From fe176caa193cc8c0907830e315788e004dab5024 Mon Sep 17 00:00:00 2001
From: feger <marc.feger@hhu.de>
Date: Tue, 21 Jan 2020 11:12:35 +0100
Subject: [PATCH] Add more text for experiments

---
 baselines.tex    |  16 ++++++++--------
 battle.tex       |   2 +-
 introduction.tex |   6 +++---
 recommender.tex  |   6 +++---
 submission.pdf   | Bin 440536 -> 440609 bytes
 submission.tex   |   1 +
 6 files changed, 16 insertions(+), 15 deletions(-)

diff --git a/baselines.tex b/baselines.tex
index 4bf1526..3b99e9a 100644
--- a/baselines.tex
+++ b/baselines.tex
@@ -20,7 +20,7 @@ It can be clearly stated that the \textit{existing baselines} have been \textit{
 \input{reported_results}
 
 \subsection{Experiment Realization}
-As the \textit{Netflix-Prize} has shown, \textit{research} and \textit{validation} is \textit{complex} even for very \textit{simple methods}. Not only during the \textit{Netflix-Prize} was intensive work done on researching \textit{existing} and \textit{new reliable methods}. The \textit{MovieLens10M-dataset} was used just as often. With their experiment the authors \textit{doubt} that the \textit{baselines} of \textit{MovieLens10M} are \textit{inadequate} for the evaluation of new methods. To test their hypothesis, the authors transferred all the findings from the \textit{Netflix-Prize} to the existing baselines of \textit{MovieLens10M}.
+As the \textit{Netflix-Prize} has shown, \textit{research} and \textit{validation} is \textit{complex} even for very \textit{simple methods}. Not only during the \textit{Netflix-Prize} was intensive work done on researching \textit{existing} and \textit{new reliable methods}. The \textit{MovieLens10M-dataset} was used just as often. With their experiment, the authors \textit{doubt} that the \textit{baselines} of \textit{MovieLens10M} are \textit{adequate} for the evaluation of new methods. To test their hypothesis, the authors transferred all the findings from the \textit{Netflix-Prize} to the existing baselines of \textit{MovieLens10M}.
 
 \subsubsection{Experiment Preparation}\label{sec:experiment_preparation}
 Before actually conducting the experiment, the authors took a closer look at the given \textit{baselines}. In the process, they noticed some \textit{systematic overlaps}. These can be taken from the \textit{table} below.
@@ -30,14 +30,14 @@ From the three aspects it can be seen that the models are fundamentally similar
 Thus, the authors examined the two learning methods \textit{stochastic gradient descent} and \textit{bayesian learning} in combination with \textit{biased matrix-factorization} before conducting the actual experiment. For $b_u = b_i = 0$ this is equivalent to \textit{regulated matrix-factorization (RSVD)}. In addition, for $\alpha = \beta = 1$ the \textit{weighted regulated matrix-factorization (WR)} is equivalent to \textit{RSVD}. Thus, the only differences are explained by the different adjustments of the methods.
 To prepare the two learning procedures they were initialized with a \textit{gaussian normal distribution} $\mathcal{N}(\mu, 0.1^2)$. The value for the \textit{standard deviation} of \textit{0.1} is the value suggested by the \textit{factorization machine libFM} as the default. In addition, \citet{Rendle13} achieved good results on the \textit{Netflix-Prize-dataset} with this value. Nothing is said about the parameter $\mu$. However, it can be assumed that this parameter is around the \textit{global average} of the \textit{ratings}. This can be assumed because \textit{ratings} are to be \textit{generated} with the \textit{initialization}.
 
