Neural axiom network for knowledge graph reasoning

2.1. Symbolic-based reasoning

Symbolic-based reasoning methods aim at inferring new knowledge or detecting noises with the help of rules or ontologies and show good reasoning ability. For example, inductive logic programing (ILP) has been used to mine logical rules and use the rules to learn a good predictor. However, it has limitations on the open-world assumption of KGs. AMIE [11] and AMIE+ [10] make up for this shortcoming by introducing an altered confidence metric based on the partial completeness assumption. With the rules generated with AMIE+, [14] proposes to discover inverse and symmetric axioms by applying the predefined reasoning rules. Due to the incompleteness of rules and axioms, and the time-consuming process of annotating them, existing methods using ontologies for reasoning are usually accompanied by the enrichment of ontology information. For example, the work [9] presents a set of inductive methods based on statistical inductive learning, consisting of correlation computing and association rule mining to enrich ontologies with disjointness axioms. It evaluates the validity of association rule mining by computing the precision and recall scores. Then [22] proposes an improvement of association rule mining for learning disjointness axioms and applies the learned axioms to inconsistency detection. In addition to disjoint axioms, enriching DBpedia ontology with domain and range restrictions and class disjointness axioms is also discussed [31]. The enhanced ontologies are further used for error detection.

2.2. Embedding-based reasoning

Knowledge graph embedding methods embed entities and relations of a KG into a continuous vector space to preserve the structure information of the KG. There are three categories of embedding models: translational distance, semantic matching and neural network models. Translational distance models learn embeddings by translating a subject entity to an object entity through a relation. For example, TransE [2] represents entities and relations in the same vector space and assumes ( s + r ) to be close to o , where s , r , o are vector embeddings for s, r and o respectively. However, it has difficulty dealing with complex relations. To overcome the flaws, TransH [36] introduces relation-specific hyperplanes to allow entities to have different embeddings in different relations. TransR [20] builds entity and relation embeddings in separate entity and relation spaces. TransD [16] constructs mapping matrices dynamically. Similarly, TorusE [8] and RotatE [29] use lie groups and rotations for translation, respectively. Semantic matching models measure plausibility by matching the implicit semantics of entities and relations. DistMult [39] uses a bilinear model to represent entities and relations. ComplEx [33] extends DistMult by introducing complex-valued embeddings to better model asymmetric relations. HolE [27] uses the circular correlation of embeddings to learn compositional representations and semantically matches circular correlation with the relation embedding. Moreover, researchers have raised interest in applying neural networks for knowledge graph reasoning. ConvE [6] and ConvKB [25] employ convolutional neural networks to achieve better link prediction performance. CapsE [26] explores a capsule network to model triples. KGTtm [17] and CKRL [38] measure the trustworthiness or confidence of triples.

2.3. Hybrid reasoning

There is also a line of work concerning hybrid methods for KG reasoning, such as the combination of symbolic- and embedding-based reasoning, and the combination of symbolic and statistical reasoning. For the former methods, TransC [21] learns the SubClassOf axiom between types by encoding each type as a sphere and each entity as a vector. Besides, SetE [42] computes the two axioms SubClassOf and SubPropertyOf in subsumption by employing linear programming methods on embeddings, focusing on domain, range or subClassOf axioms. Recently, IterE [41] iteratively learns embeddings and rules, and considers seven object property expression axioms for rule learning. It combines rule learning and embedding learning to improve the quality of sparse entity embeddings by injecting new triples about sparse entities according to the scores of the axioms. As for the latter, the statistic-based methods, such as SDType and SDValidate [28], exploit statistical distributions of types and relations. SDType deduces missing type information based on the statistical distribution of types in the subject or object position of the relation. While SDValidate measures the deviation between actual types of the subject/object entity and the apriori probabilities given by the distribution. Furthermore, [4] first exploits a combination of entailment vectors, entailment weights, and a consistency vector to encode knowledge as embeddings in ontology streams to deal with concept drifts. It takes the concepts as input. The difference is that we design the axiom modules with the help of the definition of axioms.

OPEN IN VIEWER

Fig. 2.

