added introductions to nn

.gitignore

@@ -0,0 +1,12 @@
+# weird latex files
+*.log
+*.aux
+*.toc
+*.gz
+*.xml
+TeX/auto/*
+main-blx.bib
+
+# emacs autosaves
+*.tex~
+

TeX/introduction_nn.tex

@@ -0,0 +1,124 @@
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "main"
+%%% End:
+\section{Introduction to Neural Networks}
+
+Neural Networks (NN) are a mathematical construct inspired by the
+connection of neurons in nature. It consists of an input and output
+layer with an arbitrary amount of hidden layers between them. Each
+layer consits of a numer of neurons (nodes) with the number of nodes
+in the in-/output layers corresponding to the dimensions of the
+in-/output.\par
+Each neuron recieves the output of all layers in the previous layers,
+except for the input layer, which recieves the components of the input.
+
+\tikzset{%
+  every neuron/.style={
+    circle,
+    draw,
+    minimum size=1cm
+  },
+  neuron missing/.style={
+    draw=none, 
+    scale=1.5,
+    text height=0.333cm,
+    execute at begin node=\color{black}$\vdots$
+  },
+}
+\begin{figure}[h!]
+  \center
+  
+  \fbox{
+    
+    \resizebox{\textwidth}{!}{%
+      \begin{tikzpicture}[x=1.75cm, y=1.75cm, >=stealth]
+    
+        \foreach \m/\l [count=\y] in {1,2,3,missing,4}
+        \node [every neuron/.try, neuron \m/.try] (input-\m) at (0,2.5-\y) {};
+        
+        \foreach \m [count=\y] in {1,missing,2}
+        \node [every neuron/.try, neuron \m/.try ] (hidden1-\m) at (2,2-\y*1.25) {};
+        
+        \foreach \m [count=\y] in {1,missing,2}
+        \node [every neuron/.try, neuron \m/.try ] (hidden2-\m) at (5,2-\y*1.25) {};
+        
+        \foreach \m [count=\y] in {1,missing,2}
+        \node [every neuron/.try, neuron \m/.try ] (output-\m) at (7,1.5-\y) {};
+        
+        \foreach \l [count=\i] in {1,2,3,d_i}
+        \draw [<-] (input-\i) -- ++(-1,0)
+        node [above, midway] {$x_{\l}$};
+        
+        \foreach \l [count=\i] in {1,n_1}
+        \node [above] at (hidden1-\i.north) {$\mathcal{N}_{1,\l}$};
+        
+        \foreach \l [count=\i] in {1,n_l}
+        \node [above] at (hidden2-\i.north) {$\mathcal{N}_{l,\l}$};
+        
+        \foreach \l [count=\i] in {1,d_o}
+        \draw [->] (output-\i) -- ++(1,0)
+        node [above, midway] {$O_{\l}$};
+        
+        \foreach \i in {1,...,4}
+        \foreach \j in {1,...,2}
+        \draw [->] (input-\i) -- (hidden1-\j);
+        
+        \foreach \i in {1,...,2}
+        \foreach \j in {1,...,2}
+        \draw [->] (hidden1-\i) -- (hidden2-\j);
+        
+        \foreach \i in {1,...,2}
+        \foreach \j in {1,...,2}
+        \draw [->] (hidden2-\i) -- (output-\j);
+        
+        \node [align=center, above] at (0,2) {Input\\layer};
+        \node [align=center, above] at (2,2) {Hidden \\layer $1$};
+        \node [align=center, above] at (5,2) {Hidden \\layer $l$};
+        \node [align=center, above] at (7,2) {Output \\layer};
+        
+        \node[fill=white,scale=1.5,inner xsep=10pt,inner ysep=10mm] at ($(hidden1-1)!.5!(hidden2-2)$) {$\dots$};
+    
+      \end{tikzpicture}}}
+  \caption{test}
+\end{figure}
+
+\begin{tikzpicture}[x=1.5cm, y=1.5cm, >=stealth]
+
+\foreach \m/\l [count=\y] in {1}
+  \node [every neuron/.try, neuron \m/.try] (input-\m) at (0,0.5-\y) {};
+
+\foreach \m [count=\y] in {1,2,missing,3,4}
+  \node [every neuron/.try, neuron \m/.try ] (hidden-\m) at (1.25,3.25-\y*1.25) {};
+
+\foreach \m [count=\y] in {1}
+  \node [every neuron/.try, neuron \m/.try ] (output-\m) at (2.5,0.5-\y) {};
+
+\foreach \l [count=\i] in {1}
+  \draw [<-] (input-\i) -- ++(-1,0)
+    node [above, midway] {$x$};
+
+\foreach \l [count=\i] in {1,2,n-1,n}
+  \node [above] at (hidden-\i.north) {$\mathcal{N}_{\l}$};
+
+\foreach \l [count=\i] in {1,n_l}
+  \node [above] at (output-\i.north) {};
+
+\foreach \l [count=\i] in {1}
+  \draw [->] (output-\i) -- ++(1,0)
+    node [above, midway] {$y$};
+
+\foreach \i in {1}
+  \foreach \j in {1,2,...,3,4}
+    \draw [->] (input-\i) -- (hidden-\j);
+
+\foreach \i in {1,2,...,3,4}
+  \foreach \j in {1}
+    \draw [->] (hidden-\i) -- (output-\j);
+
+\node [align=center, above] at (0,1) {Input\\layer};
+\node [align=center, above] at (1.25,3) {Hidden layer};
+\node [align=center, above] at (2.5,1) {Output\\layer};
+
+\end{tikzpicture}

TeX/main.tex

@@ -26,7 +26,10 @@
 \usepackage{makecell}
 \usepackage{dsfont}
 \usepackage{tikz}
-\usetikzlibrary{positioning}
+
+\usetikzlibrary{matrix,chains,positioning,decorations.pathreplacing,arrows}
+\usetikzlibrary{positioning,calc}
+
 \usepackage{pgfplots}
 \usepgfplotslibrary{colorbrewer}
 \usepackage{subcaption}
@@ -47,7 +50,7 @@
 \sectionfont{\centering}
  \input{insbox}
 
-\parindent0in
+%\parindent0in
 \pagestyle{plain}
 \thispagestyle{plain}
 \newtheorem{Theorem}{Theorem}[section]
@@ -83,6 +86,11 @@
 \tableofcontents 
 \newpage
 
+% Introduction Neural Networks
+\input{introduction_nn}
+
+\newpage
+
 % Theorem 3.8
 \input{theo_3_8.tex}
 

TeX/theo_3_8.tex

@@ -24,7 +24,7 @@ limes of RN as the amount of nodes is increased.
            g_{\xi}(x)\mathbb{E}\left[ v_k^2 \vert \xi_k = x \right], \forall x
            \in \mathbb{R}
   \end{align*}
-  and \(RN^{*, \tilde{\lambda}}}\), \(f^{*,\tilde{\lambda}}_{g, \pm}\)
+  and \(RN^{*, \tilde{\lambda}}\), \(f^{*,\tilde{\lambda}}_{g, \pm}\)
 as defined in ??? and ??? respectively. 
 \end{Theorem}
 In order to proof Theo~\ref{theo:main1} we need to proof a number of