-For both approaches the number of \textit{sampling steps} was then set to \textit{128}. Since \textit{SGD} has two additional \textit{hyperparameters} $\lambda, \gamma$ these were also determined. Overall, the \textit{MovieLens10M-dataset} was evaluated by a \textit{10-fold cross-validation} over a \textit{random global} and \textit{non-overlapping 90:10 split}. In each split, \textit{90\%} of the data was used for \textit{training} and \textit{10\%} of the data was used for \textit{evaluation} without overlapping. In each split, \textit{95\%} of the \textit{training data} was used for \textit{training} and the remaining \textit{5\%} for \textit{evaluation} to determine the \textit{hyperparameters}. The \textit{hyperparameter search} was performed as mentioned in \textit{section} \ref{sec:sgd} using the \textit{grid} $(\lambda \in \{0.02, 0.03, 0.04, 0.05\}, \gamma \in \{0.001, 0.003\})$. This grid was inspired by findings during the \textit{Netflix-Prize} \citep{Kor08, Paterek07}. In total the parameters $\lambda=0.04$ and $\gamma=0.003$ could be determined. Afterwards both \textit{learning methods} and their settings were compared. The \textit{RMSE} was plotted against the used \textit{dimension} $f$ of $p_u, q_i \in \mathbb{R}^f$. \textit{Figure} \ref{fig:battle} shows the corresponding results.
+For both approaches the number of \textit{sampling steps} was then set to \textit{128}. Since \textit{SGD} has two additional \textit{hyperparameters} $\lambda, \gamma$ these were also determined. Overall, the \textit{MovieLens10M-dataset} was evaluated by a \textit{10-fold cross-validation} over a \textit{random global} and \textit{non-overlapping 90:10 split}. In each step, \textit{90\%} of the data was used for \textit{training} and \textit{10\%} of the data was used for \textit{evaluation} without overlapping. In each split, \textit{95\%} of the \textit{training data} was used for \textit{training} and the remaining \textit{5\%} for \textit{evaluation} to determine the \textit{hyperparameters}. The \textit{hyperparameter search} was performed as mentioned in \textit{section} \ref{sec:sgd} using the \textit{grid} $(\lambda \in \{0.02, 0.03, 0.04, 0.05\}, \gamma \in \{0.001, 0.003\})$. This grid was inspired by findings during the \textit{Netflix-Prize} \citep{Kor08, Paterek07}. In total the parameters $\lambda=0.04$ and $\gamma=0.003$ could be determined. Afterwards both \textit{learning methods} and their settings were compared. The \textit{RMSE} was plotted against the used \textit{dimension} $f$ of $p_u, q_i \in \mathbb{R}^f$. \textit{Figure} \ref{fig:battle} shows the corresponding results.
 \input{battle}
 \newpage
 As a \textit{first intermediate result} of the preparation it can be stated that both \textit{SGD} and \textit{gibbs-samper} achieve better \textit{RMSE} values for increasing \textit{dimensional embedding}.
 
