## 5 Application Instances of Proposed SDP
### Application Instance-2: SDPTrustSVD
Compared with the original SVD++, using MAEs as examples, on each dataset with d=5, the predictive accuracies of SDPSVD++ are 7.69%, 2.84%, 1.26%, and 0.67% higher than those of SVD++. On each dataset with d=10, the improvements of SDPSVD++ are also distinct, with MAE improvements of 3.47%, 2.81%, 1.76%, and 0.65% compared with original SVD++. The greatest improvements appear in the large-scale dataset Flixster with d = 5, where the RMSE, RE, and MAE by SDPSVD++ are more than 6.8% higher than SVD++. Compared with PBESVD++, the proposed SDPSVD++ also obtains higher predictive accuracies. Continuing to use MAEs for examples as described in Table 5, for each database with d = 5, the proposed SDPSVD++ are 1.73%, 0.94%, 1.49%, and 0.65% higher than those of PBESVD++. This proves that the proposed SDP can improve the recommendation accuracy of the existing SVD++ more efficiently than PBE. Additionally, we observed that the predictive accuracies on full ratings of the small-scale dataset FilmTrust and MiniFilm are also improved greatly by SDPSVD++. On dataset FilmTrust, with d = 5 or d = 10, the RMSE, RE, and MAE are always more than 2.1% higher than SVD++. Compared with SVD++, the RMSE, RE, and MAE are 1.66%, 1.67%, and 1.76% higher, respectively, than those of the original SVD++ on MiniFilm with d = 10, proving that the SDP can be effectively combined with existing CF-based recommendation algorithms in different situations and obtain higher predictive accuracies. ### Performance of Proposed SDPTrustSVD

The purpose of this experiment was to evaluate the predictive accuracy of the proposed SDPTrustSVD on all users with full ratings using cross validation. We selected the TrustSVD [11] for comparison. Because the two need to compute the trust information, we performed the experiment on the full ratings in Flixster and FilmTrust by cross validation, with d = 5 and d = 10, respectively. The evaluation results of MAEs, RMSEs, and REs are shown in Table 6, and the visualized performance comparisons on each dataset are shown in Figure 6. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset** & **Metrics** & **SVD++** & **PBESVD++** & **SDPSVD++** & \begin{tabular}{c} **Improved** \\ **(vs. SVD++)** \\ \end{tabular} \\ \hline \multirow{3}{*}{\begin{tabular}{c} FilmTrust \\ (d = 10) \\ \end{tabular} } & RMSE & 0.81474 & 0.80754 & 0.79735 * & 2.13\% \\ \cline{2-6}  & RE & 0.25946 & 0.25717 & 0.25392 * & 2.14\% \\ \cline{2-6}  & MAE & 0.63330 & 0.61536 * & 0.61550 & 2.81\% \\ \hline \multirow{3}{*}{\begin{tabular}{c} MiniFilm \\ (d = 5) \\ \end{tabular} } & RMSE & 0.98790 & 0.98696 & 0.97232 * & 1.58\% \\ \cline{2-6}  & RE & 0.31090 & 0.31060 & 0.30598 * & 1.58\% \\ \cline{2-6}  & MAE & 0.76582 & 0.75788 & 0.75614 * & 1.26\% \\ \hline \multirow{3}{*}{\begin{tabular}{c} MiniFilm \\ (d = 10) \\ \end{tabular} } & RMSE & 0.98845 & 0.98749 & 0.97200 * & 1.66\% \\ \cline{2-6}  & RE & 0.31109 & 0.31078 & 0.30588 * & 1.67\% \\ \cline{2-6}  & MAE & 0.76924 & 0.75806 & 0.75568 * & 1.76\% \\ \hline \multirow{3}{*}{\
\begin{tabular}{c} ml\_latest\_small \\ (d = 5) \\ \end{tabular} } & RMSE & 0.87226 & 0.87089 & 0.86618 * & 0.70\% \\ \cline{2-6}  & RE & 0.23764 & 0.23727 & 0.23599 * & 0.69\% \\ \cline{2-6}  & MAE & 0.67298 & 0.67285 & 0.66847 * & 0.67\% \\ \hline \multirow{3}{*}{
\begin{tabular}{c} ml\_latest\_small \\ (d = 10) \\ \end{tabular} } & RMSE & 0.87353 & 0.87223 & 0.86813 * & 0.62\% \\ \cline{2-6}  & RE & 0.23810 & 0.23775 & 0.23663 * & 0.62\% \\ \cline{2-6}  & MAE & 0.67307 & 0.67284 & 0.66871 * & 0.65\% \\ \hline \hline \end{tabular} Note: * is the best value.

\end{table} 
Table 5: _Cont._Figure 5: Performances (MAEs, RMSEs, and REs) of predictive accuracies of different models: SVD++, PBESVD++, and proposed SDPSVD++. The results show that the proposed SDP can effectively improve the performance of SVD++.