79.7  76.8  93.0  71.5  90.0  70.5  88.9 
RESCAL  51.8  74.2  43.2  97.9  51.2  76.8  65.2  91.1  66.9  88.4  69.8  89.0 
80.9  83.4  70.2  99.5  54.9  81.0  79.2  93.2  75.0  91.5  71.3  90.0  
80.5  75.6  69.3  99.2  54.5  77.4  76.2  92.1  72.1  90.5  70.9  89.5 
The comparison of KGE models was performed in two path QA tasks: (i) ranking and (ii) binary classification tasks.
6.1.1 Path QA ranking
For the path QA ranking task, we adopted the same protocol and dataset used by Guu et al. (2015). Table 2 shows the statistics of their dataset. The dataset consists of two parts, “Base” and “Path”.
The Base part only contains facts (i.e., path queries with ), and thus it is essentially for evaluating KGC performance. Its training samples constitute the observed facts , and the facts in the entire Base part (training/validation/test sets) make the ground truth facts .
The Path part contains path queries sampled from the same and as the Base part. The test samples in the Path part is divided into “deduction” and “induction” sets. In the “deduction” set, test samples were sampled from the Base training graph . By contrast, in the “induction” set, the test samples were chosen from the ground truth graph such that none of them have a corresponding path in . Thus, the “induction” set is intended to measure how well a model generalizes to unobserved paths, whereas the “deduction” set is to test its ability to faithfully encode the observed training graph.
At the time of evaluation, for each a test sample , a candidate set
was first computed. In other words, the candidates are the entities for which (i.e., the last relation in the test query) takes as its object at least once in . Then, for each compared model, we made the ranking of the candidates entities in by the score , where is learned by the model from the training set.
The quality of the ranking was measured by two evaluation metrics: averaged mean quantile (MQ) and P@10 (percentage of correct answers ranked in the top 10). For
where , the correct answer set is the set of all entities that can be reached from by traversing over . Formally, let , and the answer set can be recursively defined: . With these definitions, MQ is computed by the following formula:(13) 
where is the set of incorrect answers. Eq. (13) cannot be computed for queries with which , and these queries were excluded from evaluation. For further details, see the original paper by Guu et al. (2015).
6.1.2 Path QA classification
In the path QA classification task, we simply report classification accuracy. After the scoring function
was trained with logistic regression, a path query
was classified as true if
, or false otherwise.Since the test and validation sets of Path in Table 2 contain only correct queries, we sampled negative ones by the following procedure: For a correct query (), we generated its reverse relation path query . If does not exist in , we used it as a negative.
6.2 Experiment setup
We compared BlockHolE with stateoftheart bilinear KGE models: DistMult, RESCAL and ComplEx. We have implemented BlockHolE in Java. BlockHolE reduces to ComplEx when , and with the imaginary parts of parameters set to , it reduces to RESCAL when and to DistMult when . For a fair run time comparison, however, we separately implemented RESCAL using jblas1.2.4 for matrix computation. Through all experiments, we optimized the logistic loss with L2 regularization on the parameters :
where denotes the truth value of a query in a training data . Given a correct query , we generated negative samples by replacing with an entity randomly sampled from .
We selected the hyperparameters via grid search such that on the validation set they maximize classification accuracy in the path QA classification task and MQ in the path QA ranking task. For all models except BlockHolE, all combinations of
, learning rate , and the embedding size were tried during grid search. For BlockHolE, all combinations of , and were tried. The maximum number of training epochs was set to 500. The number of negatives generated per positive sample was 5 during training.6.3 Results
6.3.1 Path QA ranking
Table 3 shows the results on the path QA ranking data. BlockHolE outperforms other bilinear KGE models considerably both on deductive and inductive test settings. These results strongly suggest that BlockHolE is more expressive in modeling path QA than DistMult and ComplEx, while effectively reducing redundant parameters in RESCAL which can cause model overfitting. Figure 2 shows the empirical scalability of BlockHolE. When is small, BlockHolE scales linearly in the dimension of the embedding space.
6.3.2 Path QA classification
Figure 3 shows the accuracy of path QA classification. DistMult and ComplEx were considerably worse than BlockHolE and RESCAL for both WN11 and FB13. This result confirms our claim: The noncommutativity of relation matrices plays a critical role in modeling path QA. The performance of BlockHolE () was comparable to that of RESCAL but the former was 12 times faster.
6.4 Analysis
Label  Relation Path  ComplEx  BlockHolE 

+  */parents/religion/*  96.7  100.0 
  */religion/parents/*  3.3  100.0 
The accuracies of BlockHolE and RESCAL on the path QA classification task were markedly better than those of DistMult and ComplEx. We analyzed the results further. We extracted all queries from of FB13 that consist of an interpretable relation path */parents/religion/* where denotes “can match any relation path”. For such queries , we also generated meaningless queries as negatives. Table 4 shows the classification accuracies of ComplEx and BlockHolE (). The results clearly show that ComplEx cannot correctly answer the negative queries at all due to the lack of the noncommutative property.
7 Summary
In this paper, we have pointed out the problems of existing bilinear KGE models in path QA, and proposed a new model that overcomes these problems. This model, called BlockHolE, represents relations as block circulant matrices. As a result, it respects the order of relations in path queries, while enjoying lineartime computation of scoring functions when the number
of blocks is sufficiently small. It generalizes HolE/ComplEx, and it can also be interpreted as an interpolation between RESCAL and HolE/ComplEx. Its effectiveness was shown empirically in path QA.
Our proposal can be useful in not only path QA but also many tasks such as associative rule mining Yang et al. (2015), path regularization Lin et al. (2015), and more complex QA Hamilton et al. (2018), in which composite relations need to be embedded as a vector. Other future directions include reducing the increased parameters in the proposed block circulant matrices, such as by using multiplicative L1 regularization for ComplEx Manabe et al. (2018).
Acknowledgments
We thank anonymous reviewers for helpful comments. This work was partially supported by JSPS Kakenhi Grant Numbers 19H04173, 18K11457, and 18H03288.
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