-In addition, it can be stated that learning using the \textit{bayesian approach} is better than learning using \textit{SGD}. Even if the results could be different due to more efficient setups, it is still surprising that \textit{SGD} is worse than the \textit{bayesian approach}, although the \textit{exact opposite} was reported for \textit{MovieLens10M}. For example, \textit{figure} \ref{fig:reported_results} shows that the \textit{bayesian approach BPMF} achieved an \textit{RMSE} of \textit{0.8187} while the \textit{SGD approach Biased MF} performed better with \textit{0.803}. The fact that the \textit{bayesian approach} outperforms \textit{SGD} has already been reported and validated by \citet{Rendle13}, \citet{Rus08} for the \textit{Netflix-Prize-dataset}. Looking more closely at \textit{figures} \ref{fig:reported_results} and \ref{fig:battle}, the \textit{bayesian approach} scores better than the reported \textit{BPMF} and \textit{Biased MF} for each \textit{dimensional embedding}. Moreover, it even beats all reported \textit{baselines} and new methods. Building on this, the authors have gone into the detailed examination of the methods and \textit{baselines}.
+In addition, it can be stated that learning using the \textit{bayesian approach} is better than learning using \textit{SGD}. Even if the results could be different due to more efficient setups, it is still surprising that \textit{SGD} is worse than the \textit{bayesian approach}, although the \textit{exact opposite} was reported for \textit{MovieLens10M-dataset}. For example, \textit{figure} \ref{fig:reported_results} shows that the \textit{bayesian approach BPMF} achieved an \textit{RMSE} of \textit{0.8187} while the \textit{SGD approach Biased MF} performed better with \textit{0.803}. The fact that the \textit{bayesian approach} outperforms \textit{SGD} has already been reported and validated by \citet{Rendle13}, \citet{Rus08} for the \textit{Netflix-Prize-dataset}. Looking more closely at \textit{figures} \ref{fig:reported_results} and \ref{fig:battle}, the \textit{bayesian approach} scores better than the reported \textit{BPMF} and \textit{Biased MF} for each \textit{dimensional embedding}. Moreover, it even beats all reported \textit{baselines} and new methods. Building on this, the authors have gone into the detailed examination of the methods and \textit{baselines}.
 \subsubsection{Experiment Implementation}
-For the actual execution of the experiment, the authors used the knowledge they had gained from the preparations. They noticed already for the two \textit{simple matrix-factorization models SGD-MF} and \textit{Bayesian MF}, which were trained with an \textit{embedding} of \textit{512 dimensions} and over \textit{128 epochs}, that they performed extremely well. Thus \textit{SGD-MF} achieved an \textit{RMSE} of \textit{0.7720}. This result alone was better than: \textit{RSVD (0.8256)}, \textit{Biased MF (0.803)}, \textit{LLORMA (0.7815)}, \textit{Autorec (0.782)}, \textit{WEMAREC (0.7769)} and \textit{I-CFN++ (0.7754)}. In addition, \textit{Bayesian MF} with an \textit{RMSE} of \textit{0.7653} not only beat the \textit{reported baseline BPMF (0.8197)}. It also beat the \textit{best algorithm MRMA (0.7634)}.
+For the actual execution of the experiment, the authors used the knowledge they had gained from the preparations. They noticed already for the two \textit{simple matrix-factorization models SGD-MF} and \textit{Bayesian MF}, which were trained with an \textit{embedding} of \textit{512 dimensions} and over \textit{128 epochs}, that they performed extremely well. Thus \textit{SGD-MF} achieved an \textit{RMSE} of \textit{0.7720}. This result alone was better than: \textit{RSVD (0.8256)}, \textit{Biased MF (0.803)}, \textit{LLORMA (0.7815)}, \textit{Autorec (0.782)}, \textit{WEMAREC (0.7769)} and \textit{I-CFN++ (0.7754)}. In addition, \textit{Bayesian MF} with an \textit{RMSE} of \textit{0.7633} not only beat the \textit{reported baseline BPMF (0.8197)}. It also beat the \textit{best algorithm MRMA (0.7634)}.
 As the \textit{Netflix-Prize} showed, the use of \textit{implicit data} such as \textit{time} or \textit{dependencies} between \textit{users} or \textit{items} could immensely improve existing models. In addition to the two \textit{simple matrix factorizations}, \textit{table} \ref{table:models} shows the extensions of the authors regarding the \textit{bayesian approach}.
 
 \input{model_table}
@@ -48,14 +48,14 @@ As it turned out that the \textit{bayesian approach} gave more promising results
 The first observation that emerges from \textit{figure} \ref{fig:bayes_sampling_steps} is that the \textit{increase} in \textit{sampling steps} with a \textit{fixed dimensional embedding} also results in an \textit{improvement} in \textit{RMSE} for all models. Based on this, \textit{figure} \ref{fig:bayes_dimensional_embeddings} also shows that an \textit{increase} in the \textit{dimensional embedding} for \textit{512 sampling steps} also leads to an \textit{improvement} in the \textit{RMSE} for all models. Thus, both the \textit{number of sampling steps} and the size of the \textit{dimensional embedding} are involved in the \textit{RMSE} of \textit{matrix-factorization models} when they are trained using the \textit{bayesian approach}.
 
