# Simple and Efficient Computational Intelligence Strategies for Effective Collaborative Decisions
## 3 Materials and Methods
### Multi-Task Tensor Factorization
It is assumed that, if sentiment s and item \(i\) are relevant with the \(k\)-th topic, then \(S_{s,k}\) is a Gaussian pair of \(N(I_{i,k},\sigma^{2})\) and the same is true with the other variables. Readers must note here that, \(S_{s,k}\) represents \(k\)-th characteristics of the sentiment whiles \(I_{i,k}\) is the \(k\)-th characteristic of item \(i\). Thus, the \(k\)-th character of the sentiment \(s\) as well as the \(k\)-th character of item \(i\) depicts a Gaussian-like feature. Therefore,  
\[||I_{i,k}-S_{s,k}||\sim N(0,\sigma^{2}) \tag{3}\]  
whiles \(\sigma\) represents the standard deviation of the distribution. Three pairs of the dimensions items, users and sentiments are respectively deduced assuming that,  
\[||U^{I}_{u,k}-I^{u}_{i,k}||\sim N(0,\sigma^{2})\] \[||U^{S}_{u,k}-S^{u}_{s,k}||\sim N(0,\sigma^{2})\] \[||I^{S}_{i,k}-S^{I}_{s,k}||\sim N(0,\sigma^{2})\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline U & User-tag \\ I & Item-tag matrix \\ R & Rating sentiments expressed \\ A & Tensor \\ R & Number of features in XYZ \\ C & Core tensor \\ \(R\in U\times I\times S\) & Rating history \\ PS & User item pairs \\ X & Dimension layer \\ \(\Xi\) & Set of negativities \\ \(\alpha_{out}\) & Mapping of output layer \\ Z & Observed relationship in \(\mathbf{Z}\) \\ \(\theta\) & Training parameter \\ S & tensor \\ \(c_{pi}\) & \(c\)-th neural filtering layer \\ \(V_{p^{p}}\) & first input dimensional vector \\ \(V_{i^{p}}\) & second input dimensional vector \\ \(\theta_{out}\) & interaction function for the final phase \\ \(\theta_{y}\) & interaction function for deep neural filtering phase \\ \(I_{i,k}\) & is the \(k\)-th characteristic of item \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. Based on this assumption, a triplet (\(u\); \(i\); \(t\)) as well as its potential score (\(u\); \(i\); \(t\)) is deduced as;  
\[ \begin{array}{l}f(u,i,s)=\sum\limits_{k}w_{k}^{UI}.\exp(-\frac{1}{2}\lambda|| U_{u,k}^{I}-I_{i,k}^{UI}||^{2})+\sum\limits_{k}w_{k}^{US}.\exp(-\frac{1}{2} \lambda||U_{u,k}^{S}-S_{s,k}^{UI}||^{2})\\ +\sum\limits_{k}w_{k}^{IS}.\exp(-\frac{1}{2}\lambda||I_{i,k}^{S}-S_{s,k}^{I}|| ^{2})\end{array}  \tag{4}\]  
Here, \(\lambda=1/\sigma^{2}.w_{k}^{UI}\) represents the \(k\)-th feature for the item-user features. Again, \(w_{k}^{IS}\) and \(I_{i,k}^{S}\) represent the value of the \(k\)-th topic for item-sentiment and user-sentiment interactions. The matrices \(I,U\) and \(S\) columns feature topic are fused into a feed-forward MLP layer of the deep model. On top of the input, the embedding layer maps the sparse representation to a deep vector. The resultant item or user embedding labelled as the item or user feature vector in the field of feature modelling. The item and user embedding are directly inputted into the multi-layer neural model, we name it as; deep neural filtering layer which maps the feature vectors to a recommendation field. The dimension layer \(X\) explains the potency of the model. A fully connected multilayer feed-forward networks have nodal transfer function activation flows from the input layer through a hidden layer to the output layer [48]. This is expressed as;  
\[z_{1}=f.\sum\limits_{i=1}^{N}\big{(}w_{ij}\big{)}+x_{i}+b_{j} ^{1} \tag{5}\]  
where \(x_{i}\) is one of the N inputs nodes for processing node \(j\), \(w_{ij}\) denotes the weight of the connection node, \(b\_j\) is the bias for node \(y\) and \(z_{1}\) is the output node. Each neuron is connected to each other in the forward direction within the network.