# Multi-View Network Representation Learning Algorithm Research
## 3 Our Method
### Joint Learning of MVNR  
Suppose that matrix \(M\in\mathbb{R}^{m\times n}\) admits an approximation of low rank \(k\), where \(k\ll\{m,n\}\). Based on the matrix factorization, [PERSON] et al. [33] propose a matrix factorization approach with a penalty term constraint, which aims to find \(X\in\mathbb{R}^{k\times n}\) and \(Y\in\mathbb{R}^{k\times m}\) and minimize the likelihood  
\[\underset{W,H}{\min}\|M-(X^{T}Y)\|_{F}^{2}+\frac{\alpha}{2}(\|X\|_{F}^{2}+\|Y \|_{F}^{2}), \tag{9}\]where \(\alpha\) is a harmonic factor to balance two components in Equation (9). Specifically, Equation (9) aims at factorizing \(M\in\mathbb{R}^{m\times n}\) into two matrices \(X\in\mathbb{R}^{k\times m}\) and \(Y\in\mathbb{R}^{k\times n}\), where \(M\approx XY\). Here, matrix \(M\in\mathbb{R}^{m\times n}\) can be regarded as the feature matrix of the network \(G\) for the task of network representation learning. Matrix \(X^{T}\in\mathbb{R}^{n\times k}\) can be regarded as the learnt representation matrix of the network \(G\). We do not use the model proposed by Equation (9) in MVNR algorithm.  
We use the Inductive Matrix Completion (IMC) algorithm presented by [PERSON] and [PERSON] [34], which adopts two known matrices to factorize matrix \(M\) as follows:  
\[\underset{W,H}{\text{min}}\underset{(i,j)\in\Omega}{\sum}\ (M_{ij}-(P^{T}X^{T}YQ)_{ij})^{2}+\frac{ \beta}{2}(\parallel X\parallel_{F}^{2}+\parallel Y\parallel_{F}^{2}), \tag{10}\]  
where \(\Omega\) denotes the sample set of matrix \(M\in\mathbb{R}^{m\times n}\), here, \(P\in\mathbb{R}^{p\times m}\) and \(Q\in\mathbb{R}^{q\times n}\) are two known feature matrices. \(\beta\) is the hyper-parameter to balance two components \(M_{ij}-(P^{T}X^{T}YQ)_{ij}\) and \(\parallel X\parallel_{F}^{2}+\parallel Y\parallel_{F}^{2}\). IMC aim at learning the matrix \(X\in\mathbb{R}^{d\times p}\) and matrix \(Y\in\mathbb{R}^{d\times q}\) to meet \(M\approx P^{T}X^{T}YQ\). The Equation (10) is originally applied to complete gene-disease matrix using gene and disease feature matrix. Motivated by IMC, we integrate the link weights of existing edges into Equation (10), the purpose of MVNR is to solve matrices \(X\) and \(Y\) to minimize the objective function as follows:  
\[\underset{W,H}{\text{min}}\|M^{(k)}-((X^{(k)})^{T}Y^{(k)}W^{(k)})\|_{F}^{2}+ \frac{\lambda}{2}(\left\|X^{(k)}\right\|_{F}^{2}+\left\|Y^{(k)}\right\|_{F}^{2}). \tag{11}\]  
Here, \(W^{(k)}\in\mathbb{R}^{q\times n}\) is the weight matrix of the \(k\)-th step network, \(\lambda\) is the hyper-parameter to balance two components \(M^{(k)}-((X^{(k)})^{T}Y^{(k)}W^{(k)})\) and \(\left\|X^{(k)}\right\|_{F}^{2}+\left\|Y^{(k)}\right\|_{F}^{2}\). For the Equation (10), we denote the matrix \(P\in\mathbb{R}^{p\times m}\) as the identity matrix \(E\in\mathbb{R}^{p\times m}\). Consequently, we get the Equation (11) based on Equation (10). For each \(k\)-step network, we denote \(R^{(k)}=(X^{(k)})^{T}\) as representation matrix of the \(k\)-th network. The final representation vectors can be concatenated as follows:  
\[R=R^{(1)}\oplus R^{(2)}\oplus\cdots\oplus R^{(k)}, \tag{12}\]  
where \(k\) = {1,2,..., \(K\)} and \(K\) is a pre-selected constant. As Equation (12), we concatenate all \(k\)-step representations to form the global representations, which can be used in various machine learning tasks.  
We introduce matrix \(W^{(k)}\) to adjust the weights of the different \(k\)-step networks and remedy the defect of ignoring the weight information for GraRep. For the \(k\)-step network representation \(r^{(k)}\), we apply L2 Norm to normalize the learnt network representations which can show better performance in some evaluation tasks, such as network classification and link prediction.  
We give the algorithm description of MVNR in Algorithm 1.