## 5 Discussion
### Redefining User Diversity and a New Heuristic Solution Approach  
In order to identify the problems of the proposed definition of provider diversity, we will revisit its definition as presented in the literature [7]:  
First of all, user diversity for an item is defined as the sum of dissimilarities among users to whom the specific item is recommended:  
\[Div(i)=\sum_{u\in U}\sum_{v\in U,\ v\
eq u}duv*xiu*xiv, \tag{14}\]  
while the diversity of an item provider is defined as the sum of item diversities for items that belong to the specific provider. As a next step the category diversity is normalized by dividing with the number of pairs of users to whom items of a specific provider are recommended. As a result, provider diversity per item and user pair is defined in Formula (4) or simplified (using the coverage constraint) in Formula (5). Conceptually it can be defined as:  
\[\overline{Div(c)}=\frac{2}{|Ic|}\frac{sum\ of\ all\ diversities\ for\ items\ in\ the\ specific\ category}{all\ pairs\ of\ users\ to\ whom\ items\ of\ class\ c\ are\ assigned}, \tag{15}\]  
Although this definition normalizes the category diversity according to the pairs of users its items are recommended to, it fails to capture the dynamics of most recommended items, as illustrated in the previous section. Consequently, if a provider owns x items that are recommended to y users, the provider diversity can be significantly increased if all users are recommended with the most recommended item, and this should not be happening. Figure 7: Relation between item diversity and number of users to whom the items were recommended. As a result, the definition of provider diversity per item and user pair should either normalize the diversities per item or define a new diversity for the entire set of items for the specific provider. In the first case, the definition of provider diversity per item and user pair becomes:  
\[\overline{Div\prime(c)}=\frac{2}{|lc|}\ \sum_{i\in Ic}\ \sum_{c\in U\sum_{c\in Yu \
eq v}duv*xiu*xiv}{\left(\sum_{u\in U}xiu\right)*\left(\sum_{u\in U}xiu-1 \right)}, \tag{16}\]  
Of course, with the previous definition for diversity, the optimization problem is not solvable, and of course the heuristic solution that was presented in Section 4 is not valid anymore. In the second case, the provider diversity, instead of the sum of item diversities, can be calculated for each provider by considering all the items of diversity as one super item. In this case, we should create a new matrix \(Y(c,u)\) (\(c\in C\), \(u\in U\)) that denotes the Boolean recommendation policy (if \(y(c,u)=0\) no item for category \(c\) has been recommended to user \(u\), and if \(y(c,u)=1\) one or more items of category \(c\) have been recommended to user \(u\)). As a result, category diversity can be defined as:  
\[\overline{Div\prime^{\prime\prime}(c)}=\frac{2}{|lc|}\frac{\sum_{u\in U}\sum_{ c\in V:u\
eq v}duv*ycu*ycv}{\left(\sum_{u\in U}ycu\right)*\left(\sum_{u\in U} ycu-1\right)}, \tag{17}\]  
In the previous definition, the denominator can be simplified as in [7]:  
\[\overline{Div\prime^{\prime\prime}(c)}=\frac{2}{|lc|}\ \frac{\sum_{u\in U}\sum_{c\in V:u \
eq v}duv*ycu*ycv}{Kc|lc|*\left(Kc|lc|-1\right)}, \tag{18}\]  
The last definition is much simpler (although it still places quadratic constraints on the problem, which is NP-Hard), as the number of categories is much smaller than the number of items. However, it is not suitable to be used for small systems, for which the users have rated a significant amount of items. Instead, for large-scale instances of the recommendation problem and large user-item matrices that are sparse, this definition can prove very helpful. Regarding the solution of the newly defined diversity per provider, in both cases a heuristic approach can also solve the problem.