# Graphs Regularized Robust Matrix Factorization and Its Application on Student Grade Prediction
## 4 GRMF Algorithm
We ran the three methods to recover the corrupted image in Figure 3(b), where the side-information data consists of the rows and columns of the corrupted image. Figure 5 shows the PNSRs (Peak Signal to Noise Ratios) from the three compared methods. From the curves, GRMF consistently achieves the highest PSNRs on all test cases. When the corruption ratio increases, GRMF delivers a much better result than RMF-MM. Note that GRALS has a weak performance because its reconstruction term is very sensitive to data pollutions. Figure 3(c)-e depicts the resultant images from the case of \(g=80\), using the three methods. As is clear, our GRMF produces the best visualization, while the other methods suffer a few horizontal or vertical lines. Figure 3: The effects of the parameters of GRMF on the Err1. #### 5.3.2 Salt-And-Pepper Noise

We added the salt-and-pepper noise with noise density varying from 0.05 to 0.65 with a step of 0.05 to image and obtained the corrupted image, like Figure 5(b). Then, we ran the three compared methods for denoising the corrupted image where the side-information consists of the rows and the columns of the corrupted image. Figure 7 shows the results of image denoising by GRALS, RMF-MM, and GRMF. From Figure 7 it is clear that GRMF delivers the best performance on PSNR when the noise density is less than 0.4 but drops down if the noise density is greater than 0.5, where the other two methods obtained worse results. The reason is that most pixels of the image are corrupted so that the graphs are difficult to obtain well in a noisy situation. In addition, Figure 6 shows the resultant images when the noise density is 0.4, where our method touches the highest PNSR of 22.88 dB. ### Application on Educational Data Set

The data were collected from the school of Computer Science, Northwestern Polytechnical University (NPU), across students who joined in the past five years, that is, from 2013 to 2017. We collected all the score/grade recorders before the fall of 2017, together with the side information. Figure 4: The PNSRs of Image recovery with Gaussian noise. Figure 5: Evaluation of image recovery with Gaussian noise in term of PSNR. The black line that marked with “Corrupted” means the PNSR of the corrupted images. More specifically, our dataset contains the grades, the side data of student and the side data of courses, respectively denoted by NPU-G, NPU-S, and NPU-C for short. NPU-G is composed of 1325 \(\times\) 832 grade records from 1325 students at 832 courses. NPU-S contains 25 description features of 1325 students, such as ages, gender, and department. NPU-C includes 18 description features of 882 academic/elective courses, such as hours, type, and course credit. In addition, at least 15 students enrolled and obtained grades in each course, and students starting university in 2013 and 2014 have already completed their program. SGP in our educational data set has the following challenges: (1) Data sparsity. There are 832 courses in NPU-G, but each student is only required to enroll in a small number of courses, i.e., about 85 courses in our data. (2) data corruption. Many subjective factors affect the final grade, e.g., subjective questions. (3) missing data. A few students do not attend the final exam, and thus give an empty grade in the information system. All this noisy information makes our problem very challenging. #### 5.4.1 Educational Data Preprocessing

For NPU-G, we removed the students who had lost most of the data records or had taken less than 4 courses, and then removed those courses that were taken by less than 15 students, followed by deleting the secondary courses to ensure a single record per student. Finally, we formulated the

Figure 6: The Results of Image recovery with salt-and-pepper noise. Figure 7: Evaluation of image recovery with Salt-and-pepper noise. remaining records from 882 students and 82 courses into the matrix \(\mathbf{M}\), ordered by scholar terms. In addition, we transformed the scores into grade 1-6 using the following piecewise function:

\[y\ =\