added poincare type inequality and neuron diagram

TeX/introduction_nn.tex

@@ -1,8 +1,4 @@
 
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "main"
-%%% End:
 \section{Introduction to Neural Networks}
 
 Neural Networks (NN) are a mathematical construct inspired by the
@@ -34,6 +30,8 @@ except for the input layer, which recieves the components of the input.
     
     \resizebox{\textwidth}{!}{%
       \begin{tikzpicture}[x=1.75cm, y=1.75cm, >=stealth]
+    \tikzset{myptr/.style={decoration={markings,mark=at position 1 with %
+      {\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate}}}
         
         \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) {};
@@ -48,7 +46,7 @@ except for the input layer, which recieves the components of the input.
         \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)
+        \draw [myptr] (input-\i)+(-1,0) -- (input-\i)
         node [above, midway] {$x_{\l}$};
         
         \foreach \l [count=\i] in {1,n_1}
@@ -58,20 +56,20 @@ except for the input layer, which recieves the components of the input.
         \node [above] at (hidden2-\i.north) {$\mathcal{N}_{l,\l}$};
         
         \foreach \l [count=\i] in {1,d_o}
-        \draw [->] (output-\i) -- ++(1,0)
+        \draw [myptr] (output-\i) -- ++(1,0)
         node [above, midway] {$O_{\l}$};
         
         \foreach \i in {1,...,4}
         \foreach \j in {1,...,2}
-        \draw [->] (input-\i) -- (hidden1-\j);
+        \draw [myptr] (input-\i) -- (hidden1-\j);
         
         \foreach \i in {1,...,2}
         \foreach \j in {1,...,2}
-        \draw [->] (hidden1-\i) -- (hidden2-\j);
+        \draw [myptr] (hidden1-\i) -- (hidden2-\j);
         
         \foreach \i in {1,...,2}
         \foreach \j in {1,...,2}
-        \draw [->] (hidden2-\i) -- (output-\j);
+        \draw [myptr] (hidden2-\i) -- (output-\j);
         
         \node [align=center, above] at (0,2) {Input\\layer};
         \node [align=center, above] at (2,2) {Hidden \\layer $1$};
@@ -84,7 +82,10 @@ except for the input layer, which recieves the components of the input.
   \caption{test}
 \end{figure}
 
-\begin{tikzpicture}[x=1.5cm, y=1.5cm, >=stealth]
+\begin{figure}
+  \begin{tikzpicture}[x=1.5cm, y=1.5cm]
+    \tikzset{myptr/.style={decoration={markings,mark=at position 1 with %
+      {\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate}}}
     
     \foreach \m/\l [count=\y] in {1}
     \node [every neuron/.try, neuron \m/.try] (input-\m) at (0,0.5-\y) {};
@@ -96,7 +97,7 @@ except for the input layer, which recieves the components of the input.
     \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)
+    \draw [myptr] (input-\i)+(-1,0) -- (input-\i)
     node [above, midway] {$x$};
 
     \foreach \l [count=\i] in {1,2,n-1,n}
@@ -106,19 +107,127 @@ except for the input layer, which recieves the components of the input.
     \node [above] at (output-\i.north) {};
 
     \foreach \l [count=\i] in {1}
-  \draw [->] (output-\i) -- ++(1,0)
+    \draw [myptr, >=stealth] (output-\i) -- ++(1,0)
     node [above, midway] {$y$};
     
+
     \foreach \i in {1}
     \foreach \j in {1,2,...,3,4}
-    \draw [->] (input-\i) -- (hidden-\j);
+    \draw [myptr, >=stealth] (input-\i) -- (hidden-\j);
 
