## 3 Methodology
### The NNCF Predictor  
In this study, a neural network-based collaborative filtering (NNCF) predictor is proposed to predict the ratings in the updated preference matrix \(\mathbf{R}^{\prime}\in\mathbb{R}^{N\times M}\) where \(N\) is the number of users and \(M\) is the number of items. As shown in Figure 3, the core architecture of the NNCF predictor includes an input layer, embedding layer, multilayer perceptron (MLP) layers, and output layer. The input layer is composed of two vectors \(u^{o}_{i}\) and \(v^{o}_{j}\) that are represented in one-hot encoding format after being converted from the unique identifications of user \(u_{i}\) and item \(v_{j}\), respectively. Each user vector \(u^{o}_{i}\) and item vector \(v^{o}_{j}\) rendered as a binary sparse vector is further projected to a dense vector in the embedding layer. The transformed vector is called a latent vector with \(K\)-dimension. The  
Figure 2: The schematic structure of the DAE.  
user latent vector and item latent vector are fed into the MLP layers. Each layer in the MLP layers can be customized to discover the specific latent structures of user-item interactions. The final output layer is the predicted rating \(\hat{r}^{\prime}_{ij}\). The prediction function of the NNCF predictor can be written as:  
\[\hat{r}^{\prime}_{ij}=f(\mathbf{P}^{T}\mathbf{u}^{o}_{i},\mathbf{Q}^{T}\mathbf{v}^ {o}_{j}\big{|}\mathbf{P},\mathbf{Q},\Theta_{f}) \tag{5}\]  
where \(\mathbf{P}\in\mathbb{R}^{N\times K}\) and \(\mathbf{Q}\in\mathbb{R}^{M^{\prime}\times K}\), denoting the latent factor matrix for users and items, respectively; \(\Theta_{f}\) means the model parameters of the function \(f\).  
In the NNCF predictor framework, the MLP layers consist of at least three layers of nodes, including the input layer, single or multiple hidden layers, and output layer. Except for the input nodes, each node uses a nonlinear activation function, and each layer is fully connected to the next layer except the output layer. MLP utilizes a supervised learning technique called backpropagation for training and tries to minimize the loss with actual ratings. Let the user latent vector \(\mathbf{p}_{i}\) be \(\mathbf{P}^{T}\mathbf{u}^{o}_{i}\) and item latent vector \(\mathbf{q}_{j}\) be \(\mathbf{Q}^{T}\mathbf{v}^{o}_{j}\). The MLP model under the NNCF framework can be defined as follows:  
\[\mathbf{z}_{1}=\phi_{1}(\mathbf{p}_{i^{\prime}},\mathbf{q}_{j})=\left[\begin{array} []{c}\mathbf{p}_{i}\\ \mathbf{q}_{j}\end{array}\right] \tag{6}\]  
\[\phi_{2}(\mathbf{z}_{1})=f_{2}(\mathbf{w}_{2}^{T}\mathbf{z}_{1}+\mathbf{b}_{2}) \tag{7}\]  
\[\cdots\cdots\]  
\[\phi_{L}(\mathbf{z}_{L-1})=f_{L}(\mathbf{w}_{L}^{T}\mathbf{z}_{L-1}+\mathbf{b} _{L}) \tag{8}\]  
\[\hat{r}^{\prime}_{ij}=\sigma(\mathbf{h}^{T}\phi_{L}(\mathbf{z}_{L-1})) \tag{9}\]  
where \(\mathbf{w}_{x}\), \(\mathbf{b}_{x}\), \(f_{x}\), denote the weight matrix, bias vector, activation functions for the \(x\)th layer respectively; \(\sigma\) and \(\mathbf{h}\) represent the activation function and edge weight of the output layer, respectively. In this study, the training is performed by minimizing the pointwise loss between \(\hat{r}^{\prime}_{ij}\) and \(\hat{r}^{\prime}_{ij}\) with training data. After getting the model's optimal parameters, the ratings for user \(\mathbf{u}_{i}\) and item \(\mathbf{v}_{j}\) can be predicted correctly.