# Graphs Regularized Robust Matrix Factorization and Its Application on Student Grade Prediction
## 4 GRMF Algorithm
Then we update \(\mathbf{E}\) by:

\[\mathbf{E}^{i+1}\ =\ \mathbf{W}\odot\mathbf{S}_{\sigma_{e}(i)}(\mathbf{E}^{i}- \hat{\mathbf{Y}}^{i}/\delta_{e}^{(i)})+\hat{\mathbf{w}}\odot(\mathbf{E}^{i}- \hat{\mathbf{Y}}^{i}/\delta_{e}^{(i)}), \tag{14}\]

where \(\mathbf{S}\) is the shrinkage operator [50]:

\[\mathbf{S}_{\varepsilon}\left(x\right)=max(|x|-\varepsilon,0)sgn(x), \tag{15}\]

where \(\hat{\mathbf{w}}\) is the complement of \(\mathbf{W}\). ### Updating \(\mathbf{\Delta U}\)

Updating \(\mathbf{\Delta U}\) is to solve the following problem:

\[ \begin{split}&\min_{\Delta\mathbf{U}}\frac{\lambda}{2}\parallel \mathbf{U}_{k}+\mathbf{\Delta U}\parallel_{F}^{2}\\ &+\frac{\alpha}{2}tr((\mathbf{U}_{k}^{T}+\mathbf{\Delta U}^{T}) \mathbf{H}_{u}(\mathbf{U}_{k}+\mathbf{\Delta U}))+\frac{1}{2}\parallel \mathbf{\Lambda}_{u}\Delta\mathbf{U}\parallel_{F}^{2})\\ &+\frac{\delta_{u}^{(i)}}{2}\parallel\mathbf{\Delta U}-\mathbf{ \Delta U}^{i}+\hat{\mathbf{Y}}^{i}\mathbf{v}_{k}/\delta_{u}^{(i)}\parallel_{F} ^{2}\end{split}  \tag{16}\]where \(\delta_{u}^{(i)}\ =\ \eta_{u}\beta^{i}\) and \(\eta_{u}\ =\ 3\ \|\ \mathbf{V}_{k}\ \|_{2}^{2}\ +\varepsilon\). Since all terms in (A13) is convex, (A13) is a convex problem and its closed solution can be obtained by:

\[ \begin{split}\Delta\mathbf{U}^{i+1}\ =\\ (\lambda\mathbf{I}_{m}+\alpha\mathbf{H}_{u}+\mathbf{\Lambda}_{u}^{T} \mathbf{\Lambda}_{u}+\delta_{u}^{(i)}\mathbf{I}_{m})^{-1}\\ (-\lambda\mathbf{U}_{k}-\alpha\mathbf{U}_{k}\mathbf{H}_{u}+ \delta_{u}^{(i)}\Delta\mathbf{U}^{i}-\delta_{u}^{(i)}\hat{\mathbf{Y}}^{i} \mathbf{V}_{k}/\delta_{u}^{(i)}),\end{split} \] (A14)

where \(m\) can be found in the paper [26]. ### Updating \(\mathbf{\Delta}\mathbf{V}\)

Similar to \(\mathbf{\Delta}\mathbf{U}\), updating \(\mathbf{\Delta}\mathbf{V}\) can be achieved by:

\[ \begin{split}\Delta\mathbf{V}^{i+1}=\\ (\lambda\mathbf{I}_{m}+\alpha\mathbf{H}_{v}+\mathbf{\Lambda}_{v}^{T} \mathbf{\Lambda}_{v}+\delta_{v}^{(i)}\mathbf{I}_{m})^{-1}\\ (-\lambda\mathbf{V}_{k}-\alpha\mathbf{V}_{k}\mathbf{H}_{v}+\ \delta_{v}^{(i)}\mathbf{\Delta} \mathbf{V}^{i}-\delta_{v}^{(i)}\hat{\mathbf{Y}}^{i}\mathbf{U}_{k}/\delta_{v}^ {(i)}).\end{split} \] (A15)

### Updating \(\mathbf{Y}\) and \(\beta\)

We update \(\mathbf{Y}\) and \(\beta\) as follows:

\[ \begin{split}\mathbf{Y}^{i+1}&=\mathbf{Y}^{i}+\beta ^{i}(\mathbf{E}^{i+1}+\Delta\mathbf{U}^{i+1}\mathbf{V}_{k}^{T}\\ &+\mathbf{U}_{k}\Delta\mathbf{V}^{(i+1)T}\mathbf{U}_{k}\mathbf{V }_{k}^{T}-\mathbf{M}),\end{split} \] (A16)

\[ \begin{split}\beta^{i+1}=min(\beta^{max},\rho\beta^{i}),\end{split} \] (A17)

where \(\rho\) is defined by:

\[\rho\ = \begin{cases}\rho_{0},&\text{if }\mathbf{Q}<\varepsilon_{1}\\ 1,&\text{otherwise},\end{cases} \] (A18)

and

\[ \begin{split}\mathbf{Q}=\beta^{i}\ max(\sqrt{\eta_{e}}||\mathbf{ E}^{i+1}\ -\ \mathbf{E}^{i}||_{F},\\ \sqrt{\eta_{u}}||\Delta\mathbf{U}^{i+1}\ -\ \Delta\mathbf{U}^{i}||_{F},\\ \sqrt{\eta_{v}}||\Delta\mathbf{V}^{i+1}\ -\ \Delta\mathbf{V}^{i}||_{F})/|| \mathbf{M}-\mathbf{U}_{k}\mathbf{V}_{k}^{T}||_{F}.\end{split} \] (A19)

In addition, the stopping criterion of iteration can be derived from KKT condition [45]:

\[ \begin{split}\beta^{i}\max(\sqrt{\eta_{e}}||\mathbf{E}^{i+1}\ - \mathbf{E}^{i}||_{F},\\ \sqrt{\eta_{u}}||\Delta\mathbf{U}^{i+1}\ -\ \Delta\mathbf{U}^{i}||_{F},\\ \sqrt{\eta_{v}}||\Delta\mathbf{V}^{i+1}\ -\ \Delta\mathbf{V}^{i}||_{F})/|| \mathbf{M}-\mathbf{U}_{k}\mathbf{V}_{k}^{T}||_{F}\\ <\ \varepsilon_{1},\end{split} \] (A20)

\[ \begin{split}||\mathbf{E}^{i+1}-\Delta\mathbf{U}^{i+1}\mathbf{V }_{k}^{T}-\mathbf{U}_{k}\Delta\mathbf{V}^{(i+1)T}\mathbf{U}_{k}\mathbf{V}_{k}^ {T}||_{F}\\ /||\mathbf{M}-\mathbf{U}_{k}\mathbf{V}_{k}^{T}||_{F}<\varepsilon_{2}.\end{split}