Merge pull request #64 from Satge96/master

Kapitel ausser das eigentliche K-tanh Kapitel

1 files changed, 227 insertions, 15 deletions

diff --git a/buch/papers/transfer/teil0.tex b/buch/papers/transfer/teil0.tex
index 19d4961..f8c8cb4 100644
--- a/buch/papers/transfer/teil0.tex
+++ b/buch/papers/transfer/teil0.tex
@@ -3,20 +3,232 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Teil 0\label{transfer:section:teil0}}
-\rhead{Teil 0}
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua \cite{transfer:bibtex}.
-At vero eos et accusam et justo duo dolores et ea rebum.
-Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-dolor sit amet.
-
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-At vero eos et accusam et justo duo dolores et ea rebum.  Stet clita
-kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit
-amet.
+\section{Motivation\label{transfer:section:teil0}}
+\rhead{Einleitung}
+
+Die Transferfunktion ist einer der wichtigsten Bestandteile moderner neuraler Netzwerke. Sie verleiht ihnen die nicht Linearität, die benötigt wird um komplexere Aufgaben zu lösen. Dabei kann theoretisch jede nicht lineare Funktion eingesetzt werden. In der Praxis tauchen aber nur sehr wenige Funktionen mit ähnlichen Eigenschaften auf. Einige davon sind in der Tabelle \ref{tab:aktfkt} zu sehen. In der heutigen Zeit sind vor allem die Variationen der ReLu Funktion beliebt. Der Tangens hyperbolicus wird aber dank dem Aufkommen der Recurrent Neural Networks, zum Beispiel dem Long short term memory Netzwerk, das aus Zellen wie in \ref{motivation:figure:LSTM} gezeigt bestehen, wieder vermehrt eingesetzt.
+Die klassische Berechnung ist aber sehr aufwendig und basiert auf Gleitkommaoperationen und relativ komplizierten Funktionen. Diese benötigen einen grossen Rechenaufwand. Vor allem auf Systemen die keine Gleitkommaarithmetik Hardware besitzen wie das zum Beispiel bei gewissen Mikrocontrollern der Fall ist.
+\begin{table}[h]
+	\centering
+	\begin{tabular}{llll}
+		\hline
+		\multicolumn{1}{l}{Name} & \multicolumn{1}{l}{Function} & \multicolumn{1}{l}{Figure} \\ 
+		\hline
+		Sigmoid & $\sigma(x)=\frac{1}{1+e^{-x}}$ & 
+		\begin{tikzpicture}[baseline={(0,0.2)}]
+			\draw (-1,0) -- (1,0);
+			\draw (0,0) -- (0,1);
+			\draw[red] plot[domain=-1:1,variable=\x] ({\x},{1/(1+exp(-4*\x))});
+		\end{tikzpicture}\\
+		ReLU & $f(x) =\begin{cases}
+			0 & ~\text{if}~ x<0 \\ 
+			x & ~\text{if}~x \geq 0.
+		\end{cases}$ &
+		\begin{tikzpicture}[baseline={(0,0.5)}]
+			\draw (-1,0) -- (1,0);
+			\draw (0,0) -- (0,1);
+			\draw[red] plot[domain=-1:1,variable=\x] ({\x},{ifthenelse(\x<0,0,\x)});
+		\end{tikzpicture}\\
+		Leaky ReLu & $f(x) =\begin{cases}
+			0 & ~\text{if}~ x<0 \\ 
+			x & ~\text{if}~x \geq a \cdot x.
+		\end{cases}$ &
+		\begin{tikzpicture}[baseline={(0,0.5)}]
+			\draw (-1,0) -- (1,0);
+			\draw (0,0) -- (0,1);
+			\draw[red] plot[domain=-1:1,variable=\x] ({\x},{ifthenelse(\x<0,0.1*\x,\x)});
+		\end{tikzpicture}                            
+	\end{tabular}
+	\caption{Transferfunktionen}
+	\label{tab:aktfkt}
+\end{table}
+
+\begin{figure}
+\centering
+\begin{tikzpicture}
+	\begin{axis}[
+		xmin=-2.5, xmax=2.5,
+		ymin=-1.5, ymax=1.5,
+		axis lines=center,
+		axis on top=true,
+		domain=-2.5:2.5,
+		ylabel=$y$,
+		xlabel=$x$,
+		]
+		
+		\addplot [mark=none,draw=red,ultra thick] {tanh(\x)};
+		\node [right, red] at (axis cs: 1,0.7) {$\tanh(x)$};
+		
+		%% Add the asymptotes
+		\draw [blue, dotted, thick] (axis cs:-2.5,-1)-- (axis cs:0,-1);
+		\draw [blue, dotted, thick] (axis cs:+2.5,+1)-- (axis cs:0,+1);
+	\end{axis}
+\end{tikzpicture}
+\caption{Tangens hyperbolicus
+\label{anleitung:figure:tanhyp}}
+\end{figure}
+
+\begin{figure}
+\centering
+\tikzset{
+	every node/.style={
+		font=\scriptsize
+	},
+	decision/.style={
+		shape=rectangle,
+		minimum height=1cm,
+		text width=3cm,
+		text centered,
+		rounded corners=1ex,
+		draw,
+		label={[yshift=0.2cm]left:ja},
+		label={[yshift=0.2cm]right:nein},
+	},
+	outcome/.style={
+		shape=ellipse,
+		fill=gray!15,
+		draw,
