## 4 Methodology
### Unified Hypergraph Representation Learning Network
#### 4.3.2 Across-Hyperedge Feature Propagation  
Since each node \(l\) may be connected to multiple hyperedges, we aggregate the messages from these related hyperedges to refine the node's embedding. Take the collective preference hypergraph \(\mathcal{G}_{C}\) for example; the propagation of information from hyperedges back to nodes is represented as follows:  
\[\bar{I}_{C}=\mathrm{HGCN}_{\mathcal{L}2n}(\{m_{e}|e\in\mathcal{E}\}), \tag{15}\]  
where \(\mathrm{HGCN}_{\mathcal{L}2n}(\cdot)\) is the propagation function, which aggregates the medium messages \(m_{e}\) from all hyperedges \(e\) that are related to node \(l\).  
This across-hyperedge feature propagation step enables the refinement of node embedding by incorporating information from neighboring hyperedges, capturing the multi-level relationships that exist within the hypergraph structure. The proposed disentangled hypergraph representation learning network operates iteratively, with the embeddings being refined across multiple layers. Let \(e_{C}^{(\ell)}\) denote the embedding of node \(l\) at the \(\ell\)-th layer of the model. The embedding at each layer is updated by combining the propagated message from the previous layer and the newly aggregated information:  
\[e_{C}^{(\ell)}=\mathrm{Sum}\Big{(}\bar{I}_{C}^{(\ell)},\bar{I}_{C}^{(\ell-1) }\Big{)}, \tag{16}\]  
where \(\bar{I}_{C}^{(\ell-1)}\) is the information from the previous layer, and the addition of the previous layer's embedding helps maintain residual connections. The residual connections are crucial to prevent over-smoothing, a common issue in graph neural networks (GNNs), ensuring that the node embeddings retain distinctiveness even after multiple aggregation layers.  
Finally, to generate the final representation for node \(l\), the embeddings from all layers are averaged, formulated as follows:  
\[e_{C,l}=\frac{1}{L+1}\sum_{\ell=0}^{L}e_{C}^{(\ell)}, \tag{17}\]  
The total number of layers is denoted as \(L\). After performing the multi-layer propagation and aggregation operations, we obtain the refined collective location embedding matrix \(\mathbf{M}_{C,L}=\{e_{C,1},e_{C,2},\cdots,e_{C,|L|}\}\in\mathbb{R}^{|L|\times d}\) for collective preference hypergraph \(\mathcal{G}_{C}\). These representations capture the underlying structure of the collaborative hypergraph, effectively encoding the complex interactions between users and locations.  
For the aggregation functions \(\mathrm{HGCN}_{\mathcal{L}2n}(\cdot)\) and \(\mathrm{HGCN}_{\mathcal{L}2n}(\cdot)\), we implement them using mean pooling. This choice of aggregation method has been proven to be both effective in capturing the relationships within collaborative and geographical views of the data. Specifically, mean pooling ensures that the aggregated message from each hyperedge is balanced, averaging the information from all nodes within a hyperedge without favoring any particular node.  
Similar to the collective preference hypergraph \(\mathcal{G}_{C}\), we further model geographical dependencies of locations. These modules are stacked over \(L\) layers to capture high-order neighborhood dependencies, with residual connections introduced to mitigate the over-smoothing problem.  
Through above processes, we can obtain the refined collective location embedding matrix \(\mathbf{M}_{C,L}\in\mathbb{R}^{|L|\times d}\) for the collective preference hypergraph \(\mathcal{G}_{C}\) and the refined geospatial location embedding matrix \(\mathbf{M}_{G,L}\in\mathbb{R}^{|L|\times d}\) for the geospatial context hypergraph \(\mathcal{G}_{G}\), where \(|L|\) denotes the number of locations, and \(d\) is the embedding dimension.