## 4 Methodology
### Attention-Based Adaptive Representation Fusion  
In the previous section, to comprehensively capture user preferences from diverse perspectives, we propose a disentangled hypergraph representation learning network approach that learns distinct location embeddings for three fundamental dimensions, i.e., collective preference and geospatial context. These embeddings, denoted as \(\mathbf{M}_{\mathcal{C},L}\) and \(\mathbf{M}_{G,L}\), respectively, are formulated in the latent space \(\mathbb{R}^{|L|\times d}\). Each embedding encapsulates distinct factors that influence user behavior within its respective dimension, thereby enabling the modeling of intricate user preferences. To further establish user-specific embeddings tailored to these perspectives, we use the incidence matrix \(\mathbf{H}_{\mathcal{C}}\in\mathbb{R}^{|L|\times|L|}\) of collective preference hypergraph \(\mathcal{G}_{\mathcal{C}}\), which encodes the relationships between users and locations in the collaborative dimensions. Specifically, the user embeddings \(\mathbf{E}_{\mathcal{C},U}\) and \(\mathbf{E}_{G,U}\in\mathbb{R}^{|U|\times d}\) are computed by aggregating the location embeddings \(\mathbf{M}_{\mathcal{C},L}\) and \(\mathbf{M}_{G,L}\in\mathbb{R}^{|L|\times d}\) through the incidence matrix \(\mathbf{H}_{\mathcal{C}}\). Mathematically, this process can be formulated as follows:  
\[\mathbf{E}_{\mathcal{C},U}=\mathbf{H}_{\mathcal{C}}^{\top}\mathbf{M}_{ \mathcal{C},L},\quad\mathbf{E}_{G,U}=\mathbf{H}_{\mathcal{C}}^{\top}\mathbf{M }_{G,L}, \tag{18}\]  
where \(\mathbf{H}_{\mathcal{C}}^{\top}\in\mathbb{R}^{|U|\times|L|}\) represents the transposed incidence matrix enabling the projection of location embeddings into user-specific spaces. To achieve a unified representation of user preferences that adaptively considers the relative importance of different behavioral perspectives, we introduce an attention-based fusion mechanism. Rather than using fixed scalar coefficients, we learn attention weights dynamically based on the user embeddings derived from the collective and geographical views. Formally, given the two user-specific embeddings \(\mathbf{E}_{\mathcal{C},U},\mathbf{E}_{G,U}\in\mathbb{R}^{|U|\times d}\), we compute the fused user representation \(\mathcal{R}_{U}\) as follows:  
\[\mathcal{R}_{U}=\alpha\cdot\mathbf{E}_{\mathcal{C},U}+(1-\alpha)\cdot\mathbf{ E}_{G,U}, \tag{19}\]  
where \(\alpha\in\mathbb{R}^{|U|\times 1}\) is the learned attention weight for each user, broadcasting across the embedding dimensions. We compute the attention weights \(\alpha\) using a two-layer feedforward network with non-linear activation:  
\[\alpha=\text{softmax}(\mathbf{W}_{1}\cdot\tanh(\mathbf{W}_{2}[\mathbf{E}_{ \mathcal{C},U}\,\|\,\mathbf{E}_{G,U}])), \tag{20}\]  
where \([\cdot\,\|\,\cdot]\) denotes vector concatenation, \(\mathbf{W}_{2}\in\mathbb{R}^{d_{h}\times 2d}\) and \(\mathbf{W}_{1}\in\mathbb{R}^{1\times d_{h}}\) are learnable weight matrices, and \(d_{h}\) is the hidden layer dimension. The softmax operation is applied row-wise to ensure normalized attention distribution across views. Alternatively, for a more compact implementation, a scalar attention weight per user can be computed using a single-layer attention module:  
\[\alpha=\sigma\Big{(}\mathbf{w}^{\top}[\mathbf{E}_{\mathcal{C},U}\,\|\, \mathbf{E}_{\mathcal{C},U}]+b\Big{)}, \tag{21}\]  
\[\mathcal{R}_{U}=\alpha\odot\mathbf{E}_{\mathcal{C},U}+(1-\alpha)\odot\mathbf{ E}_{G,U}, \tag{22}\]  
where \(\mathbf{w}\in\mathbb{R}^{2d}\) and \(b\in\mathbb{R}\) are learnable parameters, and \(\sigma(\cdot)\) is the sigmoid function. This attention-based fusion mechanism allows the model to dynamically weigh collaborative and geographical information according to user-specific behavioral patterns, thereby yielding more personalized and adaptive user representations. While adaptive fusion of user embeddings is critical due to the heterogeneity in user behaviors across perspectives, the aggregation of location embeddings can be simplified.