     \foreach \i in {1,2,...,3,4}
     \foreach \j in {1}
-    \draw [->] (hidden-\i) -- (output-\j);
+    \draw [myptr, >=stealth] (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}
+  \caption{Shallow Neural Network with input- and output-dimension of \(d
+    = 1\)}
+\end{figure}
+
+
+\begin{figure}
+  \begin{tikzpicture}[x=1.5cm, y=1.5cm, >=stealth]
+
+    
+    \tikzset{myptr/.style={decoration={markings,mark=at position 1 with %
+      {\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate}}}
+    
+    \node [circle, draw, fill=black, inner sep = 0pt, minimum size =
+    1.5mm, left] (i_1) at (0, 2.5) {};
+    \node [align=left, left] at (-0.125, 2.5) {\(i_1\)};
+    \node [circle, draw, fill=black, inner sep = 0pt, minimum size =
+    1.5mm] (i_2) at (0, 1.25) {};
+    \node [align=left, left] at (-0.125, 1.25) {\(i_2\)};
+    \node [neuron missing] (i_3) at (0, 0) {};
+    \node [circle, draw, fill=black, inner sep = 0pt, minimum size =
+    1.5mm] (i_4) at (0, -1.25) {};
+    \node [align=left, left] at (-0.125, -1.25) {\(i_m\)};
+    \draw[decoration={calligraphic brace,amplitude=5pt, mirror}, decorate, line width=1.25pt]
+    (-0.6,2.7) -- (-0.6,-1.45) node [black, midway, xshift=-0.6cm, left] {Input};
+
+    \node [align = center, above] at (1.25, 3) {Synaptic\\weights};
+    \node [every neuron] (w_1) at (1.25, 2.5) {\(w_{k, 1}\)};
+    \node [every neuron] (w_2) at (1.25, 1.25) {\(w_{k, 2}\)};
+    \node [neuron missing] (w_3) at (1.25, 0) {};
+    \node [every neuron] (w_4) at (1.25, -1.25) {\(w_{k, m}\)};
+
+    \node [circle, draw] (sig) at (3, 0.625) {\Large\(\sum\)};
+    \node [align = center, below] at (3, 0) {Summing \\junction};
+
+    \node [draw, minimum size = 1.25cm] (act) at (4.5, 0.625)
+    {\(\psi(.)\)};
+    \node [align = center, above] at (4.5, 1.25) {Activation \\function}; 
+
+    \node [circle, draw, fill=black, inner sep = 0pt, minimum size =
+    1.5mm] (b) at (3, 2.5) {};
+    \node [align = center, above] at (3, 2.75) {Bias \\\(b_k\)};
+
+    \node [align = center] (out) at (6, 0.625) {Output \\\(o_k\)};
+
+
+    \draw [myptr] (i_1) -- (w_1);
+    \draw [myptr] (i_2) -- (w_2);
+    \draw [myptr] (i_4) -- (w_4);
+
+    \draw [myptr] (w_1) -- (sig);
+    \draw [myptr] (w_2) -- (sig);
+    \draw [myptr] (w_4) -- (sig);
+
+    \draw [myptr] (b) -- (sig);
+
+    \draw [myptr] (sig) -- (act);
+
+    \draw [myptr] (act) -- (out);
+
+    % \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) {\(w_{k,\y}\)};
+
+    % \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);
+
+  \end{tikzpicture}
+  \caption{Structure of a single neuron}
+\end{figure}
+
+\begin{tikzpicture}
+\tikzset{myptr/.style={decoration={markings,mark=at position 1 with %
+    {\arrow[scale=2,>=stealth]{>}}},postaction={decorate}}}
+%1
+\draw [->,>=stealth] (0,.5) -- (2,.5);
+%2
+\draw [myptr] (0,0) -- (2,0);
+\end{tikzpicture}
+
+
+
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "main"
+%%% End:

TeX/main.tex

@@ -26,9 +26,10 @@
 \usepackage{makecell}
 \usepackage{dsfont}
 \usepackage{tikz}
+\usepackage{nicefrac}
 
 \usetikzlibrary{matrix,chains,positioning,decorations.pathreplacing,arrows}
-\usetikzlibrary{positioning,calc}
+\usetikzlibrary{positioning,calc,calligraphy}
 
 \usepackage{pgfplots}
 \usepgfplotslibrary{colorbrewer}
@@ -55,6 +56,7 @@
 \thispagestyle{plain}
 \newtheorem{Theorem}{Theorem}[section]
 \newtheorem{Definition}[Theorem]{Definition}
+\newtheorem{Lemma}[Theorem]{Lemma}
 \newtheorem{Algorithm}[Theorem]{Algorithm}
 \newtheorem{Example}[Theorem]{Example}
 

