## 3 Materials and Methods
### Topic Feature Learning of User and Item
#### 3.2.1. Model Description  
First, we introduce the notations of the ALDA model. Supposing that there is a collection of \(U\) users in a particular e-commerce platform: \(u\in\{1,2,\cdots,U\}\). The user \(u\) entered different queries for a particular search domain: \(q\in\{1,2,\cdots,Q\}\). There are \(P\) webpages underlying a particular query \(q\): \(p\in\{1,2,\cdots,P\}\). There are \(K\) topics that the user \(u\) is interested in: \(k\in\{1,2,\cdots,K\}\). There are \(V\) topic words in the vocabulary. \(w_{qi}\) represents the \(i\)th word in the query \(q\). \(w_{pj}\) represents the \(j\)th word in the webpage \(p\). * \(\theta_{u}\) denotes the topics probability distribution in user \(u\)'s preferences. * \(\varphi_{k}\) denotes the words probability distribution of the \(k\)th topic. * \(\alpha\) is the symmetric Dirichlet prior hyper-parameter for \(\theta_{u}\). * \(\beta\) is the symmetric Dirichlet prior hyper-parameter for \(\varphi_{k}^{q}\) and \(\varphi_{k}^{q}\). * \(z_{qi}\) denotes the topic of the \(i\)th word in query \(q\). * \(z_{pj}\) denotes the topic of the \(j\)th word in webpage \(p\). * \(w_{qi}\) denotes the \(i\)th word in the query \(q\). * \(w_{pj}\) denotes the \(j\)th word in the webpage \(p\). Formally, the generative process of query sessions and webpages based on the ALDA model is described as follows:  
Topics: We continue to work on the assumption proposed by [PERSON] and [PERSON] [49]. [PERSON] and [PERSON] [49] assumed that search query documents and webpage documents follow the same topic distributions. The topic intensities in the documents are reflected by the words displayed in the documents and each document has different topic intensities. Similar to an LDA, each topic \(k\in\{1,2,\cdots,K\}\) is represented as a topic-word distribution vector \(\varphi_{k}\). The vector \(\varphi_{k}\) follows a _Dirichlet_ distribution over \(V\) topic words in the vocabulary:  
\[\varphi_{k}\sim Dirichlet(\beta)\]  
Queries: To model the \(i\)th word \(w_{qi}\) observed in the query \(q\), ALDA sequentially samples the topic distribution of the query \(q\) and the topic assignment of the \(i\)th word in the query \(q\). The generation process of users' query online queries is as follows:  
1. For each query \(q\) (\(q\in\{1,2,\cdots,Q\}\)): 1.1 Generate topic probabilities \(\theta_{u}\) from a homogeneous Dirichlet distribution with parameter \(\alpha\): \(\theta_{u}\sim Dirichlet(\alpha)\)
2. For each topic \(k\) (\(k\in\{1,2,\cdots,K\}\)): 2.1 Generate \(\varphi_{k}\) independently from a homogeneous Dirichlet distribution with parameter \(\beta\): \(\varphi_{k}\sim Dirichlet(\beta)\)
3. For each word \(w_{qi}\) in the query \(q\): 3.1 Choose a topic \(z_{qi}\) from the \(K\) topics with probabilities given by \(\theta_{u}\): \(z_{qi}\sim Multinomial(\theta_{u})\) \begin{table}
\begin{tabular}{c l} \hline \hline
**Notations** & **Explanation** \\ \hline \(u\in\{1,2,\cdots,U\}\) & Set of users \\ \(q\in\{1,2,\cdots,Q\}\) & Set of user queries \\ \(p\in\{1,2,\cdots,P\}\) & Set of webpages \\ \(\theta_{u}\) & The vector of topic probabilities in users’ preferences \\ \(\alpha\) & Dirichlet prior distribution for \(\theta_{u}\) \\ \(\varphi_{k}\) & The vector of word probabilities for topic \(k\) \\ \(\beta\) & Dirichlet prior distribution for \(\varphi_{k}\) \\ \(z_{qi}\) & Topic assignment of the \(i\)th word in query \(q\) \\ \(z_{pj}\) & Topic assignment of the \(j\)th word in webpage \(p\) \\ \(w_{qi}\) & The \(i\)th observed word in query \(q\) \\ \(w_{pj}\) & The \(j\)th observed word in webpage \(p\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the main notations. 2 Choose a word \(w_{qi}\) from the dictionary with probabilities given by \(\varphi_{k}\): \(w_{qi}\sim Multinomial(\varphi_{k})\)  
Webpages: To model the \(j\)th word \(w_{pj}\) observed in the webpage \(p\), ALDA sequentially samples the topic distribution of the webpage \(p\) and the topic assignment of the \(j\)th word in the webpage \(p\).