 \subsubsection{Stronger Baselines}
-As a second finding, the \textit{RMSE} values of the created models can be taken from \textit{figure} \ref{fig:bayes_dimensional_embeddings}. Several points can be addressed. Firstly, it can be seen that the \textit{individual inclusion} of \textit{implicit knowledge} such as \textit{time} or \textit{user behaviour} leads to a significant \textit{improvement} in the \textit{RMSE}. For example, models like \textit{bayesian timeSVD (0.7587)} and \textit{bayesian SVD++ (0.7563)}, which already use \textit{single implicit knowledge}, beat the \textit{simple bayesian MF} with an \textit{RMSE} of \textit{0.7633}. In addition, it also shows that the \textit{combination} of \textit{implicit data} further improves the \textit{RMSE}. \textit{Bayesian timeSVD++} achieves an \textit{RMSE} of \textit{0.7523}. Finally, \textit{bayesian timeSVD++ flipped} can achieve an \textit{RMSE} of \textit{0.7485} by adding \textit{more implicit data}.
-This results in the third and most significant observation of the experiment. Firstly, the \textit{simple bayesian MF} with an \textit{RMSE} of \textit{0.7633} already beat the best method \textit{MRMA} with an \textit{RMSE} of \textit{0.7634}. Furthermore, the best method \textit{MRMA} could be surpassed with \textit{bayesian timeSVD++} by 0.0149 with respect to the \textit{RMSE}. Such a result is astonishing, as it took \textit{one year} during the \textit{Netflix-Prize} to reduce the leading \textit{RMSE} from \textit{0.8712 (progress award 2007)} to \textit{0.8616 (progress award 2008)}. Additionally, this result is remarkable as it \textit{challenges} the \textit{last five years} of research on the \textit{MovieLens10M-dataset}. Based on the results obtained, the \textit{authors} see the first problem with the \textit{results} achieved on the \textit{MovieLens10M-dataset} as being that they were \textit{compared against} too \textit{weak baselines}.
+As a second finding, the \textit{RMSE} values of the created models can be taken from \textit{figure} \ref{fig:bayes_dimensional_embeddings}. Several points can be addressed. Firstly, it can be seen that the \textit{individual inclusion} of \textit{implicit knowledge} such as \textit{time} or \textit{user behaviour} leads to a significant \textit{improvement} in the \textit{RMSE}. For example, models like \textit{Bayesian timeSVD (0.7587)} and \textit{Bayesian SVD++ (0.7563)}, which already use \textit{single implicit knowledge}, beat the \textit{simple Bayesian MF} with an \textit{RMSE} of \textit{0.7633}. In addition, it also shows that the \textit{combination} of \textit{implicit data} further improves the \textit{RMSE}. \textit{Bayesian timeSVD++} achieves an \textit{RMSE} of \textit{0.7523}. Finally, \textit{Bayesian timeSVD++ flipped} can achieve an \textit{RMSE} of \textit{0.7485} by adding \textit{more implicit data}.
+This results in the third and most significant observation of the experiment. Firstly, the \textit{simple Bayesian MF} with an \textit{RMSE} of \textit{0.7633} already beat the best method \textit{MRMA} with an \textit{RMSE} of \textit{0.7634}. Furthermore, the best method \textit{MRMA} could be surpassed with \textit{bayesian timeSVD++} by 0.0149 with respect to the \textit{RMSE}. Such a result is astonishing, as it took \textit{one year} during the \textit{Netflix-Prize} to reduce the leading \textit{RMSE} from \textit{0.8712 (progress award 2007)} to \textit{0.8616 (progress award 2008)}. Additionally, this result is remarkable as it \textit{challenges} the \textit{last five years} of research on the \textit{MovieLens10M-dataset}. Based on the results obtained, the \textit{authors} see the first problem with the \textit{results} achieved on the \textit{MovieLens10M-dataset} as being that they were \textit{compared against} too \textit{weak baselines}.
 From \textit{figure} \ref{fig:corrected_results} the \textit{improved baselines} and the \textit{results} of the \textit{new methods} can be examined.
 \input{corrected_results}
 \subsubsection{Reproducability}
 But where do these \textit{weak baselines} come from?
-In response, the authors see two main points. The first is \textit{reproducibility}. This is generally understood to mean the \textit{repetition} of an \textit{experiment} with the aim of \textit{obtaining} the \textit{specified results}. In most cases, the \textit{code} of the authors of a paper is taken and checked. Not only during the \textit{Netflix-Prize} this was a common method to compare competing methods, improve one's own and generally achieve \textit{stronger baselines}. However, the authors do not consider the \textit{simple repetition} of the experiment for the purpose of achieving the same results to be appropriate. Thus, the \textit{repetition} of the experiment only provides information about the results achieved by a specific setup. However, it does not provide deeper insights into the method, nor into its general quality. This is not only a problem of \textit{recommender-systems} but rather a general problem in the field of \textit{machine learning}. Thus, \textit{indicators} such as \textit{statistical significance}, \textit{reproducibility} or \textit{hyperparameter search} are often regarded as \textit{proof} of the quality of an experiment. But they only give information about a certain experiment, which could be performed with \textit{non-standard protocols}. The question of whether the method being used is applied and configured in a meaningful way is neglected. Thus, \textit{statistical significance} is often taken as an \textit{indication} that \textit{method A} \textit{performs better} than \textit{method B}.
+In response, the authors see two main points. The first is \textit{reproducibility}. This is generally understood to mean the \textit{repetition} of an \textit{experiment} with the aim of \textit{obtaining} the \textit{specified results}. In most cases, the \textit{code} of the authors of a paper is taken and checked. Not only during the \textit{Netflix-Prize}, this was a common method to compare competing methods, improve one's own and generally achieve \textit{stronger baselines}. However, the authors do not consider the \textit{simple repetition} of the experiment for the purpose of achieving the same results to be appropriate. Thus, the \textit{repetition} of the experiment only provides information about the results achieved by a specific setup. However, it does not provide deeper insights into the method, nor into its general quality. This is not only a problem of \textit{recommender-systems} but rather a general problem in the field of \textit{machine learning}. Thus, \textit{indicators} such as \textit{statistical significance}, \textit{reproducibility} or \textit{hyperparameter search} are often regarded as \textit{proof} of the quality of an experiment. But they only give information about a certain experiment, which could be performed with \textit{non-standard protocols}. The question of whether the method being used is applied and configured in a meaningful way is neglected. Thus, \textit{statistical significance} is often taken as an \textit{indication} that \textit{method A} \textit{performs better} than \textit{method B}.
 