The key idea of our framework. For the input triple ( s , r , o ) , we use a type embedding ( s c , o c ) and a semantic embedding ( s m , o m ) to represent the entities, and use two type embeddings ( r s , r o ) and a semantic embedding ( r m ) to represent the relation. The triple score is composed of the scores from the six modules. The domain/range axiom module calculates the compatibility between the subject/object entity type embedding and the relation embedding generated via a domain/range attention layer. The disjoint axiom module calculates the compatibility between the semantic embedding of the input relation and other relations with the same subject and object entities. The irreflexive and asymmetric axiom modules calculate the axiom scores using the type and semantic embeddings of the entities and relations. The KGE module is a knowledge graph embedding model.

3.1. Neural axiom network

We present our neural axiom network NeuRAN in Fig. 2, where the overall score of each triple is composed of a semantic score from the KGE module and five axiom scores from five axiom modules (i.e., domain, range, disjoint, irreflexive and asymmetric axiom modules). The score of each axiom module indicates the probability that the axiom holds. The assumption is that the probability values of these axioms intensify or mitigate the probability of the existence of a triple. Thus, the score of the triple ( s , r , o ) is defined as: (1) E ( s , r , o ) = E kge + λ · [ ( 1 − P dm ) + ( 1 − P rg ) + ( 1 − P dis ) + ( 1 − P irr ) + ( 1 − P asy ) ] The energy function E ( s , r , o ) consists of the score E kge from the knowledge graph module and five axiom scores from the axiom modules. λ is the weight of axiom scores. E kge is to compute a structure-based score, which can be obtained via any knowledge graph embedding models [34]. We use TransE and TransH as examples. A lower E kge indicates that the triple is more likely to be correct. The axiom scores P dm , P rg , P dis , P irr and P asy correspond to probabilities that the corresponding axioms are satisfied, where dm , rg , dis , irr and asy are respectively domain, range, disjoint, irreflexive and asymmetric axioms. Therefore, the higher P dm , P rg , P dis , P irr , P asy , and the lower ( 1 − P dm ) , ( 1 − P rg ) , ( 1 − P dis ) , ( 1 − P irr ) , ( 1 − P asy ) imply that the triple satisfies the corresponding neural axiom with a higher probability.

Following the conventional training strategy of previous models, we train NeuRAN based on the local-closed world assumption. In this case, the observed triples in KGs are regarded as positive triples, while the unobserved ones are regarded as negative triples. We utilize a margin-based ranking loss on pair-wise score functions (i.e., E ( s , r , o ) and E ( s ′ , r ′ , o ′ ) ) for training, the loss function L is defined as: (2) L = ∑ ( s , r , o ) ∈ T ∑ ( s ′ , r ′ , o ′ ) ∈ T ′ max ( 0 , E ( s , r , o ) + γ − E ( s ′ , r ′ , o ′ ) ) where γ is a margin hyper-parameter, E ( s , r , o ) and E ( s ′ , r ′ , o ′ ) are respectively the overall energy function of the positive triple ( s , r , o ) and the negative triple ( s ′ , r ′ , o ′ ) . T and T ′ are the positive and negative triple sets. As negative triples and axioms are not given, we generate negative triples by randomly corrupting the subject or object entity and make sure the replaced triples do not exist in the knowledge graph: (3) T ′ = { ( s ′ , r , o ) | s ′ ∈ E } ∪ { ( s , r , o ′ ) | o ′ ∈ E } , ( s , r , o ) ∈ T , ( s ′ , r , o ) ∉ T and ( s , r , o ′ ) ∉ T The training objective is to minimize the loss function L to learn embeddings of entities and relations and parameters involved in axiom modules. The learned embeddings and parameters are applied to complete downstream tasks, such as noise detection, triple classification and link prediction.

