diff --git a/content-based-collaborative-filtering-comparison.tex b/content-based-collaborative-filtering-comparison.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b2d0157ffefc9bf2e7f31d0d1fa56c65643b7a61
--- /dev/null
+++ b/content-based-collaborative-filtering-comparison.tex
@@ -0,0 +1,15 @@
+\begin{figure}[!ht]
+ \centering
+ \begin{subfigure}[b]{0.25\linewidth}
+ \includegraphics[width=\linewidth]{Bilder/ContendBasedFlow.jpg}
+ \caption{\textit{Content-Based}.}
+ \label{fig:cb}
+ \end{subfigure}
+ \begin{subfigure}[b]{0.25\linewidth}
+ \includegraphics[width=\linewidth]{Bilder/CollaborativeFlow.jpg}
+ \caption{\textit{Collaborative-Filtering}.}
+ \label{fig:cf}
+ \end{subfigure}
+ \caption{Overview of \textit{content-based} (left) and \textit{collaborative-filtering} (right) \textit{recommender systems}. \textit{Content-based recommender systems} work via \textit{feature vectors}. In contrast, \textit{collaborative filtering recommender systems} work over \textit{neighborhoods}.}
+ \label{fig:cbcf}
+\end{figure}
diff --git a/content-based.tex b/content-based.tex
deleted file mode 100644
index db246f6e7e0dad8657c2bfcfb175ddb88e95325b..0000000000000000000000000000000000000000
--- a/content-based.tex
+++ /dev/null
@@ -1,6 +0,0 @@
-\begin{figure}[htbp!]
- \centering
- \includegraphics[scale=0.5]{Bilder/ContendBasedFlow.jpg}
- \caption{\textit{Content-Based recommender systems} work via \textit{feature vectors}. These \textit{vectors} are learned or created using a variety of methods to model the \textit{user's preferences}. A suggestion is determined by the similarity between the \textit{feature vector} of the \textit{user} and the \textit{items}.}
- \label{img:content-based}
-\end{figure}
\ No newline at end of file
diff --git a/recommender.tex b/recommender.tex
index 11efb3f477e05a67ebe862720b67ef3bd84273c7..d07e556a8f717da7cb9ca8c9422e51f7e08ae93b 100644
--- a/recommender.tex
+++ b/recommender.tex
@@ -8,10 +8,23 @@ Each of the \textit{users} in $\mathcal{U}$ gives \textit{ratings} from a set $\
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} 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{neural networks} \citep{Lops11, Ferrari19}. 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}.
-
-\input{content-based}
+\textit{Content-based} \textit{recommender systems} 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{neural 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}, 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}.
+Missing \textit{user ratings} can be extracted by this \textit{neighbourhood} and \textit{networked} to form a whole. It is assumed that a \textit{missing rating} of the considered \textit{user} for an unknown \textit{item} $i$ will be similar to the \textit{rating} of a \textit{user} $v$ as soon as $u$ and $v$ have rated some \textit{items} similarly. The similarity of the \textit{users} is determined by the \textit{community ratings}. This type of \textit{recommender system} is also known by the term \textit{neighborhood-based recommender} \citep{DeKa11}. The main focus of \textit{neighbourhood-based methods} is on the application of iterative methods such as \textit{k-nearest-neighbours} or \textit{k-means}.
+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 items is determined by their received 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 the \textit{collaborative-filtering recommender}.
+
+\input{content-based-collaborative-filtering-comparison}
+
+\subsection{Matrix-Factorization}
+The core idea of \textit{matrix factorization} is to supplement the not completely filled out \textit{rating-matrix} $\mathcal{R}$. For this purpose the \textit{users} and \textit{items} are to be mapped to a joined \textit{latent feature space} with \textit{dimensionality} $f$. The \textit{user} is represented by the vector $p_u \in \mathbb{R}^{f}$ and the item by the vector $q_i \in \mathbb{R}^{f}$. As a result, the \textit{missing ratings} and thus the \textit{user-item interaction} are to be determined via the \textit{inner product} $\hat{r}_{ui}=q_i^Tp_u$ of the corresponding vectors \citep{Kor09}. In the following, the four most classical matrix factorization approaches are described in detail. Afterwards, the concrete learning methods with which the vectors are learned are presented. In addition, the \textit{training data} for which a \textit{concrete rating} is available should be referred to as $\mathcal{B} = \lbrace(u,i) | r_{ui} \in \mathcal{R}\rbrace$.
+
+\subsubsection{Basic Matrix-Factorization}
+The first and easiest way to solve \textit{matrix-factorization} is to connect the \textit{feature vectors} of the \textit{users} and the \textit{items} using the \textit{inner product}. The result is the \textit{user-item interaction}. In addition, the \textit{error} should be as small as possible. Therefore, $\min_{p_u, q_i}{\sum_{(u,i) \in \mathcal{B}} (r_{ui} - \hat{r}_{ui})^{2}}$ is defined as an associated \textit{minimization problem} for the \textit{RMSE}.
-\subsection{Matrix-Factorization}
\ No newline at end of file
+\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.
diff --git a/references.bib b/references.bib
index bd65f8b76cdce2862f559abaaefa4fececd655f3..df05aebe6ccf0e4450d68c164cd9d47b28fb820e 100644
--- a/references.bib
+++ b/references.bib
@@ -10,7 +10,6 @@
abstract = {A measure for proximity between documents is defined, based on data from readers. This proximity measure can be further investigated as a tool document retrieval, and as to provide data for concept formation experiments. },
year = {1990}
}
-
@article{Rendle19,
author = {Steffen Rendle and
Li Zhang and
@@ -52,7 +51,6 @@ editor = {P.B. Kantor and F. Ricci and L. Rokach and B. Shapira},
publisher={Springer},
doi = {10.1007/978-0-387-85820-3_4}
}
-
@inproceedings{Ferrari19,
author = {Maurizio Ferrari Dacrema and Paolo Cremonesi and Dietmar Jannach},
year = {2019},
@@ -61,4 +59,15 @@ pages = {},
title = {Are We Really Making Much Progress? A Worrying Analysis of Recent Neural Recommendation Approaches},
isbn = {978-1-4503-6243-6},
doi = {10.1145/3298689.3347058}
+}
+@article{Kor09,
+author = {Yehuda Koren and
+ Robert Bell and
+ Chris Volinsky},
+year = {2009},
+month = {08},
+pages = {30-37},
+title = {Matrix factorization techniques for recommender systems},
+volume = {42},
+journal = {Computer}
}
\ No newline at end of file
diff --git a/submission.pdf b/submission.pdf
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diff --git a/submission.tex b/submission.tex
index b0c8b343ac024d67a46e1065424c6b1b258669d9..6ec8f08af01ae4bee1d7b50762c8e6f9c4c44fb3 100644
--- a/submission.tex
+++ b/submission.tex
@@ -32,9 +32,10 @@
\usepackage[]{titlesec}
\titlespacing*{\section}
{0pt}{6pt}{6pt}
-\usepackage[]{titlesec}
\titlespacing*{\subsection}
{0pt}{6pt}{6pt}
+\titlespacing*{\subsubsection}
+{0pt}{6pt}{6pt}
\usepackage{footmisc}
\setlength{\abovedisplayskip}{0pt}
\renewcommand{\footrulewidth}{0.5pt}
@@ -47,7 +48,7 @@
\hypersetup{
colorlinks,
citecolor=hhuUniBlau,
- linkcolor=black,
+ linkcolor=hhuUniBlau,
urlcolor=hhuUniBlau}
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