## 2 Related Work
### DRBM-Social User Surrogate Model  
Here is a brief description of the user surrogate model proposed in [13] that integrates a trained dual Restricted Boltzmann Machine model with the social knowledge of similar users (DRBM-Social). It is selected as the base component of multidimensional preference user surrogate model proposed in this paper. Restricted Boltzmann Machine (RBM) is an energy-based stochastic neural network with a two-layer network structure, as shown in part in Figure 1, that learns the probability distribution of the input data. The visible layer v contains \(n\) units and its input is the observed data, and the hidden layer h contains \(m\) units and acts as a feature extractor. Given the state \((\text{v},\text{h})\) and the parameter set \(\theta=\{w,a,b\}\) where w is the connection weight, a and b are biases, it is easy to determine the RBM network state based on the energy function, as shown in (1). With the input observed data, all parameters in the RBM can be obtained by the Contrastive Divergence algorithm [39]:  
\[E_{\theta}(\text{v},\text{h})=-\sum_{i=1}^{n}\sum_{j=1}^{m}v_{i}w_{ij}h_{j}- \sum_{i=1}^{n}a_{i}v_{i}-\sum_{j=1}^{m}b_{j}h_{j} \tag{1}\]  
where \(v_{i}\) and \(h_{j}\) are the states of the \(i\)-th visible unit and the \(j\)-th hidden one, respectively. \(w_{ij}\) represents the connection weight between them, and \(a_{i}\) and \(b_{i}\) are their respective biases. The prediction score of DRBM-Social surrogate contains two parts: the score predicted by the dual-RBM model and the score calculated based on social knowledge. Figure 1 presents the structure of the Dual-RBM which consists of a positive RBM and a negative RBM. As a first step, the evaluated items are divided into dominant and inferior groups according to the magnitude of their scores. Then, these two groups of items are used to train the Dual-RBM separately to obtain two sets of RBM network structure parameters. The item scores predicted by DRBM-Social are derived by combining social knowledge and user preferences which contain both positive and negative preferences, as shown in (2):  
\[\hat{f}(x)=\sigma(\alpha*social(x)+preference(x)) \tag{2}\]  
where \(\alpha\) is the weight used to adjust the social knowledge contribution; \(\sigma\) is the activation function, which can be either a sigmoid or tanh function; \(social(x)\) indicates the weighted average score of \(M\) similar users on item \(x\) and will be defined in (3):\[social(x)=\sum_{j=1}^{M}Sim(u,u_{j})*R_{u_{j}x} \tag{3}\]  
where \(Sim(u,u_{j})\) is the Pearson similarity coefficient between the current user \(u\) and the neighbor \(u_{j}\), and \(R_{u_{j}x}\) is the rating of item \(x\) given by the neighbor \(u_{j}\). Then, the \(preference(x)\) score of the item \(x\) is calculated based on its positive and negative preferences obtained in the positive RBM and negative RBM, respectively, as presented in (4):  
\[preference(x)=\frac{\max(E_{\theta_{p}})-E_{\theta_{p}}(x,h)}{\max(E_{\theta_{ p}})-\min(E_{\theta_{p}})}-\beta*\frac{\max(E_{\theta_{n}})-E_{\theta_{p}}(x,h)}{ \max(E_{\theta_{n}})-\min(E_{\theta_{n}})} \tag{4}\]  
where \(E_{\theta_{p}}(x,h)\) and \(E_{\theta_{n}}(x,h)\) are the energy values of item \(x\) calculated by the hidden layer \(h\) in the positive and negative RBM models, respectively. In the positive RBM model, \(\max(E_{\theta_{p}})\) and \(\min(E_{\theta_{p}})\) are the maximum and minimum energy values of all items which are generated by the probabilistic model and then matched to the real space to obtain. \(\beta\) is used to adjust the proportion of the contribution of the negative preference. An item with \(J\) attributes is expressed as \(x=\left\{x_{1},x_{2},\cdots,x_{J}\right\}\), where \(x_{1,2,\ldots,J}\in\left\{0,1\right\}\). Furthermore, a probability model [13] deduced by positive RBM only, as expressed in (5), is designed to efficiently reflect the degree of user preference for each attribute of the item and generate a superior population of individuals.