## 2 Matrix Factorization Based Editing Expert Recommender System
### Constructing Feedback Rating Matrix  
In this section, the process of constructing \(F\) by combining \(O\) and \(B\) is described in detail. Diverse approaches can be adopted for the process as it is possible for \(u_{i}\) to leave both ordinal and binary ratings to \(e_{j}\). Moreover, there exist numerous criteria for determining the type of user feedback to employ preferentially. Therefore, we suggest four different combination methods, \(c_{o}(o_{i,j},b_{i,j})\), \(c_{b}(o_{i,j},b_{i,j})\), \(c_{l}(o_{i,j},b_{i,j})\), and \(c_{d}(o_{i,j},b_{i,j})\), for inferring \(f_{i,j}\) to identify the optimal approach by which the user's positive and negative opinions are accurately expressed. Note that the four methods are differentiated by the type of user feedback they utilize when both \(o_{i,j}\) and \(b_{i,j}\) exist, and when only one type of user feedback exists, the existing rating is employed in all methods. By \(c_{o}(o_{i,j},b_{i,j})\), \(o_{i,j}\) is employed preferentially over \(b_{i,j}\) when both \(o_{i,j}\) and \(b_{i,j}\) exist as described in Equation (2). \[c_{o}\left(o_{i,j},b_{i,j}\right)= \begin{cases}o_{i,j}&\text{if }\exists o_{i,j},b_{ i,j}\\ o_{i,j}&\text{if }\exists o_{i,j},\ \
ot\exists b_{i,j}\\ \delta_{b}(b_{i,j})&\text{if }\exists o_{i,j},\ \exists b_{i,j}\\ \mathbf{\omega}&\text{otherwise},\end{cases}  \tag{2}\]  
where \(\delta_{b}(b_{i,j})\) is a function that transforms binary ratings into 5-point scale as shown in Equation (3). \[\delta_{b}(b_{i,j})= \begin{cases}5&\text{if }b_{i,j}=1\\ 1&\text{if }b_{i,j}=-1.\end{cases}  \tag{3}\]  
The idea behind \(\delta_{b}(b_{i,j})\) is as follows. \(e_{j}\) will be recommended to \(u_{i}\) in future services if \(u_{i}\) clicked like button to \(e_{j}\), while \(e_{j}\) will be excluded for \(u_{i}\) if \(u_{i}\) clicked dislike button to \(e_{j}\). The former case indicates that \(u_{i}\) is extremely satisfied with the service provided by \(e_{j}\), and, thus, \(b_{i,j}\) is transformed into 5, which represents the most positive opinion. The latter case implies that \(u_{i}\) is absolutely unsatisfied with the service provided by \(e_{j}\), and, thus, \(b_{i,j}\) is transformed into 1, which is the most negative opinion. By \(c_{b}(o_{i,j},b_{i,j})\), \(b_{i,j}\) is employed preferentially over \(o_{i,j}\) when both \(o_{i,j}\) and \(b_{i,j}\) exist as described in Equation (4). \[c_{b}\left(o_{i,j},b_{i,j}\right)= \begin{cases}\delta_{b}(b_{i,j})&\text{if } \exists o_{i,j},b_{i,j}\\ o_{i,j}&\text{if }\exists o_{i,j},\ \
ot\exists b_{i,j}\\ \delta_{b}(b_{i,j})&\text{if }\exists o_{i,j},\ \exists b_{i,j}\\ \mathbf{\omega}&\text{otherwise}.\end{cases}  \tag{4}\]  
Next, \(c_{d}(o_{i,j},b_{i,j})\) and \(c_{l}(o_{i,j},b_{i,j})\) are approaches that employ \(b_{i,j}\) preferentially over \(o_{i,j}\) when both \(o_{i,j}\) and \(b_{i,j}\) exist like \(c_{b}(o_{i,j},b_{i,j})\). However, they employ \(b_{i,j}\) over \(o_{i,j}\) only when \(b_{i,j}\) is a certain value,  
Figure 4: Overview of the proposed non-negative matrix factorization based editing expert recommender system. like or dislike. Specifically, by \(c_{d}(o_{i,j},b_{i,j})\), \(b_{i,j}\) is selected over \(o_{i,j}\) only when \(b_{i,j}\) is -1 (dislike), and \(o_{i,j}\) is employed otherwise as shown in Equation (5). \[c_{d}\left(o_{i,j},b_{i,j}\right)= \begin{cases}\delta_{d}(o_{i,j},b_{i,j})& \text{if }\exists o_{i,j},b_{i,j}\\ o_{i,j}&\text{if }\exists o_{i,j},\ \exists b_{i,j}\\ \delta_{b}(b_{i,j})&\text{if }\
ot\exists o_{i,j},\ \exists b_{i,j}\\ \mathbf{o}&\text{otherwise},\end{cases}  \tag{5}\]  
where \(\delta_{d}(o_{i,j},b_{i,j})\) determines the utilization of \(b_{i,j}\) according to the value of \(b_{i,j}\) as described by Equation (6). \[\delta_{d}(o_{i,j},b_{i,j})= \begin{cases}o_{i,j}&\text{if }b_{i,j}=1\\ \delta_{b}(b_{i,j})&\text{if }b_{i,j}=-1.\end{cases}  \tag{6}\]  
On the contrary, by \(c_{l}(o_{i,j},b_{i,j})\), \(b_{i,j}\) is selected over \(o_{i,j}\) only when \(b_{i,j}\) is 1 (like), and \(o_{i,j}\) is employed otherwise as shown in Equation (7).