# Graphs Regularized Robust Matrix Factorization and Its Application on Student Grade Prediction
## 4 GRMF Algorithm  
In this section, we use a majorization-minimization algorithm to solve problem (7). Suppose that we already have obtained \((\mathbf{U}_{k},\mathbf{V}_{k})\) after the \(k\)-th iterations. We split \((\mathbf{U},\mathbf{V})\) as the sum of \((\mathbf{U}_{k},\mathbf{V}_{k})\) and the unknown residue \((\Delta\mathbf{U}_{k},\Delta\mathbf{V}_{k})\):  
\[(\mathbf{U}_{k+1},\mathbf{V}_{k+1})=(\mathbf{U}_{k},\mathbf{U}_{k})+(\Delta \mathbf{U}_{k},\Delta\mathbf{V}_{k}). \tag{8}\]  
The task can now be finding the small increment \((\Delta\mathbf{U}_{k},\Delta\mathbf{V}_{k})\) in the \(k\)-th iteration such that the objective function keeps decreasing. To seek the best \((\Delta\mathbf{U}_{k},\Delta\mathbf{V}_{k})\), we employ the linearized Direction Method with Parallel Splitting and Adaptive Penalty (LADMPSAP) [45]. We made the detailed procedure of this optimization in Appendix A. We summarize the main flow of GRMF to make the paper self-contained in Algorithm 1, shown as below:```
Input:\(\mathbf{M}\in\mathbb{R}^{n\times m},\alpha\), and \(\lambda\) Output: U and \(\mathbf{V}\) Method:  Initialize \(\mathbf{U}_{0}\) and \(\mathbf{V}_{0}\) with using SVD on \(\mathbf{M}\); \(\Delta\mathbf{U}^{0}=\ \Delta\mathbf{V}^{0}=0\); \(\varepsilon_{1}=\varepsilon_{2}=1e-6\). While not converged when we arrived \((\mathbf{U}_{k},\mathbf{V}_{k})\), do  Let \(t=1\); While not converged, do  Update \(\Delta\mathbf{U}^{t}\) and \(\Delta\mathbf{V}^{t}\) via LADMPSAP; \(t=t+1\); End while \((\Delta\mathbf{U}_{k},\Delta\mathbf{V}_{k})=(\Delta\mathbf{U}_{t},\Delta \mathbf{V}_{t})\);  Update \(\mathbf{U}\) and \(\mathbf{V}\) in parallel: \(\mathbf{U}_{k+1}=\ \mathbf{U}_{k}\ +\ \Delta\mathbf{U}_{k}\); \(\mathbf{V}_{k+1}=\mathbf{V}_{k}+\Delta\mathbf{V}_{k}\);  Check the convergence coditions, if \(\mathbf{V}_{k+1}\ -\mathbf{V}_{k}<\varepsilon_{1}\) and \(\mathbf{U}_{k+1}\ -\mathbf{U}_{k}<\varepsilon_{2}\); End while. ```

**Algorithm 1** Graph regularized Robust Matrix Factorization (GRMF) by Majorization Minimization

## 5 Experimental Results

In order to evaluate the performance of GRMF, we conducted the following experiments: (1) testing GRMF, RMF-MM and on MOVIELENS 100k datasets and a public image data; (2) comparing GRMF with several fashion methods for student grade prediction, including RMF-MM [26], GRALS [29], MF [46], NMF [22], PMF [47], KNN(\(k\)-Nearest Neighbor) [48] and column mean [49] using the real educational dataset from our university. Note that MF is the standard matrix factorization solved with gradient descent; column-mean is the mean scores of historical grades of target course; and for KNN-mean, we obtained the \(k\) neighbor students and then computed the grade mean. The code and data sets are available on our website, [[https://github.com/ypzhaang/student-performance-prediction](https://github.com/ypzhaang/student-performance-prediction)]([https://github.com/ypzhaang/student-performance-prediction](https://github.com/ypzhaang/student-performance-prediction)). ### Evaluation Metric

Three metrics are used for evaluating the results: Root Mean Squared Error (RMSE), \(L_{1}\)-norm Error (Err1) [26], PSNR (Peak Signal to Noise Ratio) and Acc (Accuracy rate). Especially, in our paper, Acc is computed as follows:

\[Acc=\frac{\sum_{i=1}^{n}\Delta g_{i}}{n}, \tag{9}\]

where

\[\Delta g\ = \begin{cases}1,&|\ (g_{re}-g)\ |\geq 0.5\\ 0,&|\ (g_{re}-g)\ |<0.5^{\prime}\end{cases}  \tag{10}\]

in which \(g_{re}\) is the predicted grade while \(g\) is the true grade and \(n\) is the number of grades. ### Test on a Toy Data from Movie Dataset

MovieLens data sets were collected by the GroupLens Research Project at the University of Minnesota. These data sets consist of 100,000 ratings (1-5) from 943 users on 1682 movies, background information from users (e.g., age, occupation, and zip code) and movies (e.g., title, release date, and genre). Besides, users who have less than 20 ratings or do not have completed demographic information were removed.