# Graphs Regularized Robust Matrix Factorization and Its Application on Student Grade Prediction
## 4 GRMF Algorithm
GRMF integrates the side information with the robust objective function of matrix factorization, which can be effectively solved by the MM optimization algorithm. The extensive experiments are conducted on movie data, image data, and our education data for testing the performance on rate prediction, image recovery, and SGP. The evaluation results by the used matrices show that GRMF can deliver a better performance than all compared methods. In SGP, GRMF can achieve the highest accuracy of about 65.4%. However, it is still weak in our challenging data. We will improve GRMF and try other fashionable methods to pursue a higher accuracy, while boosting a personalized education. In addition, a function \(f\) that maps from \(\mathbf{U}\) and \(\mathbf{V}\) to the grade matrix \(\mathbf{G}\) could be used to achieve a better prediction model, due to the gap between the predicted grade and the real grade. That is because the noise is often caused by accidental events, like exam slipping and guessing. Our study has this limitation on considering this noise in grade prediction. Adding this map \(f\) may help to obtain more accurate results in the real-world environment. We leave this study for future work. Conceptualization, Y.Z. and Y.Y. ; Data curation, Y.Y. and J.C.; Formal analysis, Y.Z. ; Funding acquisition, Y.Z. and X.S. ; Investigation, Y.Y. and H.D. ; Methodology, Y.Z. ; Resources, J.C. and X.S. ; Software, Y.Z. and Y.Y. ; Supervision, X.S. ; Validation, H.D. and J.C.; Writing--original draft, Y.Z. and Y.Y. ; Writing--review & editing, H.D., J.C. and X.S. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Grants No. 61802313, 61772426, U1811262) and the Fundamental Research Funds for Central Universities (Grant No. G2018 KY0301). We thank the editors and any reviewers for their helpful comments. The authors declare no conflict of interest. Figure 8: The effects of the parameters of GRMF on the Err1 and Acc. ## Appendix A Objective Minimization

Suppose that we already have obtained \(\left(\mathbf{U}_{k},\mathbf{V}_{k}\right)\) after the \(k\)th iterations. We split \(\left(\mathbf{U},\mathbf{V}\right)\) as the sum of \(\left(\mathbf{U}_{k},\mathbf{U}_{k}\right)\) and the unknown residue \(\left(\Delta\mathbf{U},\Delta\mathbf{V}\right)\). \[\left(\mathbf{U},\mathbf{V}\right)=\left(\mathbf{U}_{k},\mathbf{U}_{k}\right) +\left(\Delta\mathbf{U},\Delta\mathbf{V}\right) \tag{10}\]

In a similar way, the graph regularization of (6) can be rewritten as follows:

\[ \begin{split} L(\Delta\mathbf{U},\ \Delta\mathbf{V}\ )&=tr\left(\left(\mathbf{U}^{T}+ \Delta\mathbf{U}^{T}\right)\mathbf{H}_{u}\left(\mathbf{U}+\Delta\mathbf{U} \right)\right)\\ tr&\left(\left(\mathbf{V}^{T}\ \Delta\mathbf{V}^{T} \right)\mathbf{H}_{v}\left(\mathbf{V}+\Delta\mathbf{V}\right)\right)\end{split}  \tag{11}\]

With (7) and (8), our task is to minimize the following:

\[ \begin{split}\min_{\Delta\mathbf{U},\Delta\mathbf{V}}H_{k}\left( \Delta\mathbf{U},\Delta\mathbf{V}\right)=\\ \min_{\Delta\mathbf{U},\Delta\mathbf{V}}&\parallel \mathbf{W}\odot\left(\mathbf{M}-\left(\mathbf{U}_{k}+\Delta\mathbf{U}\right) \left(\mathbf{V}_{k}^{T}+\Delta\mathbf{V}_{T}\right)\right)\ \parallel_{1}\\ &+\frac{\lambda}{2}\left(\parallel\mathbf{U}+\Delta\mathbf{U} \parallel_{F}^{2}+\parallel\mathbf{V}+\Delta\mathbf{V}\parallel_{F}^{2}\right) \\ &+\frac{\alpha}{2}L(\Delta\mathbf{U},\ \Delta\mathbf{V}\ )\end{split}  \tag{12}\]

Now our task is to find a small increment \(\left(\Delta\mathbf{U},\ \Delta\mathbf{V}\right)\) such that the objective function keeps decreasing. Inspired by [26], we try to relax (9) to a convex surrogate. By using the triangular inequality of norms, we arrive at the following inequality:

\[