# Graphs Regularized Robust Matrix Factorization and Its Application on Student Grade Prediction
## 4 GRMF Algorithm
\begin{split} H_{k}&\left(\Delta\mathbf{U},\Delta \mathbf{V}\right)\\ &\leq\parallel\mathbf{W}\odot\left(\mathbf{M}-\mathbf{U}_{k} \mathbf{V}_{k}^{T}-\Delta\mathbf{U}\mathbf{V}_{k}^{T}-\mathbf{U}_{k}\Delta \mathbf{V}^{T}\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\left(\Delta\mathbf{U},\Delta\mathbf{V}\right)+ \parallel\mathbf{W}\odot\Delta\mathbf{U}\Delta\mathbf{V}^{T}\ \parallel_{1}.\end{split}  \tag{13}\]

Besides, we can introduce the following relaxation:

\[ \begin{split}&\parallel\mathbf{W}\odot\left(\Delta\mathbf{U} \Delta\mathbf{V}^{T}\right)\ \parallel_{1}\\ &\leq\frac{1}{2}\parallel\mathbf{\Lambda}_{u}\Delta\mathbf{U} \parallel_{F}^{2}+\frac{1}{2}\parallel\mathbf{\Lambda}_{v}\Delta\mathbf{V} \parallel_{F}^{2}.\end{split}  \tag{14}\]

For simplicity, we define \(J_{k}\left(\Delta\mathbf{U},\Delta\mathbf{V}\right)\) as follows:

\[ \begin{split} J_{k}&\left(\Delta\mathbf{U},\Delta \mathbf{V}\right)\\ =&\parallel\mathbf{W}\odot\left(\mathbf{M}-\mathbf{U}_{k} \mathbf{V}_{k}^{T}-\Delta\mathbf{U}\mathbf{V}_{k}^{T}-\mathbf{U}_{k}\Delta \mathbf{V}^{T}\right)\ \parallel_{1}\\ &+\frac{\lambda}{2}\left(\parallel\mathbf{U}+\Delta\mathbf{U} \parallel_{F}^{2}+\parallel\mathbf{V}+\Delta\mathbf{V}\parallel_{F}^{2}\right) \\ &+\frac{\alpha}{2}L(\Delta\mathbf{U},\ \Delta\mathbf{V}).\end{split}  \tag{15}\]Then we have the relaxed function of \(H_{k}\left(\Delta\mathbf{U},\Delta\mathbf{V}\right)\). Our optimization problem can be recast as:

\[ \begin{split} F_{k}\left(\Delta\mathbf{U},\Delta\mathbf{V}\right)& =J_{k}\left(\Delta\mathbf{U},\Delta\mathbf{V}\right)\\ &+\frac{1}{2}\parallel\mathbf{\Lambda}_{u}\Delta\mathbf{U} \parallel_{F}^{2}+\frac{1}{2}\parallel\mathbf{\Lambda}_{v}\Delta\mathbf{V} \parallel_{F}^{2}.\end{split}  \tag{10}\]

Thus, our optimization problem (9) can be further rewritten as:

\[ \begin{split}&\min_{\mathbf{E},\Delta\mathbf{U},\Delta\mathbf{V}} \parallel\mathbf{W}\odot\mathbf{E}\parallel_{1}\\ &+(\frac{\lambda}{2}\parallel\mathbf{U}+\Delta\mathbf{U} \parallel_{F}^{2}+\frac{1}{2}\parallel\mathbf{\Lambda}_{u}\Delta\mathbf{U} \parallel_{F}^{2})\\ &+(\frac{\lambda}{2}\parallel\mathbf{V}+\Delta\mathbf{V} \parallel_{F}^{2}+\frac{1}{2}\parallel\mathbf{\Lambda}_{v}\Delta\mathbf{V} \parallel_{F}^{2})\\ &+\frac{\alpha}{2}L(\Delta\mathbf{U},\Delta\mathbf{V})\\ & s.t.\quad\mathbf{M}-\mathbf{U}_{k}\mathbf{V}_{k}^{T}=\mathbf{E }+\Delta\mathbf{U}\mathbf{V}_{k}^{T}+\mathbf{U}_{k}\Delta\mathbf{V}^{T},\end{split}  \tag{11}\]

where \(\mathbf{\Lambda}_{u},\mathbf{\Lambda}_{v}\) are diagonal matrices. We optimize the objective by the Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty (LADMPSAP) [45], as follows. ### Updating \(E\)

Fixing other variables, updating \(\mathbf{E}\) is equivalent to the following problem:

\[\min_{\mathbf{E}}\parallel\mathbf{W}\odot\mathbf{E}\parallel_{1}+\parallel \mathbf{E}-\mathbf{E}^{i}+\hat{\mathbf{Y}}^{i}/\delta_{e}^{(i)}\parallel_{F}^ {2}. \tag{12}\]

where

\[\hat{\mathbf{Y}}^{i}=\mathbf{Y}^{i}+\beta^{i}(\mathbf{E}^{i}+\Delta\mathbf{U} ^{i}\mathbf{V}_{k}^{T}+\mathbf{U}_{k}\Delta\mathbf{V}^{iT}-\mathbf{M}+\mathbf{ U}_{k}\mathbf{V}_{k}^{T}), \tag{13}\]

and \(\delta_{e}^{(i)}\ =\ \eta_{e}\beta^{i}\), \(\eta_{e}\ =\ 3L_{e}+\varepsilon\), where 3 is the number of variables which have to be updated in parallel, such as \(\mathbf{E}\), \(\Delta\mathbf{U}^{i}\), and \(\Delta\mathbf{V}^{i}\). Specially, \(L_{e}\) is the squared spectral norm of the linear mapping on \(\mathbf{E}\), which is equal to 1, and \(\varepsilon\) is a small positive scalar.