 \subsubsection{Inadequate validations}
-The authors do not doubt the relevance of such methods. They even consider them \textit{necessary} but \textit{not meaningful enough} for the \textit{general goodness} of an \textit{experiment}. Thus, their preparation, which takes up the above mentioned methods, shows that they can achieve meaningful results.
+The authors do not doubt the relevance of such methods. They even consider them \textit{necessary} but \textit{not meaningful enough} for the \textit{general goodness} of an \textit{experiment}. Thus, their preparation, which takes up the above mentioned methods shows, that they can achieve meaningful results.
 Therefore the authors see the second point of criticism of the results obtained on the \textit{MovieLens10M-dataset} as the \textit{wrong understanding} of \textit{reliable experiments}. The \textit{main reason} given is the \textit{difference} between \textit{scientific} and \textit{industrial work}. For example, during the\textit{ Netflix-Prize}, which represents \textit{industrial work}, \textit{audible sums} were \textit{awarded} for the best results. This had several consequences. Firstly, a \textit{larger community} was addressed to work on the solution of the \textit{recommender problem}. On the other hand, the high number of \textit{competitors} and the \textit{simplicity} in the formulation of the task encouraged each participant to investigate the \textit{simplest methods} in \textit{small steps}. The \textit{small-step approach} was also driven by the \textit{standardized guidelines} for the \textit{evaluation} of the methods given in \textit{section} \ref{sec:netflix} and by the \textit{public competition}. Thus, a better understanding of the \textit{basic relationships} could be achieved through the \textit{miniscule evaluation} of hundreds of models. All in all, these insights led to \textit{well-understood} and \textit{sharp baselines} within a \textit{community} that \textit{continuously} worked towards a \textit{common goal} over a total of \textit{three years}. Such a \textit{motivation} and such a \textit{target-oriented competitive idea} is mostly not available in the \textit{scientific field}. Thus, publications that achieve \textit{better results} with \textit{old methods} are considered \textit{unpublishable}. Instead, experiments are \textit{not questioned} and their \textit{results} are \textit{simply transferred}. In some cases experiments are \textit{repeated exactly as specified} in the instructions. Achieving the \textit{same result} is considered a \textit{valid baseline}. According to the authors, such an approach is \textit{not meaningful} and, by not questioning the \textit{one-off evaluations}, leads to \textit{one-hit-wonders} that \textit{distort} the \textit{sharpness} of the \textit{baselines}. Therefore, the \textit{MovieLens10M-dataset} shows that the main results of the last \textit{five years} were \textit{measured} against too \textit{weak baselines}.
diff --git a/battle.tex b/battle.tex
index e2ee3d8..e733f91 100644
--- a/battle.tex
+++ b/battle.tex
@@ -1,7 +1,7 @@
 \begin{figure}[!ht]
   \centering
     \includegraphics[scale=0.60]{Bilder/battle.png}
-  \caption{Comparison of \textit{matrix-factorization} learned by \textit{gibbs-sampling (bayesian learning)} and \textit{stochastic gradient descent (SGD)} for an \textit{embedding dimension} from \textit{16} to \textit{512} with \textit{128} \textit{sampling-steps}.
+  \caption{Comparison of \textit{matrix-factorization} learned by \textit{gibbs-sampling (bayesian learning)} and \textit{stochastic gradient descent (SGD)} for an \textit{embedding dimension} from \textit{16} to \textit{512} with \textit{128} \textit{sampling steps}.
 }
 \label{fig:battle}
 \end{figure}
diff --git a/introduction.tex b/introduction.tex
index 8a317ef..4e78991 100644
--- a/introduction.tex
+++ b/introduction.tex
@@ -8,15 +8,15 @@ Since \citet{JuKa90} first presented \textit{recommender systems} as a kind of i
 The most diverse subject areas were not only illuminated by the industry.
  A whole new branch of research also opened up for science.
  
