You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
234 lines
7.6 KiB
TeX
234 lines
7.6 KiB
TeX
|
|
\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]
|
|
\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) {};
|
|
|
|
\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 [myptr] (input-\i)+(-1,0) -- (input-\i)
|
|
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 [myptr] (output-\i) -- ++(1,0)
|
|
node [above, midway] {$O_{\l}$};
|
|
|
|
\foreach \i in {1,...,4}
|
|
\foreach \j in {1,...,2}
|
|
\draw [myptr] (input-\i) -- (hidden1-\j);
|
|
|
|
\foreach \i in {1,...,2}
|
|
\foreach \j in {1,...,2}
|
|
\draw [myptr] (hidden1-\i) -- (hidden2-\j);
|
|
|
|
\foreach \i in {1,...,2}
|
|
\foreach \j in {1,...,2}
|
|
\draw [myptr] (hidden2-\i) -- (output-\j);
|
|
|
|
\node [align=center, above] at (0,2) {Input\\layer};
|
|
\node [align=center, above] at (2,2) {Hidden \\layer $1$};
|
|
\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{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) {};
|
|
|
|
\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 [myptr] (input-\i)+(-1,0) -- (input-\i)
|
|
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 [myptr, >=stealth] (output-\i) -- ++(1,0)
|
|
node [above, midway] {$y$};
|
|
|
|
|
|
\foreach \i in {1}
|
|
\foreach \j in {1,2,...,3,4}
|
|
\draw [myptr, >=stealth] (input-\i) -- (hidden-\j);
|
|
|
|
\foreach \i in {1,2,...,3,4}
|
|
\foreach \j in {1}
|
|
\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:
|