It is worth mentioning that, we attempt to introduce type embeddings in the design of the model. Considering domain/range axioms are associated with type compatibility, and type information is not provided, we distinguish type embeddings from semantic embeddings. Following the type-sensitive models TypeDM and TypeComplex [15], each entity is represented as two vectors (a type embedding and a semantic embedding), and each relation is represented as three vectors (two type embeddings and a semantic embedding). Take the triple ( s , r , o ) as an example. For entities, we use type embeddings ( s c and o c ) and semantic embeddings ( s m and o m ) to represent the subject entity s and the object entity o. For the relation, r s and r o represent subject type and object type embeddings expected by the relation r, and r m is the semantic embedding of r.

3.2. KG embedding module

The knowledge graph embedding module of the framework concerns the learning of a function E kge , which is designed to score each triple based on the structural information in KGs. Our framework can use any knowledge graph embedding model as the knowledge graph embedding module. In the experiments, we take the two embedding models TransE and TransH as examples to verify our method.

3.3. Five axiom modules

In addition to focusing on structural information, we consider the inherent implicit axioms in triples. Although axioms are not pre-given, it is intuitive that a correct triple is also a logically consistent triple. Any correct triple satisfies all the five axioms, including domain, range, disjoint, irreflexive, and asymmetric axioms. For example, in the case of missing domain and range of the relation nationality and types of the entities J. K. Rowling and United Kingdom, we can infer the triple (J. K. Rowling, nationality, United Kingdom) satisfies domain and range axioms by its correctness.

The design of the axiom modules relies on the definition of these axioms. To be specific, for domain/range axioms, we use the type embedding of the subject/object entity expected by the relation r s / r o and the type embedding of the subject/object entity s c / o c to calculate a type compatibility score. For disjoint axiom, we explore the semantic compatibility of any two relations with the same subject and object entities. We assume that ( o m − s m ) represents the shared semantic embedding of the relation for the subject to be s and object to be o. Then the similar score of r m and ( o m − s m ) will be calculated to reflect how well the disjoint axiom is satisfied. For irreflexive axiom, whether the relation is irreflexive and whether the subject entity is the same as the object entity ( s = o ? ) are considered. For asymmetric axiom, whether the relation is asymmetric and whether the symmetric triple of the input triple exists ( ( o , r , s ) ∈ G ? ) are concerned. We then introduce these typical axioms in detail.

4.1. Experimental settings

Datasets. In this paper, we use two popular benchmark datasets: FB15K237 [32] and WN18RR [6] to evaluate NeuRAN. They are constructed from FB15K and WN18 by removing inverse relations to solve test leakage. FB15K is a relatively dense subset extracted from the collaborative knowledge graph Freebase [1], which consists of billions of real-world facts. WN18 is a subset of WordNet [24] that describes relations between words.

Error Imputation. Since KGs are constructed in an automated or semi-automated way, noises can not be avoided. However, existing knowledge graph reasoning methods assume that triples in KGs are positive triples. Thus there are no pre-given noisy triples in FB15K237 and WN18RR. In order to verify our method, we generate new datasets with different noise rates based on the two datasets to simulate the real noisy knowledge graphs. Before generating noises, for each dataset with training, validation and test sets, we generate negative triples for the validation and test sets. The positive and negative triples in the validation set are used to find the optimum thresholds for each relation. To evaluate the performance of NeuRAN on triple classification, the positive and negative triples in the test set are classified as positive or negative triples based on both the triple scores and the thresholds. Here, we directly use the negative triples generated in OpenKE.2

Baselines. We choose TransE or TransH as the knowledge graph embedding module and compare our methods (i.e., NeuRAN(TransE), NeuRAN(TransH)) with them. Results of TransE and TransH are produced by running OpenKE [12]. We also consider CKRL(TransE) and CKRL(TransH) as baselines, which introduce path information for knowledge graph reasoning in noisy KGs. Results of CKRL(TransE) and CKRL(TransH) are reproduced by us. In the following tasks, the results of TransE, TransH, CKRL(TransE) and CKRL(TransH) are all obtained in this way.

Training Details. We use SGD [7] or Adam [18] to optimize the model for different tasks on different datasets. We select the learning rate from {0.01, 0.1, 0.5, 1}, the margin among {2,4,6,8,10}, the batch size from {100, 500}, dimension of the type from {20, 50, 100, 200}, dimension of the semantic from {50, 100, 200, 300}, the combination weight of the axiom scores λ from {0.01, 0.05, 0.1, 0.5, 1}. The number of training epochs is set as 1000.

