## 5 Application Instances of Proposed SDP
### Application Instance-1: SDPSVD++  
(1) Improved by SDP  
SVD++ extends SVD using feedback from user ratings. The detailed improvement is that a free factor-user vector \(p_{u}\) is complemented by \(|l_{u}|^{-\frac{1}{2}}\left(\sum_{j\in l_{u}}y_{j}\right)\), and a user \(u\) is modeled as \(\left(p_{u}+|l_{u}|^{-\frac{1}{2}}h\left(\sum_{j\in l_{u}}y_{j}\right)\right)\). The equation of SVD++ is:  
\[\hat{r}_{ui}=b_{u}+b_{i}+\mu+q_{i}^{T}\left(p_{u}+|l_{u}|^{-\frac{1}{2}}\cdot \left(\sum_{j\in l_{u}}y_{j}\right)\right)\text{,} \tag{12}\]  
where \(l_{u}\) represents the set of item ratings by user \(u\); \(y_{j}\in Y\) represents the implicit influence of items rated by user \(u\) in the past on the ratings of unknown items in the future. Based on the SDP model, we improved the baseline estimation using a proportion of \(\left(\frac{\sigma_{u}}{\sigma}\right)\) for SVD++, leading to the following model:  
\[\hat{r}_{ui}=\left(\frac{\sigma_{u}}{\sigma}\right)b_{i}+b_{u}+\mu+q_{i}^{T} \left(p_{u}+|I_{u}|^{-\frac{1}{2}}t\sum\
olimits_{j\in I_{u}}y_{j}\right). \tag{13}\]  
All of these parameters have the same meanings as in SVD++. We named the new SVD++ model (13) SDPSVD++. (2) Model learning  
The involved parameters in Equation (13) of SDPSVD++ are learned by minimizing the regularized squared error function associated with (14):  
\[ \begin{array}{l}\mathcal{L}=\min\limits_{q_{u}p_{u},y_{u}>b_{u},\sigma_{u}, }\left(\frac{1}{2}\sum\
olimits_{u}\sum\
olimits_{i\in I_{u}}\left(\hat{r}_{ ui}-\left(\frac{\sigma_{u}}{\sigma}\right)b_{i}-b_{u}-\mu\right)^{2}\right.\\ \left.+\frac{\lambda}{2}\sum\
olimits_{u}\left|I_{u}\right|^{-\frac{1}{2}}({b_ {u}}^{2}+\sigma_{u}{}^{2}+\left\|p_{u}\right\|_{F}^{2})+\right.\\ \left.\frac{\lambda}{2}\sum\
olimits_{i}\left|U_{i}\right|^{-\frac{1}{2}}\left( {b_{i}}^{2}+\left\|q_{i}\right\|_{F}^{2}\right)+\frac{\lambda}{2}\sum\
olimits_ {j}\left|\left|U_{j}\right|^{-\frac{1}{2}}\left\|y_{j}\right\|_{F}^{2}\right), \end{array}  \tag{14}\]  
where \(U_{i}\) and \(U_{j}\) are the set of users who rate items \(j\) and \(i\), respectively; \(\left\|\ \right\|\) denotes the Frobenius norm; and \(\lambda\) alleviates operational complexity and avoids overfitting. To obtain a local minimization of the objective function \(\mathcal{L}\), we performed the following gradient descents on \(b_{u},b_{i},\sigma_{u},p_{u},q_{i},\ \text{and}\ y_{j}\) for all the users and items in each given training dataset:  
\[ \begin{array}{l}\frac{\partial\mathcal{L}}{\partial b_{u}}=\sum\
olimits_{ i\in I_{u}}e_{ui}+\lambda\left|I_{u}\right|^{-\frac{1}{2}}\cdot b_{u}\\ \frac{\partial\mathcal{L}}{\partial b_{i}}=\sum\
olimits_{u\in I_{u}}\left( \frac{\sigma_{u}}{\sigma}\right)te_{ui}+\lambda\left|U_{i}\right|^{-\frac{1}{2 }}\cdot b_{i}\\ \frac{\partial\mathcal{L}}{\partial u_{i}}=\sum\limits_{i\in I_{u}}\left( \frac{b_{i}}{\sigma}\right)te_{ui}+\lambda\left|I_{u}\right|^{-\frac{1}{2}} \cdot\sigma_{u}\\ \frac{\partial\mathcal{L}}{\partial p_{u}}=\sum\limits_{i\in I_{u}}e_{ui}q_{i} +\lambda\left|I_{u}\right|^{-\frac{1}{2}}p_{u}\\ \frac{\partial\mathcal{L}}{\partial q_{i}}=\sum\limits_{u\in I_{i}}e_{ui}(p_{ u}+\left|I_{u}\right|^{-\frac{1}{2}}\sum\limits_{j\in I_{u}}y_{j})+\lambda\left|U_{i} \right|^{-\frac{1}{2}}q_{i}\\ \forall j\in I_{u},\ \frac{\partial\mathcal{L}}{\partial y_{j}}=\sum\
olimits_{ i\in I_{u}}e_{ui}[I_{u}]^{-\frac{1}{2}}q_{i}+\lambda\left|U_{j}\right|^{-\frac{1}{2}}y_{j}. \end{array} \]  
During the above gradient descent, there is \(e_{ui}=\hat{r}_{ui}-r_{ui}\). Processes of \(\frac{\partial\mathcal{L}}{\partial b_{u}}\), \(\frac{\partial\mathcal{L}}{\partial p_{u}}\), \(\frac{\partial\mathcal{L}}{\partial\hat{q}_{i}}\), and \(\frac{\partial\mathcal{L}}{\partial y_{j}}\) are the same as those in SVD++, while \(\frac{\partial\mathcal{L}}{\partial b_{i}}\) adds factor \(\left(\frac{\sigma_{u}}{\sigma}\right)\) to \(e_{ui}\), and \(\frac{\partial\mathcal{L}}{\partial\hat{q}_{u}}\) is added during gradient descent according to \(\mathcal{L}_{SDP}\) in (10).