# New Vector-Space Embeddings for Recommender Systems
## 4 Evaluation and Results
### Prediction
Consequently, \(U\) and \(V\) are reduced, obtaining \(R_{k}=U_{k}\cdot S_{k}\cdot V_{k}^{\prime}\). For user \(u\) and item \(i\), the rating prediction is computed as the dot product between the \(u\)-th row of \(U_{k}S_{k}^{1/2}\) and \(i\)-th column of \(S_{k}^{1/2}V_{k}^{\prime}\) followed by adding the user average back. * SVD++ [49]: This is an extension of the SVD approach, where implicit feedback is incorporated. An implicit feedback refers to the fact that a user \(u\) rated an item \(i\) without regard to the value of the given rating. * Non-negative Matrix Factorization (NMF) [50]: This approach is similar to SVD; however, stochastic gradient descent (SGD) is used for the optimization procedure. SGD is carried out with a specific choice of step size ensuring the non-negativity of factors given that the factors' initial values are positive. * Slope One [51]: The technique relies on the SlopeOne algorithm, where predictors are used to compute the average difference between the ratings of one item and another for users who rated both. The rating prediction is defined as: \(r_{ui}^{\prime}=\mu_{u}+\frac{1}{card(R_{i}(u))}\sum_{i\
eq R_{i}(u)}dev(i,j)\), where \(\mu_{u}\) is the average of ratings given by user \(u\). \(R_{i}(u)\) is the set of relevant items: the set of items \(j\) rated by user \(u\) that also have at least one common user with \(i\). _card_ is the number of elements in a set. \(dev(i,j)\) is the average deviation of ratings of items \(i\) with respect to items \(j\), given by: \(dev(i,j)=\frac{1}{card(U_{ij})}\sum_{u\
eq U_{ij}}r_{ui}-r_{uj}\)
* K-Nearest Neighbors (k-NN) [24]: The predictions of ratings are performed based on the ratings given by the user's nearest-neighbors. In such models, known as memory-based models, the stored ratings are directly used in the prediction process. * Centered k-NN [24]: This is a k-NN technique that takes into account the average ratings of each user to provide a corrective action to the prediction. k-NN Baseline [52]: This is a k-NN technique that incorporates baseline estimates. These estimates account for data tendencies such as the likelihood that some users give higher ratings than other users or that some items receive higher ratings than other items. * Co-Clustering [53]: Clusters are built simultaneously for users and items. The predictions are based on the average ratings of the co-clusters (user-item neighborhoods). Formally, rating prediction is given as: \(\hat{r_{ui}}=\overline{C_{ui}}+(\mu_{u}-\overline{C_{u}})+(\mu_{i}-\overline{C _{i}})\), where \(\overline{C_{ui}}\) is the average rating of co-cluster \(C_{ui}\) for user \(u\) and item \(i\). \(\overline{C_{u}}\) is the average rating of user \(u\)'s cluster. \(\overline{C_{i}}\) is the average rating of item \(i\)'s cluster. * Baseline [52]: In this technique, a simple baseline estimate is computed for the rating predictions, taking into account the aforementioned data tendencies. Formally, rating prediction is defined as: \(\hat{r_{ui}}=b_{ui}=\mu+b_{u}+b_{i}\), where \(\mu\) is the overall average rating. Parameters \(b_{u}\) and \(b_{i}\) are determined by solving a least squares problem. Table 2 shows the errors of the proposed models as well as the competing approaches, using the five fold validation test. Missing entries mean that the system crashed and was not able to produce any results. From Table 2, one can observe that among a total of 11 methods, our approach obtained the best RMSE for the MovieLens 1 M dataset, the second best RMSE (0.923) for the MovieLens 100 K dataset (being very close to the best RMSE (0.92) obtained by the SVD++ method) and the best MAE for ModCloth and Amazon MS datasets. Overall, the best two models are our proposed Sphere model and SVD++. We suggest these models be the candidates to be tested when designing a recommender system and possibly using these approaches in a hybrid model. One can observe that all the approaches produced higher errors for the ModCloth dataset. This is due to the fact that ModCloth is a sparser dataset.