From bdab8079c343b1ef796c3380c08dcf54026f4233 Mon Sep 17 00:00:00 2001
From: feger <marc.feger@hhu.de>
Date: Thu, 30 Jan 2020 17:55:30 +0100
Subject: [PATCH] Refactor

---
 recommender.tex |   2 +-
 submission.pdf  | Bin 454569 -> 454591 bytes
 2 files changed, 1 insertion(+), 1 deletion(-)

diff --git a/recommender.tex b/recommender.tex
index 7e46978..e330ff7 100644
--- a/recommender.tex
+++ b/recommender.tex
@@ -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}.
 
 \subsubsection{Alternating Least Square}
diff --git a/submission.pdf b/submission.pdf
index d1069448e38f245ae380fe82dfd6fa922d86cea7..07e416a844f6afd944bec38723c494246c7a9756 100644
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