## 1 Introduction  
Knowledge graph (KG) embeddings are low-dimensional representations for entities and relations. This approach can benefit a range of downstream tasks, such as semantic parsing [1, 2], knowledge reasoning [3], and question answering [4, 5]. Embeddings are supposed to contain semantic information and should be able to deal with multiple linguistic relations. At present, research on knowledge graph embedding occurs mainly along three main lines. One of these lines of research includes studies based on translation. TransE [6] was the first model to introduce translation-based embedding, which represents entities and relationships in the same space, and regards the relationship vector r as the translation between the head entity vector h and the tail entity vector t, that is, h + r \(\approx\) t. Since transE cannot handle one-to-many, many-to-one, and many-to-many relationships (1-to-N, N-to-1, N-to-N), TransH [7] is proposed to enable an entity to have different representations when involved in various relations. In the TransR model [8], an entity is a complex of multiple attributes, and different relationships focus on different attributes of the entity. Another line of research includes studies based on semantic matching. RESCAL [9] obtains its latent semantics by using a vector to represent each entity. Each relationship is represented as a matrix that is used to model the interaction of potential relationships. It defines the scoring function of the triple (h, r, t) as a bilinear function. DistMult [10] simplifies RESCAL by restricting the relationship matrix to a diagonal matrix, which greatly improves training efficiency. ComplEx [11] extends DistMult by introducing complex number domain embedding to better model asymmetric relationships. In ComplEx, the embedding of entities and relationships no longer exists in real space, but in complex space. The third line of research includes studies based on graph convolutional neural networks. ConvE [12] employs a multi-layer convolutional network, which enables expressive feature learning, while remaining highly parameter-efficient. Unlike previous works, which focused on shallow, fast models that can scale to large knowledge graphs, ConvE uses 2D convolution over embeddings and multiple layers of nonlinear features to model KGs. Subsequently, the ConvKB [13] model has been used to explore the global relationships among same-dimensional entries of the entity and relation embeddings. However, neither of them models the interactions between various positions of entities and relations. R-GCN [14] is another convolutional network designed for KBs, generalized from GCN [15] for uni-relational data. A typical KG embedding technique has two necessary elements: (i) an encoder to generate KG embeddings and (ii) a scoring function to measure plausibility for each fact. Entities are usually represented as vectors in low-dimensional space, whereas relations are represented as an operation between entities, resulting in vectors for translational operations [6] or matrices for linear transformation [16]. By doing so, the embedding of KG components can be used to enhance the performance in many downstream tasks. Despite the success those previous algorithms have achieved, we note that those methods have the following defects(1) the \(n\)-dimensional representation of the KG component can be regarded as a single point, neglecting the uncertainties for entities and relations; (2) they represent entities as vectors located in low-dimensional space and relations as an operation between entities [6; 16], thus ignoring the affinities between entities and relations as they are embedded in different semantic spaces. To address the issues mentioned above, we propose a co-embedding model for KG, learning low-dimensional representations for entities and relations in the same semantic space so that the affinities between them can be effectively captured. Moreover, we introduce a variational auto-encoder to infer the representations of KG components as Gaussian distributions.