# A Location-Time-Aware Factorization Machine Based on Fuzzy Set Theory for Game Perception
## 3 Problem Formulation and Algorithm
### LTFM Based on Fuzzy Set Theory  
In this section, an LTFM model under the FM framework is proposed as a solution to the user game perception evaluation problem, as it has the advantage of capturing nonlinear feature interactions efficiently for sparse data. Unlike the standard FM model, LTFM takes the projection of location and time information into account. We construct an LTFM model that can effectively capture the projection information of location and time information as additional information without introducing additional information, and the similarity between feature interactions and labels obtained based on fuzzy set theory to improve the predictive strength of the model for user game perception valuation. The structure of FM is shown in Equation (3). Inspired by this, our data will be encoded as \(\mathbf{x}=[\mathbf{x}_{users},\mathbf{x}_{services}]\). Each element of \(\mathbf{x}_{users}\) represents a user, and each element of \(\mathbf{x}_{new\ series}\) is a game service after time and location projections. The projection information of location and time information is integrated into the new service matrix, and the new data is represented as \(\mathbf{x}=[\mathbf{x}_{users},\mathbf{x}_{new\ series}]\), which is presented into the FM model's pairwise feature interaction module, as shown below. \[y_{FM}(\mathbf{x},\Theta) =w_{0}+\sum\limits_{i=1}^{N}\mathbf{w}_{i}\mathbf{x}_{i}+\sum \limits_{i=1}^{N}\sum\limits_{j=1}^{N}\left\langle\mathbf{v}_{i},\mathbf{v}_{j} \right\rangle\cdot\mathbf{x}_{i}\mathbf{x}_{j} \tag{4}\] \[y_{new}(x) =\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\left\langle\mathbf{ v}_{i},\mathbf{v}_{j}\right\rangle\cdot\mathbf{x}_{users}\mathbf{x}_{new}\text{ { services}} \tag{5}\]  
At this point, the LTFM can be expressed as  
\[y_{\text{LTFM}}(\mathbf{x};\Theta) =\phi_{\text{LTFM}}(y|\mathbf{x},\Theta) \tag{6}\] \[=y_{FM}(\mathbf{x};\Theta)+y_{new}(\mathbf{x};\Theta)\] \[=w_{0}+\mathbf{w}_{users}\cdot\mathbf{x}_{users}+\mathbf{w}_{ serrives}\cdot\mathbf{x}_{services}\] \[+\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\left\langle\mathbf{ v}_{i},\mathbf{v}_{j}\right\rangle\cdot\mathbf{x}_{users}\mathbf{x}_{services}+\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N} \left\langle\mathbf{v}_{i},\mathbf{v}_{j}\right\rangle\cdot\mathbf{x}_{users} \mathbf{x}_{new}\text{ { services}}\]  
where \(w_{0}\) is the weight of the global bias, and \(\mathbf{v}_{i}\) is the hidden vector of the \(i\)-th variable in \(\mathbf{x}_{user}\). \(\mathbf{v}_{j}\) is the hidden vector of the \(j\)-th variable in \(\mathbf{x}_{services}\). \(\left\langle\mathbf{v}_{i},\mathbf{v}_{j}\right\rangle\) is the inner product of the hidden vectors of the user matrix and the service matrix. The inner product value can represent the pairwise interactions between the users and the services. As introduced in the discussion in Section 3.3, feature interactions that are less predictable are given lower weights since they contribute less to the game perception evaluation. The lack of ability to distinguish the predicted strength of pairwise feature interactions may lead to additional computational resources and suboptimal predictions. It means that embeddings of less important feature interactions are ignored. Thus, the similarity between pairwise interaction features and labels is captured in the output weights to reformat the pairwise part of FM as follows. \[\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}u_{ij}\left\langle\mathbf{v}_{i}, \mathbf{v}_{j}\right\rangle\cdot\mathbf{x}_{users}\mathbf{x}_{services} \tag{7}\]  
The calculation of \(u_{ij}\) is given in Equation (2), which considers the influence of the correlation between pairwise interaction features and labels on the predicted strength of the user's game perception evaluation. The same can be applied to new feature interactions. Combined with the above description, the final network architecture of our LTFM algorithm is shown in Figure 2, which consists of three layers: the input layer, the middle layer, and the output layer.