- In their work ``\textit{On the Diffculty of Evaluating Baselines A Study on Recommender Systems}`` \citet{Rendle19} show that current research on the \textit{MovieLens10M-dataset} leads in a wrong direction.
+In their work ``\textit{On the Diffculty of Evaluating Baselines A Study on Recommender Systems}`` \citet{Rendle19} show that current research on the \textit{MovieLens10M-dataset} leads in a wrong direction.
  In addition to general problems, they particulary list wrong working methods and missunderstood \textit{baselines} by breaking them by a number of simple methods such as \textit{matrix-factorization}.
  
- They were able to beat the existing \textit{baselines} by not taking them for granted.
+They were able to beat the existing \textit{baselines} by not taking them for granted.
  On the contrary, they questioned them and transferred well evaluated and understood properties of the \textit{baselines} from the \textit{Netflix-Prize} to them.
 
 As a result, they were not only able to beat the \textit{baselines} reported for the \textit{MovieLens10M-dataset}, but also the newer methods from the last five years of research. Therefore, it can be assumed that the current and former results obtained on the \textit{MovieLens10M-dataset} were not sufficient to be considered as a true \textit{baseline}. Thus they show the \textit{community} a critical error on which can be found not only in the evaluation of \textit{recommender systems} but also in other scientific areas.
 
-The first problem, the authors point out that scientific papers whose focus is on better understanding and improving existing \textit{baselines} do not receive recognition because they do not seem innovative enough. In contrast to industry, which tenders horrendous prizes for researching and improving such \textit{baselines}, there is a lack of such motivation in the scientific field. From the authors point of view, the scientific work on the \textit{MovieLens10M-dataset} is misdirected, because \textit{one-off evaluations} leading to \textit{one-hit-wonders}, which are then used as a starting point for further work. Thus \citet{Rendle19} points out as a second point of criticism that the need for further basic research for the \textit{MovieLens10M-dataset} is not yet exhausted.
+The first problem the authors point out that, scientific papers whose focus is on better understanding and improving existing \textit{baselines} do not receive recognition because they do not seem innovative enough. In contrast to industry, which tenders horrendous prizes for researching and improving such \textit{baselines}, there is a lack of such motivation in the scientific field. From the authors point of view, the scientific work on the \textit{MovieLens10M-dataset} is misdirected, because \textit{one-off evaluations} leading to \textit{one-hit-wonders}, which are then used as a starting point for further work. Thus \citet{Rendle19} points out as a second point of criticism, that the need for further basic research for the \textit{MovieLens10M-dataset} is not yet exhausted.
 
 This submission takes a critical look at the topic presented by \citet{Rendle19}. In addition, basic terms and the results obtained are presented in a way that is comprehensible to the non-experienced reader.
 For this purpose, the submission is divided into three subject areas. First of all, the non-experienced reader is introduced to the topic of \textit{recommender systems} in the section ``\textit{A Study on Recommender Systems}``. Subsequently, building on the first section, the work in the section ``\textit{On the Diffculty of Evaluating Baselines}`` is presented in detail. The results are then evaluated in a critical discourse.
\ No newline at end of file
diff --git a/recommender.tex b/recommender.tex
index d5e5710..3d92371 100644
--- a/recommender.tex
+++ b/recommender.tex
@@ -3,12 +3,12 @@ This section explains the basics of \textit{recommender systems} necessary for t
 
 \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 $p(u,i)$.
+Each of the \textit{users} in $\mathcal{U}$ gives \textit{ratings} from a set $\mathcal{S}$ of possible \textit{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.
 
