# Discretly Exploiting Inter-Session Information for Session-Based Recommendation
## 3 Preliminaries
### Disentangled Represent Learning  
The primary objective of DRL is to acquire factor-level embeddings through the process of re-embedding the original item-level embedding into multiple distinct factor spaces. These factor-level embeddings function as representations of latent factors that encompass more intricate and nuanced semantics. It is worth noting that these latent factors correspond to abstracted aspects of the targets rather than concrete factors such as brand and color. Nevertheless, they can still encapsulate certain aspects of the target information.  
Let \(\mathbf{e}_{c}\in\mathbb{R}^{d}\) represent an item-level embedding with dimension \(d\). Then DRL computes \(\mathcal{K}\) factor-level embeddings from \(c\) by re-embedding. The factor-level embedding \(\mathbf{e}_{c}^{(f,k)}\) on factor \(k\) is computed with Equation (1).  
\[\mathbf{e}_{c}^{(f,k)}=\sigma(\mathbf{W}_{k}^{f}\mathbf{e}_{c})+\mathbf{b}_{k}^{f}(1\leq k\leq K) \tag{1}\]  
Here, \(\mathbf{W}_{k}^{f}\in\mathbb{R}^{d_{f}\times d}\) and \(\mathbf{b}_{k}^{f}\in\mathbb{R}^{d_{f}}\) are the weight matrix and bias on factor \(k\), with \(\sigma\) representing the activation function. Additionally, \(d_{f}=\lfloor\frac{d}{k}\rfloor\) is the dimension of factor-level embedding.  
Given that different factors are not mutually independent, it is possible to encounter duplicated information between two latent factors, thereby potentially impeding the accurate assessment of preference for a specific factor in subsequent stages. To address this concern, we incorporate a distance correlation-based method into our proposed DRL framework to promote factor independence and mitigate the aforementioned issue. By optimizing this learning objective in Equation (2), the DRL framework can effectively reduce the correlation between factors and minimize the generation of redundant information, as follows:  
\[\mathcal{L}_{d}=\sum_{k}^{K}\sum_{j\
eq k}^{K}dCor(\mathbf{e}_{c}^{(f,k)},\mathbf{e}_ {c}^{(f,j)}), \tag{2}\]  
\begin{table}
\begin{tabular}{l l l l} \hline \hline \(\mathbf{e}_{s}\) & item-level embedding of session s & \(\mathbf{e}_{s}^{(f,k)}\) & factor-level embedding of s on factor k \\ \hline \(\mathbf{e}_{(s,i)}\) & item-level embedding of i-th item in s & \(\mathbf{e}_{(s,i)}^{(f,k)}\) & factor-level embedding of i-th item in s on factor k \\ \hline \(\mathbf{\hat{y}}\) & scores of all candidate items & \(U_{(s,i)}\) & i-th interest unit in s \\ \hline \(\mathcal{L}_{d}\) & loss from disentangled representation learning & \(Sta_{s}^{(f,1)}\) & interest stability of s on factor k \\ \hline \(\mathcal{L}_{c}\) & loss from contrastive learning & \(\mathbf{\overline{e}_{s}^{s}}\) & user’s interest learned from inter-session channel \\ \hline \(\mathcal{L}_{p}\) & loss from prediction & \(\mathbf{\overline{e}_{s}^{s}}\) & user’s interest learned from intra-session channel \\ \hline \(\mathcal{L}\) & total loss & \(\mathbf{\overline{e}_{s}}\) & final user’s interest \\ \hline \hline \end{tabular}
\end{table}
Table 1: Cont.  
where \(dCor\) is a function to measure the correlation between variables in different spaces. For other details, please refer to [32].