Knowledge Graphs (KG), composed of entities and relations, provide a structured representation of knowledge. For easy access to statistical approaches on relational data, multiple methods to embed a KG as components of R^d have been introduced. We propose TransINT, a novel and interpretable KG embedding method that isomorphically preserves the implication ordering among relations in the embedding space. TransINT maps set of entities (tied by a relation) to continuous sets of vectors that are inclusion-ordered isomorphically to relation implications. With a novel parameter sharing scheme, TransINT enables automatic training on missing but implied facts without rule grounding. We achieve new state-of-the-art performances with signficant margins in Link Prediction and Triple Classification on FB122 dataset, with boosted performance even on test instances that cannot be inferred by logical rules. The angles between the continuous sets embedded by TransINT provide an interpretable way to mine semantic relatedness and implication rules among relations. Recently, learning distributed vector representations of multi-relational knowledge has become an active area of research (Bordes et al.; Nickel et al.; Kazemi & Poole; Wang et al.; Bordes et al.) . These methods map components of a KG (entities and relations) to elements of R d and capture statistical patterns, regarding vectors close in distance as representing similar concepts. However, they lack common sense knowledge which are essential for reasoning (Wang et al.; Guo et al.; Nickel & Kiela) . For example, "parent" and "father" would be deemed similar by KG embeddings, but by common sense, "parent ⇒ father" yet not the other way around. Thus, one focus of current research is to bring common sense rules to KG embeddings (Guo et al.; Wang et al.; Wei et al.( . Some methods impose hard geometric constraints and embed asymmetric orderings of knowledge (Nickel & Kiela; Vendrov et al.; Vilnis et al.( . However , they only embed hierarchy (unary Is_a relations), and cannot embed n-ary relations in KG's. Moreover , their hierarchy learning is largely incompatible with conventional relational learning, because they put hard constraints on distance to represent partial ordering, which is a common metric of similarity/ relatedness in relational learning. We propose TransINT, a new KG embedding method that isomorphically preserves the implication ordering among relations in the embedding space. TransINT restrict entities tied by a relation to be embedded to vectors in a particular region of R d included isomorphically to the order of relation implication. For example, we map any entities tied by is_father_of to vectors in a region that is part of the region for is_parent_of; thus, we can automatically know that if John is a father of Tom, he is also his parent even if such a fact is missing in the KG. Such embeddings are constructed by sharing and rank-ordering the basis of the linear subspaces where the vectors are required to belong. Mathematically, a relation can be viewed as sets of entities tied by a constraint (Stoll) . We take such a view on KG's, since it gives consistancy and interpretability to model behavior. Furthermore, for the first time in KG embedding, we map sets of entitites under relation constraint to a continuous set of points (whose elements are entity vectors) -which learns relationships among not only individual entity vectors but also sets of entities. We show that angles between embedded relation sets can identify semantic patterns and implication rules -an extension of the line of thought as in word/ image embedding methods such as Mikolov et al., Frome et al. to relational embedding. Such mining is both limited and less interpretable if embedded sets are discrete (Vilnis et al.; Vendrov et al.) or each entitity itself is embedded to a region, not a member vector of it (Vilnis et al.) . 1 TransINT's such interpretable meta-learning opens up possibilities for explainable reasoning in applications such as recommender systems (Ma et al.) and question answering (Hamilton et al. We presented TransINT, a new KG embedding method that embed sets of entities (tied by relations) to continuous sets in R d that are inclusion-ordered isomorphically to relation implications. Our method achieved new state-of-the-art performances with signficant margins in Link Prediction and Triple Classification on the FB122 dataset, with boosted performance even on test instances that are not affected by rules. We further propose and interpretable criterion for mining semantic similairty among sets of entities with TransINT.
We propose TransINT, a novel and interpretable KG embedding method that isomorphically preserves the implication ordering among relations in the embedding space in an explainable, robust, and geometrically coherent way.
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