 \subsection{Content-Based}
-\textit{Content-based} \textit{recommender systems (CB)} work directly with \textit{feature vectors}. Such a \textit{feature vector} can, for example, represent a \textit{user profile}. In this case, this \textit{profile} contains information about the \textit{user's preferences}, such as \textit{genres}, \textit{authors}, \textit{etc}.  This is done by trying to create a \textit{model} of the \textit{user}, which best represents his preferences. The different \textit{learning algorithms} from the field of \textit{machine learning} are used to learn or create the \textit{models}. The most prominent \textit{algorithms} are: \textit{tf-idf}, \textit{bayesian learning}, \textit{Rocchio's algorithm} and \textit{neuronal networks} \citep{Lops11, Ferrari19, DeKa11}. Altogether the built and learned \textit{feature vectors} are compared with each other. Based on their closeness, similar \textit{features} can be used to generate \textit{missing ratings}. Figure \ref{fig:cb} shows a sketch of the general operation of \textit{content-based recommenders}.
+\textit{Content-based} \textit{recommender systems (CB)} work directly with \textit{feature vectors}. Such a \textit{feature vector} can, for example, represent a \textit{user profile}. In this case, this \textit{profile} contains informations about the \textit{user's preferences}, such as \textit{genres}, \textit{authors}, \textit{etc}.  This is done by trying to create a \textit{model} of the \textit{user}, which best represents his preferences. The different \textit{learning algorithms} from the field of \textit{machine learning} are used to learn or create the \textit{models}. The most prominent \textit{algorithms} are: \textit{tf-idf}, \textit{bayesian learning}, \textit{Rocchio's algorithm} and \textit{neuronal networks} \citep{Lops11, Ferrari19, DeKa11}. Altogether the built and learned \textit{feature vectors} are compared with each other. Based on their closeness, similar \textit{features} can be used to generate \textit{missing ratings}. Figure \ref{fig:cb} shows a sketch of the general operation of \textit{content-based recommenders}.
 
 \subsection{Collaborative-Filtering}
 Unlike the \textit{content-based recommender (CF)}, the \textit{collaborative-filtering recommender} not only considers individual \textit{users} and \textit{feature vectors}, but rather a \textit{like-minded neighborhood} of each \textit{user}.
@@ -16,7 +16,7 @@ Missing \textit{user ratings} can be extracted by this \textit{neighbourhood} an
 A \textit{neighborhood-based recommender} can be viewed from two angles: The first and best known problem is the so-called \textit{user-based prediction}.  Here, the \textit{missing ratings} of a considered \textit{user} $u$ are to be determined from his \textit{neighborhood} $\mathcal{N}_i(u)$. 
 $\mathcal{N}_i(u)$ denotes the subset of the \textit{neighborhood} of all \textit{users} who have a similar manner of evaluation to $u$ via the \textit{item} $i$. The second problem is that of \textit{item-based prediction}. Analogously, the similarity of the \textit{items} are determined by their received \textit{ratings}.
 This kind of problem consideres the \textit{neighborhood} $\mathcal{N}_u(i)$ of all \textit{items} $i$ which were similar rated via the \textit{user} $u$. The similarity between the objects of a \textit{neighborhood} is determined by \textit{distance functions} such as \textit{mean-squared-difference}, \textit{pearson-correlation} or \textit{cosine-similarity}.
-Figure \ref{fig:cf} shows a sketch of the general operation of \textit{content-based} and \textit{collaborative-filtering} \textit{recommender}.
+Figure \ref{fig:cf} shows a sketch of the general operation of \textit{collaborative-filtering} \textit{recommender}.
 
 \input{content-based-collaborative-filtering-comparison}
 
diff --git a/submission.pdf b/submission.pdf
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diff --git a/submission.tex b/submission.tex
index fecf820..f5036da 100644
--- a/submission.tex
+++ b/submission.tex
@@ -36,6 +36,7 @@
 {0pt}{6pt}{6pt}
 \titlespacing*{\subsubsection}
 {0pt}{6pt}{6pt}
+
 \usepackage{footmisc}
 \setlength{\abovedisplayskip}{0pt}
 \renewcommand{\footrulewidth}{0.5pt}
-- 
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