# New Vector-Space Embeddings for Recommender Systems
## 4 Evaluation and Results
### Prediction
It has many fewer reviews per user than the other datasets. Another point worth mentioning is that for all KNN-based algorithms, the RAM (12 GB) crashed while running for ModCloth dataset, that is, such algorithms consume huge memory sizes and do not scale well when increasing the number of users as in this dataset. Furthermore, our algorithm converges in a relatively low number of cycles. The average number of cycles are 10, 23, 11 and 7 for MovieLens 100 K, MovieLens 1 M, ModCloth and Amazon MS datasets, respectively. The algorithm's computational complexity is detailed in Appendix A. Our algorithm possesses other benefits as well. It does not suffer as much as SVD++ and k-NN from memory problems. In the latter two approaches, a huge user-item ratings \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**MovieLens 100 K**} & \multicolumn{3}{c}{**MovieLens 1 M**} & \multicolumn{3}{c}{**ModCloth**} & \multicolumn{2}{c}{**Amazon MS**} \\ \hline
**Approach** & **RMSE** & **MAE** & **RMSE** & **MAE** & **RMSE** & **MAE** & **RMSE** & **MAE** \\ \hline Sphere (our proposed method) & 0.923 & 0.738 & **0.857** & 0.684 & 1.165 & **0.783** & 1.03 & **0.605** \\ SVD & 0.934 & 0.737 & 0.873 & 0.686 & 1.056 & 0.828 & 0.971 & 0.724 \\ SVD++ & **0.92** & **0.722** & 0.862 & **0.673** & 1.06 & 0.829 & **0.941** & 0.689 \\ NMF & 0.963 & 0.758 & 0.916 & 0.724 & 1.2 & 0.948 & 1.035 & 0.759 \\ Slope One & 0.946 & 0.743 & 0.907 & 0.715 & 1.150 & 0.87 & 0.996 & 0.637 \\ k-NN & 0.98 & 0.774 & 0.923 & 0.727 & - & - & 1.156 & 0.771 \\ Centered k-NN & 0.951 & 0.749 & 0.929 & 0.738 & - & - & 0.958 & 0.611 \\ k-NN Baseline & 0.931 & 0.733 & 0.895 & 0.706 & - & - & 1.038 & 0.699 \\ Co-Clustering & 0.963 & 0.753 & 0.915 & 0.717 & 1.137 & 0.856 & 0.986 & 0.619 \\ Baseline & 0.944 & 0.748 & 0.909 & 0.719 & **1.041** & 0.82 & 1.002 & 0.766 \\ Random & 1.514 & 1.215 & 1.504 & 1.206 & 1.398 & 1.06 & 1.437 & 1.059 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Prediction Recommender Systems Results. matrix has to be stored, even though it is sparse. In our approach, in contrast, only the user-item review pairs have to be stored and used in the algorithm, which provides memory savings. Another advantage is that our algorithm is incremental in nature. So it can add any extra user or item without having to retrain the entire vectors' space. For example, it can begin the retraining process from the previous state of the vectors instead of starting with random vectors and retraining entirely from scratch. This incremental feature may be lacking in the SVD-type methods and it is not clear how one can start from a previous solution if extra data arrive. However, the major advantage of our approach is its versatility. One can add user-user and item-item similarities to the existing user-item similarities. This makes the approach more general and allows us to add content-based, demographic and other similarities to create one large model that takes into account all available information. On the other hand, a model such as SVD is designed to handle only user-item similarities.