## 3 Method
### Model Training  
The embedding vector representations of users and items, the user's preference coefficients for different feature combinations, and the weight coefficients of combinations among different features are the parameters to be trained for the proposed MIMA model. The model weights are optimized by minimizing the loss functions through negative sampling, and the user's preferences for positive samples and negative samples are computed separately. And then, the trained parameters are updated based on the differences between negative and positive samples. The loss function is Equation (9). Figure 3: The aggregation layer of the MIMA algorithm. \[\mathcal{L}=-\sum_{(u,v)\in\Omega}\log\sigma(h_{u}^{T}\cdot h_{v})-\sum_{(u^{T},v^{ \prime})\in\Omega^{-}}\log\sigma(-h_{u}^{T}\cdot h_{v^{\prime}}) \tag{9}\]  
where \(\Omega\) is the set of observed (positive) interaction records between nodes, and \(\Omega^{-}\) is a set of negative node pairs randomly selected from all unobserved node pairs. The Adam optimizer [63] is used for model training. Figure 4 illustrates the comprehensive framework of the proposed MIMA algorithm, including the input layer, embedding layer, feature interaction layer, meta-path aggregation layer, and prediction layer. Algorithm 1 demonstrates the pseudocode of the MIMA algorithm. ```
Input: heterogeneous graph with attributes \(G=(V,E)\), node feature \(\{e_{i},\forall i\in V\}\), meta-path set \(\mathbb{P}=\{P_{1},...,P_{J}\}\), input vector \(\mathbf{x}\), number of attention head \(H\), the training rounds \(L\), and meta-path neighbor set of \(v\) as \(N_{v}^{P}\) Output: the embedding vector representation of users \(U\) and items \(V\) for Node feature \(\{e_{i},\forall i\in V\}\)do  Node content transformation \(\mathbf{h}_{i}\gets w_{i}\cdot x_{i}\forall i\in V\) endfor for\(l=1\)to\(L\)do forNode \(e_{i}\in V\)do forMeta-path \(P\in\mathbb{P}\)do  Calculate the embedding representation of each node feature \(e_{i}^{Res}\leftarrow\sum_{k=1}^{V}a_{ik}^{h}(W_{i}^{h}e_{k})+W_{Res}e_{i}\)  Calculate the vector representation of the internal aggregation of each target node's meta-path and the aggregation between meta-paths \(h_{v}\gets\sigma(W_{o}\cdot\sum_{P_{j}\in\mathbb{P}}\frac{\exp{(e_{j})}}{ \sum_{P_{i}\in\mathbb{P}}\exp{(e_{j})}}\cdot\sigma(\sum_{u\in N_{v}^{P}}a_{uu} ^{P}\cdot h_{p(v,u)}))\) endfor endfor
```  
**Algorithm 1** Multi-feature interaction meta-path aggregation heterogeneous graph neural network (MIMA). **MIMA algorithm complexity**: The input vector \(e\) with dimension \(d\) is transformed into an embedding representation with dimension \(\tilde{d}\); the dimension of the transformation matrix is \(d\times\tilde{d}\). The computational complexity of generating the embedding representations for diverse node categories can be denoted as \(\mathcal{O}(N\tilde{d})\), where \(N\) refers to the total number of nodes. When determining the attention coefficient, it is necessary to compute  
Figure 4: The comprehensive model architecture diagram illustrating the proposed MIMA algorithm. the user-item graph repeatedly, with the number of evaluations matching the number of edges. The computational complexity associated with this operation is \(\mathcal{O}(h(E\times\tilde{d}))\), where \(E\) signifies the quantity of edges in the heterogeneous graph, and \(h\) represents the count of heads within the multi-head attention module. The aggregation calculation of node characteristics usually involves the information propagation and feature aggregation operations of neighbor nodes. Considering different numbers of neighbor nodes and varying feature dimensions, the time complexity of aggregation operations is generally \(\mathcal{O}(\tilde{d}\times N)\). In general, the computational complexity of aggregation between meta-paths can be represented as \(\mathcal{O}(p\times\tilde{d})\), where p is the numerical value of the meta-paths. The upper limit of iterations for the MIMA algorithm is \(L\).