# Graphs Regularized Robust Matrix Factorization and Its Application on Student Grade Prediction
## 4 GRMF Algorithm
In this test experiment, we draw out a toy data set from MovieLens to probe the effectiveness, convergence, and parameter effects of GRMF. And in the toy data set, the user ids are less than 200, and the movie ids are less than 300. #### 5.2.1 Rating Prediction and Algorithm Convergence

We divided the toy data set into a training set and test set by random sampling. To evaluate the small toy data, we employed a five-fold cross validation that trains models on four-fold samples and tests on the remaining samples. Whereby we constructed two five-nearest neighborhood graphs from the background data of both users and movies. We chose suitable parameters for achieving best performance using all the mentioned methods. Note that the optimal parameters of GRALS were selected in Reference [29]. Table 1 shows the prediction results from using four methods on the toy data set. It is easy to observe that: (1) MF is better than RMF-MM and GRALS in terms of RMSE, but worse than the two latter compared to Err1. (2) RMF-MM has better performance on Err1 than GRALS, which is more robust to evaluate. (3) Overall, our method delivers the best results using either RMSE or Err1. All the above says that GRMF can benefit from the side information data to enhance rate prediction performance. In addition, Figure 2 displays the convergence proceeding of GRMF on the toy data. As is shown, GRMF can converge to stable Err1 after about 16 iterations. With more observations on other data sets, Algorithm 1 can have a fast convergence and arrive at an effective solution. #### 5.2.2 The Effects of Parameters on Rating Prediction

We have the graph regularization parameter \(\alpha\), the regularization parameter \(\lambda\) and the rank of factorized matrices \(k\) in the objective (7) of GRMF. We here discuss the effects of these three parameters on the prediction performance utilizing the above toy data set on our prediction experiment. \begin{table}
\begin{tabular}{l l l} \hline \hline  & **Err1** & **RMSE** \\ \hline GRMF & **0.735** & **0.957** \\ MF & 1.549 & 1.007 \\ GRALS & 0.770 & 1.008 \\ RMF-MM & 0.776 & 1.104 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Err1 and RMSE on toy dataset. Figure 2: The value of Err1 versus iterations of Graph Regularized Robust Matrix Factorization (GRMF) on toy dataset. The two parameters of \(\alpha\) and \(\lambda\) vary in wide ranges as is shown in Figure 2(b). Figure 2(b) shows the 3D curve of Err1 created under the effects of \(\alpha\) and \(\lambda\). From the curve, we observe that there is a broad range of parameter pairs that can be available for producing decent prediction results. Besides, we also probe the effect of the parameter \(k\), shown in Figure 2(a). The results show that GRMF has the most stable performance under varying \(k\), while the RMF-MM has the worst performance. ### Evaluation on Image Data Set

The problem of image recovery is often formulated as matrix completion. Since the top singular values dominate the main information, most of the images could be regarded as a low-rank matrix. Hence, we apply the proposed method to recover the image from its noisy version. This test aims to recast the experiment conducted in the work of [PERSON] et al. (2016). Concretely, we pollute the images ([[https://sites.google.com/site/zjuyaohu/](https://sites.google.com/site/zjuyaohu/)]([https://sites.google.com/site/zjuyaohu/](https://sites.google.com/site/zjuyaohu/))) with Gaussian noise or salt-and-pepper noise, then recover the images from the noisy version in comparison with the methods of RMF-MM and GRALS. #### 5.3.1 Gaussian Noise

We added Gaussian noise with the variance being 1 and mean being 0 to \(g\) percent of the observed pixels, where \(g\) is the corruption ratio. Figure 3(b) shows the example image which was corrupted with Gaussian noise. \(g\) was varied in the range of (45; 90) to observe the performance in various situations.