# 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
Figure 2: The network architecture of the proposed location-time-aware factorization machine (LTFM) based on fuzzy set theory. The input layer: LTFM performs feature embedding based on the embedding matrix. Every feature is represented by dense vectors. It also can be seen as a fully connected layer, which implements array lookup. In the input layer, the key point is able to multiply the matrix fully. The inputs multiply with the weight matrix, which also takes the bias vector into consideration so that the spatial dimensionality and the projection of sparse to dense representations can be handled efficiently. The middle layer: The model is inspired by the FM model. LTFM also needs to find how to represent the latent relationship in pairwise features which can be solved by the inner product. The set of pairwise feature interactions is represented as \(\mathcal{P}=\left\{\left(v_{j},v_{j^{{}^{\prime}}}\right)x_{j}x_{j^{{}^{ \prime}}}\middle|j,j^{{}^{\prime}}=1,2,\ldots j;j\
eq j^{{}^{\prime}}\right\}\) depending only on the embedding matrix. The middle layer has listed the potential vectors in interactions. The input layer has given feature embedding vectors. Each latent vector is the product of previous embedding vectors. Therefore, we use the idea of using neural networks to design and build into this layer to roughly express the FM model. The output layer: We compress all feature interactions based on distinguishing their importance in the embedding space. Then, we project them onto the final perceptual prediction scores. The predicted similarity of a paired feature interaction is calculated based on Equation (2). In conclusion, the LTFM proposed in this paper can be expressed as follows. \[ \begin{array}{ll}y_{LTFM}(\mathbf{x};\Theta)&=\phi_{LTFM}(y|\mathbf{x}, \Theta)\\ &=w_{0}+\mathbf{w}_{users}\cdot\mathbf{x}_{users}+\mathbf{w}_{series}\cdot \mathbf{x}_{series}\\ &+\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}u_{ij}\langle\mathbf{v}_{i}, \mathbf{v}_{j}\rangle\cdot\mathbf{x}_{users}\mathbf{x}_{series}+\sum\limits_{ i=1}^{N}\sum\limits_{j=1}^{N}u_{ij}\langle\mathbf{v}_{i},\mathbf{v}_{j}\rangle \cdot\mathbf{x}_{users}\mathbf{x}_{new\ series}\end{array}  \tag{8}\]  
In order to evaluate the parameters in Equation (8), the main method is to minimize the sum of losses on the observed dataset. \[\text{min}R_{LTFM}(\Theta)=\sum\limits_{i=1}^{N}\mathcal{L}(\phi_{LTFM}(y_{i}| x_{i},\Theta),y_{i})+R(\Theta) \tag{9}\]  
where \(\mathcal{L}\) denotes the loss function and \(R\) denotes the regularization term on \(\Theta\), which is usually used to avoid overfitting. The overfitting problem cannot be ignored when optimizing machine learning models and FM models [26]. In this study, our regularization term is \(R(\Theta)=\frac{1}{2}\|\theta\|^{2}\). Neurons in paired interaction layers can easily cooperate with each other to adapt, which leads to overfitting. In this study, the LTFM model captures interactions that are predictive and useful. We control the regularization strength by the L2 regularization method and further prevent overfitting of the LTFM model by the dropout method on paired interaction layers [27]. The dropout method prevents the complex cooperative adaptation of neurons to the training data. The main idea is to randomly drop out some neurons along the connections during the training process. The dropout module is applied for model training when the entire network architecture is used for prediction and is disabled for testing. Dropout has the additional side effect of using smaller neural networks for averaging, which may improve performance. Therefore, the dropout method is used in the middle-paired interaction layer of the LTFM model to deal with the overfitting problem. The complexity analysis of our proposed LTFM model has the following two points. First, the parameters of the feature embedding matrix \(\mathbf{v}\in\mathbb{R}^{K\times J}\) are \(K\times J\).