# Simple and Efficient Computational Intelligence Strategies for Effective Collaborative Decisions
## 3 Materials and Methods
### Multi-Task Tensor Factorization  
As in the case of MF in Recommender Systems (Rss), TF produces a predictive model by revealing patterns from the data. The major advantage of a tensor based approach is the ability to take into account a multifaceted nature of user-item interactions [43]. In the Google's wide and deep model [44], a generalized linear model was used to capture latent features in the wide perspective. In this paper, tensor factorization model which is non-linear in nature is chosen to be a platform model as a result of the fact that it has appealing property to efficiently impose structure on the vector space representation of the data as propounded by [37]. We will regard an array of numbers with more than 2 dimensions as a tensor. This is a natural extension of matrices to a higher order case. A tensor with \(N\) distinct dimensions is called an \(N\)-way tensor or a tensor of order \(N\). Tucker Decomposition (TD) factorizesa multi-dimensional array into a main tensor multiplied by a matrix [20]. Taking a 3 rd-order tensor \(A\in\Re^{l\times l\times K}\) for instance,  
\[A_{i,j,k}=\sum_{r_{1}=1}^{R_{1}}\sum_{r_{2}=1}^{R_{2}}\sum_{r_{3}=1}^{R_{3}}C_{r _{1},r_{2},r_{3},\mathbf{x}}X_{j,r_{1}\bullet}Y_{j,r_{2}\bullet}Z_{k,r_{3}} \tag{1}\]  
In that, \(X\in\Re^{l\times R_{1}},Y\in\Re^{l\times R_{2}},Z\in\Re^{K\times R_{3}}\) and \(C\in\Re^{R_{1}\times R_{2}\times R_{3}}\) are core tensors. \(R_{1},R_{2},R_{3}\) represent latent characteristics. However, CD factorizes a multi-way data which is a tensor into a summation of rank-one modes. Let \(A\in\Re^{l\times l\times K}\) be a tensor and it can be written as;  
\[A_{i,j,k}=\sum_{r=1}^{R}X_{i,r}.Y_{j,r}.Z_{k,r} \tag{2}\]  
where \(X\in\Re^{l\times R},Y\in\Re^{l\times R},Z\in\Re^{K\times R}\) and \(R\) represents the number of features. \(X,Y,Z\) in Equations (1) and (2) can be equated to the factor matrices SVD and core tensor C can be said to be the eigenvalues [45]. Probabilistic Tensor factorization is applied to model the reviews sentiment and rating scores with their item (IDs) which are the explicit user feedback [46] which are in line with [12]. We treat dataset as three-dimensional tensor structure. We adapt notations of [12; 47] for  
Figure 1: General Framework. the formulation of our model. If I is the set of items, \(U\) is the set of users whiles is the set of review sentiments, then the process that reviewer of an item \(S\) reviews or scores with sentiment s and how the user u rates an item is symbolized by a triple \((u,i,s)\). \(R\in U\times I\times X\) denotes the set of review history (Table 1). If a triplet \((u,i,s\in R)\) then the interactions between items, users and review sentiment for personalized tag recommendation can be represented by a three-order tensor, which is depicted in Figure 1. A ranking scheme propounded by [35] for personalized recommendation is employed in this paper. If \(PS\) denotes the item-user pairs \((u,i)\) in \(S\), then a multi-task tensor factorization (MTF) using Gaussian Kernel for the filtering job can be fitted. The triplicate interactions between items, users and review sentiment is presented by a 3 rd-order tensor with \(|\,\mathrm{I}|\) items, \(|\,\mathrm{U}|\) users and \(|\,\mathrm{S}|\) sentiment votes [40]. The tensor can then be factorized into three components, an item matrix \(I\in\mathbb{Z}^{|I|\times Z}\), a user matrix \(U\in\mathbb{Z}^{|U|\times Z}\) and a sentiment matrix \(S\in\mathbb{Z}^{|s|\times z}\), where \(R\) is the number of features. The entry of the \(i\)-th row of the matrix \(I\), matrix \(U\) and matrix \(S\) denotes the latent variables of the \(i\) = item, \(i\)-th user and \(i\)-th sentiment vote, respectively. The columns of matrices \(I,U\) and \(S\) could be referenced as a feature topic. Three paired relations are modelled with respect to this paper; user-sentiment, user-item and user ratings.