@@ -28,3 +28,15 @@ The first and easiest way to solve \textit{matrix-factorization} is to connect t
\subsubsection{Regulated Matrix-Factorization}
This problem extends the \textit{basic matrix factorization} by a \textit{regulation factor}$\lambda$ in the corresponding \textit{minimization problem}. Since $\mathcal{R}$ is thinly occupied, the effect of \textit{overfitting} may occur due to learning from the few known values. The problem with \textit{overfitting} is that the generated \textit{ratings} are too tight. To counteract this, the magnitudes of the previous vectors is taken into account. High magnitudes are punished by a factor $\lambda(\lVert q_i \rVert^2+\lVert p_u \lVert^2)$ in the \textit{minimization problem}. Overall, the \textit{minimization problem}$\min_{p_u, q_i}{\sum_{(u,i)\in\mathcal{B}}(r_{ui}-\hat{r}_{ui})^{2}}+\lambda(\lVert q_i \rVert^2+\lVert p_u \lVert^2)$ is to be solved.
A \textit{regulation factor}$\lambda$ is introduced in analogy to \textit{regulated matrix-factorization}. Additional \textit{weights}$\alpha$ and $\beta$ are introduced to take into account the individual magnitude of a vector. The \textit{minimization problem} then corresponds to $\min_{p_u, q_i}{\sum_{(u,i)\in\mathcal{B}}(r_{ui}-\hat{r}_{ui})^{2}}+\lambda(\alpha\lVert q_i \rVert^2+\beta\lVert p_u \lVert^2)$.
\subsubsection{Biased Matrix-Factorization}
A major advantage of \textit{matrix-factorization} is the ability to model simple relationships according to the application. Thus, an excellent data source cannot always be assumed. Due to the \textit{natural interaction} of the \textit{users} with the \textit{items}, \textit{preferences} arise. Such \textit{preferences} lead to \textit{behaviour patterns} which manifest themselves in the form of a \textit{bias} in the data. In principle, a \textit{bias} is not bad, but it must be taken into account when modeling the \textit{recommender system}.
The most popular model that takes \textit{bias} into account is called \textit{biased matrix-factorization}.
In addition, the \textit{missing rating} is no longer determined only by the \textit{inner product} of the two vectors $q_i$ and $p_u$. Rather, the \textit{bias} is also considered. Accordingly, a \textit{missing rating} is calculated by $\hat{r}_{ui}= b_{ui}+ q_i^Tp_u$, where $b_{ui}$ is the \textit{bias} of a \textit{user}$u$ and an \textit{item}$i$. The \textit{bias} is determined by $b_{ui}=\mu+ b_u + b_i$. The parameter $\mu$ is the \textit{global average} of all \textit{ratings}$r_{ui}\in\mathcal{R}$.
Furthermore, $b_u =\mu_u -\mu$ and $b_i =\mu_i -\mu$.
Here $\mu_u$ denotes the \textit{average} of all \textit{assigned ratings} of the \textit{user}$u$. Similarly, $\mu_i$ denotes the \textit{average} of all \textit{received ratings} of an \textit{item}$i$.
Thus $b_u$ indicates the \textit{deviation} of the \textit{average assigned rating} of a \textit{user} from the \textit{global average}. Similarly, $b_i$ indicates the \textit{deviation} of the \textit{average rating} of an item from the \textit{global average}.
In addition, the \textit{minimization problem} can be extended by the \textit{bias}. Accordingly, the \textit{minimization problem} is then $\min_{p_u, q_i}{\sum_{(u,i)\in\mathcal{B}}(r_{ui}-\hat{r}_{ui})^{2}}+\lambda(\lVert q_i \rVert^2+\lVert p_u \lVert^2+ b_u^2+ b_i^2)$.