To verify the capability of our method on noise detection task, we follow the setting of KG noise detection proposed in [38]. It aims to detect possible noises in noisy KGs according to the scores of triples and can be viewed as triple classification task on the training set.

Evaluation Protocol. First of all, we compute the score of the triple ( s , r , o ) via the energy function E ( s , r , o ) = E kge + λ · [ ( 1 − P dm ) + ( 1 − P rg ) + ( 1 − P dis ) + ( 1 − P irr ) + ( 1 − P asy ) ] . Then all triples in the training set will be ranked based on the scores. The lower the triple score, the more valid the triple is. Triples with higher values of the energy function tend to be noises. We consider the evaluation indicator the Area Under the ROC Curve (auc value) to examine how well the method classifies the noises as errors. Before calculating the AUC metric, we normalize the energy function scores into the [ 0 , 1 ] interval. Values close to 0 indicate correct triple, and values close to 1 indicate incorrect triples.

Result Analysis. Evaluation results on noisy datasets generated based on FB15K237 and WN18RR can be found in Table 4. We observe that: (1) Regardless of whether the embedding module is TransE or TransH, our models achieve comparable performance or slightly outperform TransE, CKRL(TransE), TransH, and CKRL(TransH) on the two datasets WN18RR and FB15K237 with different noise rates (i.e., WN18RR-10%, WN18RR-20%, WN18RR-40%, FB15K237-10%, FB15K237-20%, and FB15K237-40%). (2) When the complex relations are well encoded, for example, in TransH. Our model with axiom information performs better than path information on noise detection on WN18RR- and FB15K237-based datasets. (3) With the increase of noises, the ability of baselines and our models to detect noises decreases on WN18RR-based datasets. However, it may increase on FB15K237-based datasets, which indicates that the more relations and triples in noisy datasets, the more valid information may be introduced, even though these triples may be noises.

We can thus conclude that implicit axiom information is helpful for noise detection and is better reflected on datasets with a large number of relations and triples.

Triple classification aims to judge whether a triple in the test set is correct or not, according to triple scores calculated by the energy function E ( s , r , o ) = E kge + λ · [ ( 1 − P dm ) + ( 1 − P rg ) + ( 1 − P dis ) + ( 1 − P irr ) + ( 1 − P asy ) ] , which can be viewed as a binary classification task on the test set.

Evaluation Protocol. As the test sets of the datasets used for triple classification only have correct triples, we generate negative triples by randomly corrupting the subject or object entity of correct triples. For the validation and test sets, the number of negative triples is the same as the number of positive triples. Thus there are labeled positive and negative triples in the two sets. For example, for WN18RR-based datasets, the number of triples is 6068 in the validation set and 6268 in the test set. As for triple classification, we learn a relation-specific threshold δ r for every relation. δ r is optimized by maximizing classification accuracies on the validation set. Given a triple ( s , r , o ) , if the score obtained by the energy function is below δ r , it is classified as positive, otherwise negative. We use accuracy(ACC), precision(P) and recall(R) as the evaluation metrics.

Result Analysis. Table 5 and 6 show the detailed evaluation results of triple classification. From the two tables, we can observe that: (1) Regarding the three metrics, our method outperforms baselines on the WN18RR-based datasets and achieves the best results. It confirms that learning knowledge representations with axiom information can help triple classification. (2) On the FB15K237-based dataset, the results are comparable with baselines. The improvements on WN18RR-10%, WN18RR-20% and WN18RR-40% are more evident than on FB15K237-10%, FB15K237-20% and FB15K237-40%. It demonstrates that implicit axiom information is more effective on a dataset with a smaller number of relations and triples. (3) Compared with TransE, TransH, CKRL(TransE) and CKRL(TransH), the higher the noise rate, the smaller the decrease in the accuracy metric of our method on WN18RR-10%, WN18RR-20% and WN18RR-40%. It indicates that on noisy datasets, triple classification results of NeuRAN can be more robust than baselines on small datasets.