TeX/theo_3_8.tex

@@ -29,3 +29,70 @@ as defined in ??? and ??? respectively.
 \end{Theorem}
 In order to proof Theo~\ref{theo:main1} we need to proof a number of
 auxilary Lemmata first.
+
+\begin{Definition}[Sobolev Norm]
+  \label{def:sobonorm}
+  The natural norm of the sobolev space is given by
+  \[
+    \norm{f}_{W^{k,p}(K)} =
+    \begin{cases}
+      \left(\sum_{\abs{\alpha} \leq k}
+        \norm{f^{(\alpha)}}^p_{L^p}\right)^{\nicefrac{1}{p}},&
+      \text{for } 1 \leq p < \infty \\
+      max_{\abs{\alpha} \leq k}\left\{f^{(\alpha)}\right\},& \text{for
+      } p = \infty
+    \end{cases}
+    .
+  \]
+\end{Definition}
+
+\begin{Lemma}[Poincar\'e typed inequality]
+  Let \(f:\mathbb{R} \to \mathbb{R}\) differentiable with \(f' :
+  \mathbb{R} \to \mathbb{R}\) Lesbeque integrable. Then for \(K=[a,b]
+  \subset \mathbb{R}\) with \(f(a)=0\) it holds that
+  \begin{equation}
+    \label{eq:pti1}
+    \exists C_K^{\infty} \in \mathbb{R}_{>0} :
+    \norm{f}_{w^{1,\infty}(K)} \leq C_K^{\infty}
+    \norm{f'}_{L^{\infty}(K)}.
+  \end{equation}
+  If additionaly \(f'\) is differentiable with \(f'': \mathbb{R} \to
+  \mathbb{R}\) Lesbeque integrable then additionally
+  \begin{equation}
+    \label{eq:pti2}
+    \exists C_K^2 \in \mathbb{R}_{>0} : \norm{f}_{W^{1,\infty}(K)} \leq
+    C_K^2 \norm{f''}_{L^2(K)}.
+  \end{equation}
+  \proof
+  With the fundamental theorem of calculus, if
+  \(\norm{f}_{L^{\infty}(K)}<\infty\) we get
+  \begin{equation}
+    \label{eq:f_f'}
+    \norm{f}_{L^{\infty}(K)} = \sup_{x \in K}\abs{\int_a^x f'(s) ds} \leq
+    \sup_{x \in K}\abs{\int_a^x \sup_{y \in K} \abs{f'(y)} ds} \leq \abs{b-a}
+    \sup_{y \in K}\abs{f'(y)}.
+  \end{equation}
+  Using this we can bound \(\norm{f}_{w^{1,\infty}(K)}\) by
+  \[
+    \norm{f}_{w^{1,\infty}(K)} \stackrel{\text{Def~\ref{def:sobonorm}}}{=}
+    \max\left\{\norm{f}_{L^{\infty}(K)},
+      \norm{f'}_{L^{\infty}(K)}\right\}
+    \stackrel{(\ref{eq:f_f'})}{\leq} max\left\{\abs{b-a},
+      1\right\}\norm{f'}_{L^{\infty}(K)}.
+  \]
+  With \(C_k^{\infty} \coloneqq max\left\{\abs{b-a}, 1\right\}\) we
+  get (\ref{eq:pti1}).
+  By using the Hölder inequality, we can proof the second claim.
+  \begin{align*}
+    \norm{f'}_{L^{\infty}(K)} &= \sup_{x \in K} \abs{\int_a^bf''(y)
+      \mathds{1}_{[a,x]}(y)dy} \leq \sup_{x \in
+      K}\norm{f''\mathds{1}_{[a,x]}}_{L^1(K)}\\
+                              &\hspace{-6pt} \stackrel{\text{Hölder}}{\leq} sup_{x
+                                \in
+                                K}\norm{f''}_{L^2(K)}\norm{\mathds{1}_{[a,x]}}_{L^2(K)}
+                                = \abs{b-a}\norm{f''}_{L^2(K)}.
+  \end{align*}
+  Thus (\ref{eq:pti2}) follows with \(C_K^2 \coloneqq
+  \abs{b-a}C_K^{\infty}\).
+  \qed
+\end{Lemma}