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$$\mathcal{L}=\sum_{(h,r,t) \in S} \sum_{(h^{'},r^{'},t^{'}) \in S^{'}_{(h,r,t)}} \[ \gamma + d(h + r, t) - d(h^{'} + r^{'}, t^{'}) \]_{+} \\ S^{'}_{(h,r,t)}=\{(h^{'},r,t)|h^{'} \in E\} \cup \{(h,r,t^{'})|t^{'} \in E\} \\ d(h + r, t)=||h + r - t||^{2}_{2}$$Latex Code
\mathcal{L}=\sum_{(h,r,t) \in S} \sum_{(h^{'},r^{'},t^{'}) \in S^{'}_{(h,r,t)}} \[ \gamma + d(h + r, t) - d(h^{'} + r^{'}, t^{'}) \]_{+} \\ S^{'}_{(h,r,t)}=\{(h^{'},r,t)|h^{'} \in E\} \cup \{(h,r,t^{'})|t^{'} \in E\} \\ d(h + r, t)=||h + r - t||^{2}_{2}
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Introduction
Equation
Latex Code
\mathcal{L}=\sum_{(h,r,t) \in S} \sum_{(h^{'},r^{'},t^{'}) \in S^{'}_{(h,r,t)}} \[ \gamma + d(h + r, t) - d(h^{'} + r^{'}, t^{'}) \]_{+} \\ S^{'}_{(h,r,t)}=\{(h^{'},r,t)|h^{'} \in E\} \cup \{(h,r,t^{'})|t^{'} \in E\} \\ d(h + r, t)=||h + r - t||^{2}_{2}
Explanation
Given a training set S of triplets (h, l, t) composed of two entities h, t ? E (the set of entities) and a relationship l ? L (the set of relationships), our model learns vector embeddings of the entities and the relationships. The embeddings take values in Rk (k is a model hyperparameter) and are denoted with the same letters, in boldface characters. The basic idea behind our model is that the functional relation induced by the l-labeled edges corresponds to a translation of the embeddings, i.e. we want that h + l ? t when (h, l, t) holds (t should be a nearest neighbor of h + l), while h + l should be far away from t otherwise. Following an energy-based framework, the energy of a triplet is equal to d(h + l, t) for some dissimilarity measure d, which we take to be either the L1 or the L2 -norm. To learn such embeddings, we minimize a margin-based ranking criterion over the training set. See paper Translating Embeddings for Modeling Multi-relational Data for more details.
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