@@ -76,7 +76,7 @@ The best known and most common method when it comes to \textit{machine learning}
\end{algorithmic}
\end{algorithm}
At the beginning, the matrices $\mathcal{P}, \mathcal{Q}$ are filled with \textit{random numbers}. According to \citet{Funk06} this can be done by a \textit{gaussian-distribution}. Then, for each element in the \textit{training set}, the entries of the corresponding vectors $p_u \in\mathcal{P}, q_i \in\mathcal{Q}$ are recalculated on the basis of the \textit{error} that occurred in an \textit{epoch}. The parameters $\mu, \gamma$ are introduced to avoid \textit{over}- and \textit{underfitting}. These can be determined using \textit{grid-search} and \textit{k-fold cross-validation}. For the \textit{optimization} of the parameters $\mu$ and $\gamma$ the so-called \textit{grid-search} procedure is used. A \textit{grid} of possible parameters is defined before the analysis. This \textit{grid} consists of the sets $\Lambda$ and $\Gamma$. The \textit{grid-search} method then trains the algorithm to be considered with each possible pair of $(\lambda\in\Lambda, \gamma\in\Gamma)$. The models trained in this way are then tested using a \textit{k-fold cross-validation}. The data set is divided into $k$-equally large fragments. Each of the $k$ parts is used once as a test set while the remaining ($k-1)$ parts are used as training data. The average error is then determined via the $k$-\textit{folds} and entered into the \textit{grid}. Thus the pair $(\lambda\in\Lambda, \gamma\in\Gamma)$ can be determined for which the \textit{error} is lowest.
At the beginning, the matrices $\mathcal{P}, \mathcal{Q}$ are filled with \textit{random numbers}. According to \citet{Funk06} this can be done by a \textit{gaussian-distribution}. Then, for each element in the \textit{training set}, the entries of the corresponding vectors $p_u \in\mathcal{P}, q_i \in\mathcal{Q}$ are recalculated on the basis of the \textit{error} that occurred in an \textit{epoch}. The parameters $\lambda, \gamma$ are introduced to avoid \textit{over}- and \textit{underfitting}. These can be determined using \textit{grid-search} and \textit{k-fold cross-validation}. For the \textit{optimization} of the parameters $\lambda$ and $\gamma$ the so-called \textit{grid-search} procedure is used. A \textit{grid} of possible parameters is defined before the analysis. This \textit{grid} consists of the sets $\Lambda$ and $\Gamma$. The \textit{grid-search} method then trains the algorithm to be considered with each possible pair of $(\lambda\in\Lambda, \gamma\in\Gamma)$. The models trained in this way are then tested using a \textit{k-fold cross-validation}. The data set is divided into $k$-equally large fragments. Each of the $k$ parts is used once as a test set while the remaining ($k-1)$ parts are used as training data. The average error is then determined via the $k$-\textit{folds} and entered into the \textit{grid}. Thus the pair $(\lambda\in\Lambda, \gamma\in\Gamma)$ can be determined for which the \textit{error} is lowest.
This approach is also called \textit{Funk-SVD} or \textit{SVD} in combination with section \ref{subsec:rmf} and \ref{subsec:bmf}\citep{Rendle19}. The algorithm shown above can also be extended. Thus procedures like in section \ref{subsec:amf} can be solved. The second method from section \ref{subsec:amf} is then also called \textit{SVD++}. A coherent \textit{SGD} approach was given by \citet{Kor11}.