+		text width=1.5cm,
+		text centered
+	},
+	decision tree/.style={
+		edge from parent path={[-latex] (\tikzparentnode) -| (\tikzchildnode)},
+		sibling distance=4cm,
+		level distance=1.5cm
+	}
+}
+
+\begin{tikzpicture}
+	
+	\node [decision] { $x>k \cdot \frac{\ln 10}{2}$ }
+	[decision tree]
+	child { node [outcome] { $+1$ } }
+	child { node [decision] { $x<-k \cdot \frac{\ln 10}{2}$} 
+		child { node [outcome] { $-1$ } }
+		child { node [decision] { $-0,1<x<+0,1$ } 
+			child { node [outcome] { $\frac{\sinh x}{e^{x}-\sinh x}$ } }
+			child { node [outcome] { $\frac{e^{2 x}-1}{e^{2 x}+1}$ } }
+		}
+	};
+\end{tikzpicture}
+\caption{Annäherung für Tangens hyperbolicus
+\label{anleitung:figure:approxtanhhypalgo}}
+\end{figure}
+
+
+\begin{figure}
+\centering
+\newcommand{\empt}[2]{$#1^{\langle #2 \rangle}$}
+	
+\begin{tikzpicture}[
+		% GLOBAL CFG
+		font=\sf \scriptsize,
+		>=LaTeX,
+		% Styles
+		cell/.style={% For the main box
+			rectangle, 
+			rounded corners=5mm, 
+			draw,
+			very thick,
+		},
+		operator/.style={%For operators like +  and  x
+			circle,
+			draw,
+			inner sep=-0.5pt,
+			minimum height =.2cm,
+		},
+		function/.style={%For functions
+			ellipse,
+			draw,
+			inner sep=1pt
+		},
+		ct/.style={% For external inputs and outputs
+			circle,
+			draw,
+			line width = .75pt,
+			minimum width=1cm,
+			inner sep=1pt,
+		},
+		gt/.style={% For internal inputs
+			rectangle,
+			draw,
+			minimum width=4mm,
+			minimum height=3mm,
+			inner sep=1pt
+		},
+		mylabel/.style={% something new that I have learned
+			font=\scriptsize\sffamily
+		},
+		ArrowC1/.style={% Arrows with rounded corners
+			rounded corners=.25cm,
+			thick,
+		},
+		ArrowC2/.style={% Arrows with big rounded corners
+			rounded corners=.5cm,
+			thick,
+		},
+		]
+		
+		%Start drawing the thing...    
+		% Draw the cell: 
+		\node [cell, minimum height =4cm, minimum width=6cm] at (0,0){} ;
+		
+		% Draw inputs named ibox#
+		\node [gt] (ibox1) at (-2,-0.75) {$\sigma$};
+		\node [gt] (ibox2) at (-1.5,-0.75) {$\sigma$};
+		\node [function, draw=red!60, fill=red!5] (ibox3) at (-0.5,-0.75) {$\tanh$};
+		\node [gt] (ibox4) at (0.5,-0.75) {$\sigma$};
+		
+		% Draw opérators   named mux# , add# and func#
+		\node [operator] (mux1) at (-2,1.5) {$\times$};
+		\node [operator] (add1) at (-0.5,1.5) {+};
+		\node [operator] (mux2) at (-0.5,0) {$\times$};
+		\node [operator] (mux3) at (1.5,0) {$\times$};
+		\node [function, draw=red!60, fill=red!5] (func1) at (1.5,0.75) {$\tanh$};
+		
+		% Draw External inputs named as basis c,h,x
+		\node[ct, label={[mylabel]}] (c) at (-4,1.5) {\empt{c}{t-1}};
+		\node[ct, label={[mylabel]}] (h) at (-4,-1.5) {\empt{h}{t-1}};
+		\node[ct, label={[mylabel]}] (x) at (-2.5,-3) {\empt{x}{t}};
+		
+		% Draw External outputs? named as basis c2,h2,x2
+		\node[ct, label={[mylabel]}] (c2) at (4,1.5) {\empt{c}{t}};
+		\node[ct, label={[mylabel]}] (h2) at (4,-1.5) {\empt{h}{t}};
+		\node[ct, label={[mylabel]}] (x2) at (2.5,3) {\empt{h}{t}};
+		
+		% Start connecting all.
+		%Intersections and displacements are used. 
+		% Drawing arrows    
+		\draw [ArrowC1] (c) -- (mux1) -- (add1) -- (c2);
+		
+		% Inputs
+		\draw [ArrowC2] (h) -| (ibox4);
+		\draw [ArrowC1] (h -| ibox1)++(-0.5,0) -| (ibox1); 
+		\draw [ArrowC1] (h -| ibox2)++(-0.5,0) -| (ibox2);
+		\draw [ArrowC1] (h -| ibox3)++(-0.5,0) -| (ibox3);
+		\draw [ArrowC1] (x) -- (x |- h)-| (ibox3);
+		
+		% Internal
+		\draw [->, ArrowC2] (ibox1) -- (mux1);
+		\draw [->, ArrowC2] (ibox2) |- (mux2);
+		\draw [->, ArrowC2] (ibox3) -- (mux2);
+		\draw [->, ArrowC2] (ibox4) |- (mux3);
+		\draw [->, ArrowC2] (mux2) -- (add1);
+		\draw [->, ArrowC1] (add1 -| func1)++(-0.5,0) -| (func1);
+		\draw [->, ArrowC2] (func1) -- (mux3);
+		
+		%Outputs
+		\draw [-, ArrowC2] (mux3) |- (h2);
+		\draw (c2 -| x2) ++(0,-0.1) coordinate (i1);
+		\draw [-, ArrowC2] (h2 -| x2)++(-0.5,0) -| (i1);
+		\draw [-, ArrowC2] (i1)++(0,0.2) -- (x2);
+		
+\end{tikzpicture}
+\caption{Long short term memory cell
+\label{motivation:figure:LSTM}}
+\end{figure}
+
+
+