# Graphs Regularized Robust Matrix Factorization and Its Application on Student Grade Prediction
## 2 Related Works
### Matrix Factorization  
Letting \(\mathbf{M}\in\mathbb{R}^{m\times n}\) be the given matrix, MF aims to seek two latent feature matrices \(\mathbf{U}\in\mathbb{R}^{m\times k}\) and \(\mathbf{V}\in\mathbb{R}^{m\times k}\) to approximate \(\mathbf{M}\). The traditional MF can be written as:  
\[\min_{\mathbf{U},\mathbf{V}}||\mathbf{M}\ -\ \mathbf{U}\mathbf{V}^{T}||_{F}, \tag{1}\]  
where \(k\) is the number of latent features predefined in \(\mathbf{U}\) and \(\mathbf{V}\), and \(||\cdot||_{F}\) is the Frobenius norm. Optimization problem (1) can be solved by various algorithms, such as Majorization Minimization (MM) [35], alternating the direction of the method of multipliers (ADMM) [36], simulated annealing (SA) [37]. Besides, many variants of MF have been proposed, including LRMF [25], NMF [22] and TF [24].  
To enhance the robustness, robust matrix factorization via majorization minimization (RMF-MM) employs \(L_{1}\)-norm instead of \(L_{F}\)-norm as the reconstruction term [26]. The objective problem of RMF-MM is:  
\[\min_{\mathbf{U},\mathbf{V}}||\mathbf{W}\odot\left(\mathbf{M}-\mathbf{U} \mathbf{V}^{T}\right)||_{1}\ +\ \frac{\lambda}{2}||\mathbf{U}||_{F}^{2}\ +\ \frac{\lambda}{2}||\mathbf{V}||_{F}^{2}\, \tag{2}\]  
where \(||\cdot||_{1}\) is \(L_{1}\)-norm of matrix, \(\lambda>0\) is a regularization parameter and \(\mathbf{W}\) is defined as follows:  
\[\mathbf{W}_{ij}=\begin{cases}0,&\text{the value of $\mathbf{M}_{ij}$ is missing}\\ 1,&\text{otherwise}.\end{cases} \tag{3}\]  
The problem above can be effectively optimized by MM algorithm. The results in the experiments by [PERSON] et al. show that RMF-MM is robust to high missing rate or severe data corruption [26].  
Since RMF-MM can effectively learn the features from noisy data and then uses the features for prediction, we reformulate the Problem 1 for employing this novel technique, as follows:  
**Problem 2. (SGP-MF):** Given a student grade matrix \(\mathbf{M}\), SGP-MF aims to extract \(\mathbf{U}\) for students and \(\mathbf{V}\) for courses such that \(\mathbf{M}=\mathbf{U}\mathbf{V}\). Then the target grade is predicted by  
\[g(s,c)=\mathbf{M}_{s,c}=\mathbf{u}_{s}^{T}\mathbf{v}_{c}, \tag{4}\]  
where \(g(s,c)\) is the grade of student \(s\) on course \(c\), \(\mathbf{u}_{s}\) is the s-th row of \(\mathbf{U}\) and \(\mathbf{v}_{c}\) is the c-th row of \(\mathbf{V}\). And \(\mathbf{M}_{s,c}\) is the element of \(s\)-th row, \(c\)-th column of matrix \(\mathbf{M}\).  
The reason we consider the Formula (4) is the fact that a student enrolls on a course and obtains a grade. This fact motivates us to obtain the student's features and course's features, given the grade matrix. In this paper, we consider this problem using the matrix factorization (MF) method. As in Formula (4), each grade \(M_{s,c}\) is made by \(\mathbf{u}_{s}^{T}\mathbf{v}_{c}\) to obtain the latent features  
However, RMF-MM fails to consider the side-information data that is often available. The method of graph matrix factorization (GMF) is an approach to integrate the neighborhood structure of \(\mathbf{M}\), but it does not work for matrix completion [38]. Based on GMF, we here solve the SGP by combining two side information graphs with RMF-MM.