## 2 Background and Related Work
### Collaborative Filtering
\[\omega\)\((u_{a},u_{i}\,)=\frac{\sum_{j}\bigl{(}r_{aj}-\overline{r}_{a}\bigr{)}\bigl{(}r _{ij}-\overline{r}_{i}\bigr{)}}{\sqrt{\sum_{j}\bigl{(}r_{aj}-\overline{r}_{a} \bigr{)}^{2}\bigl{(}r_{ij}-\overline{r}_{i}\bigr{)}^{2}}}\) (1)where \(r_{aj}\) and \(r_{ij}\) are the ratings that \(u_{a}\) and \(u_{i}\) give to item \(i_{j}\) respectively, and \(\overline{r}_{a}\) and \(\overline{r}_{i}\) are the average ratings of \(u_{a}\) and \(u_{i}\), respectively. The Pearson coefficient can represent inverse and direct correlation, with values in the interval [1; 1], where the value 0 corresponds to the absence of a correlation.  
Another commonly used similarity metric is cosine, which is given by the dot product of the vectors representing the preferences of two given users \(u_{a}\) and \(u_{i}\) in the Euclidean space. The cosine similarity between those user, \(cosSim(u_{a},u_{i})=cos\Big{(}R_{u_{a}},R_{u_{i}}\Big{)}\), is computed according to Equation (2) [8], where \(R_{u_{a}}\) and \(R_{u_{i}}\) are the vectors containing the ratings given to items by users \(u_{a}\) and \(u_{i}\) respectively.  
\[cos\left(R_{u_{a}},R_{u_{i}}\right)=\frac{R_{u_{a}}\cdot R_{u_{i}}}{\|R_{u_{a} }\|\ \|R_{u_{i}}\|}=\frac{\sum_{j=1}^{n}r_{aj}\ \!r_{ij}}{\sqrt{\sum_{j=1}^{n}r_{aj}^{2}} \sqrt{\sum_{j=1}^{n}r_{ij}^{2}}} \tag{2}\]  
The ratings of the \(k\) of most similar users are used to estimate the preferences of the active users about items that they have not rated. The rating \(pr_{aj}\) estimated for the active user \(u_{a}\) and an item \(i_{j}\) is computed using the following equation [8]:  
\[pr_{aj}=\overline{r}_{a}+\frac{\sum_{i=1}^{k}sim(u_{a},u_{i})\Big{(}r_{ij}- \ \overline{r}_{i}\Big{)}}{\sum_{i=1}^{k}\big{|}sim(u_{a},u_{i})\big{|}} \tag{3}\]  
where \(sim(u_{a},u_{i})\) is the similarity between the active user and his neighbor \(u_{i}\), which could be obtained by using the Pearson coefficient, cosine or any other similarity metric. There are other ways of computing ratings from similarities, especially when they are used for top-N recommendations. These are given in the form of a ranked list containing N items. In this context, the normalizing denominator can be eliminated, since there is no need for the ratings to be adjusted to a range. Rating averages can also be replaced by bias for the user-item pairs [9] or eliminated [10]. Other memory-based CF methods involve changes in the prediction function, such as rating normalizations using Z-score [11]. However, most of the proposals are hybrid approaches [12] that use the ratings and information from users or items. This additional information can be item content data (e.g., the genre of a movie) [13] or user data such as demographic attributes [14], user social data [15; 16], and other types of user information [17] which is usually used in combination with ratings to compute similarity. In the category of model-based methods, there are many proposals in the literature. Some of these focus on replacing the use of the lazy k-nearest neighbor technique with other machine learning algorithms [18], such as associative classification [19], rule-based methods [20; 21], Bayes classifiers [22], support vector machines [23], and neural networks [24]. Latent factor models [25] are more widely used than approaches based on machine learning. This group includes matrix factorization [26], which is a procedure for dimensionality reductions that generate latent factors for each user and each item. It consists of decomposing the rating matrix \(R\in\mathbb{R}^{m\times m}\) with many empty elements into two less scarce matrices, a user matrix \(U\in\mathbb{R}^{m\times d}\) and an item matrix \(V\in\mathbb{R}^{m\times d}\), so that the product \(UV^{T}\) is an approximation of R. The latent factors of \(U\) and \(V\) will be those that minimize the function \(\sum_{i,j}\!\left(R_{i,j-}(U_{i},V_{j})\right)^{2}\). A popular factorization technique in the area of recommender systems is singular value decomposition (SVD) [27].