From the triple classification results, we can conclude that the combination of implicit axiom and structural information reflected by existing triples in knowledge graphs works better than using only structural information on datasets with a small number of relations and triples. In comparison, path information is more helpful when the number of relations and triples is large.

To show that axiom information could improve the embedding learning of entities and relations and further help complete knowledge graphs, we conduct link prediction to evaluate the performance of knowledge graph completion. This task aims to predict the missing entity when given one entity and one relation of a triple, including subject entity prediction ( ? , r , o ) and object entity prediction ( s , r , ? ) .

Evaluation Protocol. For each test triple, suppose the subject entity prediction ( ? , r , o ) with the correct subject entity s. We first take all entities e ∈ E in the dataset as candidate predictions, and then replace the missing part with each entity e and calculate scores for the triples in T = { ( e , r , o ) | e ∈ G } . Subsequently, we rank these scores in ascending order, and the rank of the correct entity is stored. The object entity prediction is in the same way. The evaluation metrics are MRR and Hits@N, where MRR is the mean reciprocal rank of the ranks of all test triples, and Hits@N ( N = 1 , 3 ) is the proportion of ranks within N of all the test triples. A higher MRR and a higher Hits@1, 3 should be achieved by a good embedding model. We call this ‘raw’ setting. If we filter out the corrupted triples that exist in the training, validation or test set before ranking, the evaluation setting is called ‘filter’. In this paper, we report the evaluation results of the filter setting.

Result Analysis. Link prediction results are shown in Table 7 and 8. We analyze the results as follows: (1) The link prediction results of our method are improved compared with baselines on WN18RR-10%, WN18RR-20% and WN18RR-40% datasets, as well as on FB15K237-10%, FB15K237-20% and FB15K237-40%. It confirms that the learned knowledge graph embeddings’ quality is better and could help complete KGs. Besides, it indicates axiom information can be more useful than path information on noisy datasets. (2) On WN18RR-10%, WN18RR-20% and WN18RR-40% datasets, our method achieves the best performance on all metrics. The improvements are significant on all metrics, especially on Hits@1. It demonstrates that axiom information is of great help in improving the predictive ability of a missing triple when the dataset has fewer relations and triples. (3) On FB15K237-10%, FB15K237-20% and FB15K237-40%, although the improvements of the results are less prominent compared with WN18RR-based datasets, the results are better than baselines. It reaffirms that our method can improve link prediction, and the more relations and triples, the more information and noises brought by axiom information. Therefore, the advantages of implicit axiom information would not be as significant as in small-scale datasets.

Thus we can conclude that implicit axiom information encoded by neural axiom networks helps to improve the quality of learned embeddings of entities and relations and link prediction results. Moreover, such information is more effective on datasets with relatively few relations and triples.

We conduct ablation studies on link prediction to assess the effectiveness of NeuRAN. As our model is composed of a knowledge graph embedding module and five neural axiom modules, we add each axiom module to the knowledge graph embedding module to investigate the contributions of the axiom module. Specifically, we use the score function and loss function defined in equation (1) and (2) to train our model. TransE is taken as the knowledge graph embedding module. For evaluation, we set the score function as E ( s , r , o ) = E kge + λ · E a to illustrate the impact of each axiom module. E ( s , r , o ) is the score of the triple, and E kge is the score from the knowledge graph embedding module. E a means the score of the selected axiom and can be ( 1 − P dm ) , ( 1 − P rg ) , ( 1 − P dis ) , ( 1 − P irr ) or ( 1 − P asy ) .

From Table 9, we can observe that adding these five axioms can improve link prediction results on WN18RR-10%. The disjoint and irreflexive modules work better than other modules. Notably, the irreflexive axiom module has substantially improved on MRR and Hits@1 metrics. As the number of relations is small and most relations are asymmetric on WN18RR%, using the attention mechanism to aggregate relation representations or adding the asymmetric module has a small gain. For the results on the FB15K237-10% dataset with more relations, the improvement of the disjoint module increases compared to on WN18RR%. Although the domain, range and asymmetric modules are not as efficient as the other modules, we consider them for a comprehensive exploration.