# A Top-N Movie Recommendation Framework Based on Deep Neural Network with Heterogeneous Modeling
## 3 Our Approach: MFDNN  
### Problem Definition  
In this subsection, we first provide the related preliminary definition and then provide a formal problem definition.  
**Definition 1**.: _Heterogeneous information network (HIN). HINs were first defined in [34]. A directed graph \(G=\langle V,E\rangle\) is defined to present an information network, where \(V\) is the set of objects, \(E\) is the set of relations, the object-type mapping function is \(\phi:V\to A\), and the relation-type mapping function is \(\psi:E\to R\). A network is called a heterogeneous information network when the types of objects \(|A|>1\) or the types of relations \(|R>1|\); otherwise, it is a homogeneous information network._  
A bibliographic information network [63] is a typical HIN that contains three types of objects: author, venue, and paper, and two types of relations: publish and write. Other examples of HINs are shown in [64]. We mainly focus on the bipartite network--a special HIN that has two types of objects.  
**Definition 2**.: _Bipartite Network. A bipartite network is a special HIN that has two types of objects._  
**Problem 1**.: \(U=\{u_{1},u_{2},\cdots,u_{m}\}\) _is a user set of size \(\mathbf{m}\), and \(I=\{i_{1},i_{2},\cdots,i_{n}\}\) is an item set of size \(\mathbf{n}\). We first analyze and select a reasonable meta-path that can help find the most similar user or item according to the meta-path. An example of a selected meta-path is shown in Figure 1._  
In this study, we consider the meta-path link information as well as the attribute information of the user and item. According to the \(U1\) rating meta-path, we define the user-item interaction matrix \(Y^{-}\in R^{m\times n}\) and the user-item rating matrix \(Y^{+}\in R^{m\times n}\) according to the historical rating record as Equation (1) and (2):  
\[y^{-}_{ui}=\begin{cases}1&\text{if interaction (user $u$, item $i$) is observed(score>2);}\\ 0&\text{otherwise.}\end{cases} \tag{1}\]  
\[y^{+}_{ui}=\begin{cases}Y^{+}_{ui}&\text{the rating (user $u$, item $i$);}\\ 0&\text{otherwise.}\end{cases} \tag{2}\]  
Figure 1: The selected meta-path.  
Here, a value of 1 for \(y_{ud}^{+}\) denotes that \(u\) and \(i\) have an interaction; however, this does not necessarily mean that \(u\) actually likes \(i\). Similarly, a value of 0 for \(y_{ud}^{-}\) also does not indicate that user \(u\) dislikes item \(i\); perhaps user \(u\) is not aware of item \(i\) at all. In other words, observed relations reflect the users' preferences on items, while unobserved relations can result from missing data. For example, when shopping online, the rating value of an item is affected by factors other than the item itself, such as delivery speed and service attitude.  
In such cases, the final rating may not indicate whether the user likes the item. However, if the user buys the item, it is certain that one of the characteristics of the item attracts the user. Therefore, we simply conclude that the level of the rating reflects the preferences of the user. It becomes a challenge to learn users' intentions from the historical rating record since it contains various noisy data indicating users' preferences. We often cannot obtain explicit feedback information directly, and the data is sparse.  
In contrast, we can easily obtain implicit feedback information, and the data covers most users and objects; thus, it can mitigate the problem of sparse data to some extent [65; 66; 67]. We obtain a top-N recommendation list via modeling with a recommendation algorithm according to historic explicit and implicit feedback information.