## 3 Methodology
### Ensemble Model  
In the first part of the model, we use householder transformation to realize projections and rotations. The householder transformation is an orthogonal transformation that can transform an n-dimensional vector \(x\) to any n-dimensional vector \(\hat{x}\). From a geometric perspective, the householder transformation reflects \(x\) relative to \(\hat{x}\) via the hyperplane between \(x\) and \(\hat{x}\). Assuming that the source vector is \(x\), given the unit vector \(e\) and identity matrix \(E\), a new vector, \(\hat{x}\), can be obtained using householder matrix \(Matrix(e)\), where \(Matrix(e)=E-2 ee^{T}\). The matrix \(Matrix(e)\) takes \(e\) as a variable. The basic transformation of the householder transformation is described as follows:  
\[\hat{x}=Matrix(e)x=x-2<x,e>e, \tag{1}\]  
where \(<x,e>\) represents the dot product of \(x\) and \(e\). Many research studies have been conducted with respect to entity projection, and entity projections allow entities to produce relationship-specific representations in order to solve complex relationship-mapping properties. We use householder transformations to realize householder projections. The householder projection can be realized by fine-tuning the householder matrix \(Matrix(e)\). We add a scalar, \(\omega\), to control the distance amplitude of the projection. The projection can better change the relative distance between two points and ensure that it produces a robust relationship-specific representation. The projection matrix is modified to \(PMatrix(e,\omega)=E-\omega ee^{T}\). We denote a head entity in the first part as \(h_{1}\), \(h_{1}\in\mathbb{R}^{k\times m}\), where \(k\) represents the embedding size and \(m\) represents the dimension of each row vector,  
Figure 1: An illustration of the knowledge graph completion process. while \(t_{1}\) represents a tail entity in the first part, \(t_{1}\in\mathbb{R}^{k\times m}\). For each relationship, two types of parameters are used to represent the projection of the relationship, namely, \(P\in\mathbb{R}^{k\times c\times m}\) and \(W\in\mathbb{R}^{k\times c}\). Each row \(P[i]\in\mathbb{R}^{c\times m}\) comprises \(c\) m-dimensional unit vectors. Each row of \(W\) consists of \(c\) real numbers. Each dimension of the projected head entity \(h_{project}\) and tail entity \(t_{project}\) can be calculated in the following form:  
\[\left\{ \begin{array}{l}h_{project}[i]=Project(h_{1}[i],P_{1}[i],W_{1}[i])= \prod_{j=1}^{c}PMatrix(P_{1}[i][j],W_{1}[i][j])h_{1}[i]\\ t_{project}[i]=Project(t_{1}[i],P_{2}[i],W_{2}[i])=\prod_{j=1}^{c}PMatrix(P_{2 }[i][j],W_{2}[i][j])t_{1}[i],\end{array} \right. \tag{2}\]  
where \(1\leq i\leq k\) and \(1\leq j\leq c\). Based on Formula (1), the rotation matrix is set to \(RMatrix(e)=E-2 ee^{T}\). The householder rotation operation can be performed on the source vector. [PERSON] et al. (2018) proved that the composition of \(2\lfloor m/2\rfloor\) householder reflections can realize any m-dimensional transformation; thus, we denote a relation in the first part as \(r_{1}\), \(r_{1}\in\mathbb{R}^{k\times 2n\times m}\), where \(n=\lfloor m/2\rfloor\) and each row of \(r_{1}\) consists of a series of m-dimensional unit vectors. Formula (3) shows the rotation of each row vector \(h_{project}[i]\):  
\[h_{rotate}[i]=Rotate(h_{project}[i],r_{1}[i])=\prod_{j=1}^{2n}RMatrix(r_{1}[i ][j])h_{project}[i], \tag{3}\]  
where \(1\leq i\leq k\) and \(1\leq j\leq 2n\), while \(h_{rotate}\) represents the rotation of the projected entity \(h_{project}\). Relation-specific householder rotations are applied to each row of the projected head embedding. There is an expectation that the rotated result \(h_{rotate}\) should be close to the projected tail embedding \(t_{project}\); thus, \(dist_{1}\) can be calculated as follows:  
\[dist_{1}=\sum_{i=1}^{k}\|h_{rotate}[i]-t_{project}[i]\|_{2}. \tag{4}\]  
In addition to using householder reflections, we achieve projection and rotation operations in a complex vector space.