## 5 Application Instances of Proposed SDP
### Application Instance-2: SDPTrustSVD
Movielens (10 M) has more than 70 K users and 10 K movies with 10 M ratings. Movielens (Latest Small) is the latest small dataset of Movielens and has 700 users and 10 K movies with 100 K ratings. FilmTrust is relatively small and has 1508 users and 2071 items with 35,497 ratings and 1853 trust ratings. The smallest dataset is MiniFilm, which has only 55 users and 334 items with 1K ratings. Flixster and FilmTrust can be used to evaluate the performance of TrustSVD and SDPTrustSVD, which require latent factors from trust ratings. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**DataSets** & **\#Users** & **\#Items** & **\#Ratings** & **\#Density** & **Rating Scale** & **\#Trust** \\ \hline Flixster & 147,612 & 48,794 & 8,196,077 & 0.11\% & [0.5,5] & 11,794,648 \\ FilmTrust & 1508 & 2071 & 35,497 & 1.14\% & [0.5,4] & 1853 \\ MiniFilm & 55 & 334 & 1000 & 5.44\% & [0.5,4] & 0 \\ ml-10 m & 71,567 & 10,681 & 10,000,054 & 1.31\% & [1,5] & 0 \\ ml-latest-small & 700 & 10,000 & 100,000 & 1.43\% & [1,5] & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Details of the datasets for the experiments. ### Performance of Proposed SDP

The purpose of this experiment was to evaluate the baseline estimation accuracy of SDP on all users with full ratings using cross validation. We selected the classical baseline estimation (BE) described in (2) and the PBEModel [14] for baseline estimation performance comparison. We performed the experiment on the full ratings in each dataset. The evaluation results of MAEs, RMSEs, and REs are shown in Table 4 where PBE is short for PBEModel, and the visualized performance comparisons are shown in Figure 4.

The proposed SDP exhibits the best performances on Flixster, MiniFilm, Movielens (10 M), and Movielens (Latest Small), and second on FilmTrust. Compared with classical BE, the baseline predictive accuracies of the proposed SDP are superior. The MAEs of SDP are 5.26%, 1.35%, 2.11%, 0.81%, and 2.15% higher than classical BE on the datasets Flixster, FilmTrust, MiniFilm, Movielens (10 M), and Movielens (Latest Small), respectively. On the large-scale dataset Flixster, compared with BE, the RMSE and RE are also improved by 2.96% and 2.95%, respectively, by SDP. On another large-scale dataset, MovieLens (10 M), the improvements of RMSE, RE, and MAE are 0.84%, 1.19%, and 0.81%, respectively. Compared with classical PBE, the predictive accuracies of SDP are also superior on Flixster, MiniFilm, Movielens (10 M), and Movielens (Latest Small), except for the small dataset FilmTrust. The results prove that the proposed SDP could effectively improve the baseline estimation accuracy, and all MAEs, RMSEs, and REs are superior to BE. ### Performance of Proposed SDPSVD++

The purpose of this experiment was to evaluate the predictive accuracy of the proposed SDPSVD++ on all users with full ratings using cross validation. SDPSVD++ is an improved recommendation algorithm based on SVD++ [5] by the proposed baseline estimation model SDP. PBESVD++ [14] is also an improved SVD++ based on the baseline estimation model PBE. Therefore, we selected SVD++ and PBESVD++ for comparisons. We performed the experiment on the full ratings in four datasets by

\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Data Sets**} & \multirow{2}{*}{**Metrics**} & \multirow{2}{*}{**BE**} & \multirow{2}{*}{**PBE**} & \multirow{2}{*}{**Proposed SDP**} & \multirow{2}{*}{**Improved (vs. BE)**} \\ \cline{3-3} \cline{5-6}  & RMSE & 0.98813 & 0.95974 & 0.95890 * & 2.96\% \\ \cline{2-6} \cline{2



cross validation, including Flixster, FilmTrust, MiniFilm, and Movielens (Latest Small), with 5 and 10 dimensions, respectively. The evaluation results of MAEs, RMSEs, and REs are shown in Table 5, and the visualized performance comparisons on each dataset are shown in Figure 5. The results show that the predictive accuracies (RMSEs, REs, and MAEs) of the proposed SDPSVD++ are superior to those of SVD++ and PBESVD++.