## 5 Application Instances of Proposed SDP
### Application Instance-2: SDPTrustSVD
\]  
During the above process, \(e_{ui}\) is also equal to \((\hat{t}_{ui}-r_{ui})\), and \(\frac{\partial\mathcal{L}}{\partial b_{u}}\), \(\frac{\partial\mathcal{L}}{\partial\hat{p}_{u}}\), \(\frac{\partial\mathcal{L}}{\partial q_{i}}\), \(\frac{\partial\mathcal{L}}{\partial y_{j}}\), and \(\frac{\partial\mathcal{L}}{\partial\hat{w}_{v}}\) are the same as the related processes in TrustSVD, and \(\frac{\partial\mathcal{L}}{\partial\hat{p}_{i}}\) is renewed by factor \(\big{(}\frac{\partial\mathcal{L}}{\partial\hat{w}_{v}}\big{)}\), while \(\frac{\partial\mathcal{L}}{\partial\hat{w}_{v}}\) is calculated during SDPTrustSVD according to \(\mathcal{L}_{SDP}\) in (10). As a result, with the minimization convergence of the error function \(\mathcal{L}\), latent factor matrices for the user and item, \(P\) and \(Q\), are output. Algorithm 2 with pseudocode shows the learning processes during predicting \(\hat{r}_{ui}\). ``` Input: user-item rating matrix \(\mathbb{R}\), user-user Trust matrix \(T\), regularization parameters \(\lambda\) and \(\lambda_{t}\), learning rate \(\gamma_{1}\) and \(\gamma_{2}\) Output: Rating prediction \(\hat{r}_{ui}\) (1) Initialize \(\varepsilon\) with very small positive value; (2) Calculate global deviation \(\sigma\); (3) Calculate local standard deviation \(\sigma_{u}\); (4) if\(\sigma_{u}\leq\sigma\)then (5) Begin learning with loss function \(\mathcal{L}\) according to Equation (17); (6) while\(\varepsilon_{ui}>\varepsilon\)do (7) \(b_{u}\gets b_{u}-\gamma_{1}\big{(}\sum_{



## 6 Experiments and Analysis

In this section, we performed a series of experiments to evaluate the efficiencies of the proposed SDP and its application instances (SDPSVD++ and SDPTrustSVD) on five real datasets, including Flixster, FilmTrust, MiniFilm, Movielens-10 M, and Movielens-latest-small (100K). The evaluation performance of predictive accuracies was measured by the metrics mean absolute error (MAE), root mean square error (RMSE), and relative error (RE), respectively, where a lower value indicates better predictive accuracy. It was assumed \(\tau=\{(u,i)\big{|}\exists_{t}((u,i,r)\in\mathbb{R})\}\), and \(N\) is the number of observed ratings. The evaluation matrices are as follows:

\[MAE=\frac{\sum_{(u,i)\in\tau}\lvert r_{ui}-\hat{r}_{ui}\rvert}{N}, \tag{18}\]

\[RMSE=\sqrt{\frac{\sum_{(u,i)\in\tau}\left(r_{ui}-\hat{r}_{ui}\right)^{2}}{N}}, \tag{19}\]

\[RE=\sqrt{\frac{\sum_{(u,i)\in\tau}\left(r_{ui}-\hat{r}_{ui}\right)^{2}}{\sum_ {(u,i)\in\tau}\left(r_{ui}\right)^{2}}}. \tag{20}\]

We designed three experiments to evaluate the performance. * Experiment (1): The first experiment evaluates the baseline estimation accuracy of the proposed SDP on the five datasets, compared with classical baseline estimation in (2) and the PBEModel [14]. * Experiment (2): The second experiment evaluates the improved accuracies of SDPSVD++ and proves the efficiency of SDP on improving existing CF recommendations. * Experiment (3): The third experiment evaluates the performance of the proposed SDPTrustSVD based on the Flixster and FilmTrust datasets, which have trust ratings. Each dataset in the experiments was separated into five parts. To perform a full evaluation, we used a cross-validation method during experiments in which four parts of a given dataset were used for training while the remaining part was used for prediction. As a result, each part was used for both training and prediction by cross validation. The final performance on a given dataset was expressed as the average of the cross-prediction results. As mentioned above, our experiments were performed on five real datasets, and Table 3 shows their statistics. These datasets are all about movies. Flixster, Movielens (10 M), and Movielens (Latest Small) are generally large datasets, while FilmTrust and MiniFilm are relatively small. All of these could be found in related websites or www.librec.net. Flixster has 147 K users and 48 K movies with 409 K ratings and more than 11 M trust ratings with friendship links.