This section explains the basics of \textit{recommender systems} necessary for the essential understanding of the paper presented. Besides the general definition of the \textit{recommender problem}, the corresponding solution approaches are presented. Furthermore, the main focus will be on the solution approach of \textit{matrix factorization}.
\subsection{Recommender Problem}
The \textit{recommender problem} consists of the entries of the sets $\mathcal{U}$ and $\mathcal{I}$, where $\mathcal{U}$ represents the set of all \textit{users} and $\mathcal{I}$ the set of all \textit{items}.
Each of the \textit{users} in $\mathcal{U}$ gives \textit{ratings} from a set $\mathcal{S}$ of possible scores for the available \textit{items} in $\mathcal{I}$. The resulting \textit{rating-matrix}$\mathcal{R}$ is composed of $\mathcal{R}=\mathcal{U}\times\mathcal{I}$. The entries in $\mathcal{R}$ indicate the \textit{rating} from \textit{user}$u \in\mathcal{U}$ to \textit{item}$i \in\mathcal{I}$. This entry is then referred to as $r_{ui}$. Due to incomplete \textit{item-ratings}, $\mathcal{R}$ may also be incomplete. In the following, the subset of all \textit{users} who have rated a particular \textit{item}$i$ is referred to as $\mathcal{U}_i$. Similarly, $\mathcal{I}_u$ refers to the subset of \textit{items} that were rated by \textit{user}$u$. Since $\mathcal{R}$ is not completely filled, there are missing values for some \textit{user-item relations}. The aim of the \textit{recommender system} is to estimate the missing \textit{ratings}$\hat{r}_{ui}$ using a \textit{prediction-function}$p(u,i)$. The \textit{prediction-function} consists of $p: \mathcal{U}\times\mathcal{I}\rightarrow\mathcal{S}$\citep{DeKa11}. In the further course of the work different methods are presented to determine $p(u,i)$.
In the following, the two main approaches of \textit{collaborative-filtering} and \textit{content-based}\textit{recommender systems} will be discussed. In addition, it is explained how \textit{matrix factorization} can be integrated into the two ways of thinking.
