From eb18ac33b2da5b9bb556068d34166c6aed271ae7 Mon Sep 17 00:00:00 2001 From: Tobias Arndt Date: Mon, 20 Apr 2020 15:33:58 +0200 Subject: [PATCH] added introductions to nn --- .gitignore | 12 ++++ TeX/introduction_nn.tex | 124 ++++++++++++++++++++++++++++++++++++++++ TeX/main.pdf | Bin 0 -> 306982 bytes TeX/main.tex | 14 ++++- TeX/theo_3_8.tex | 2 +- 5 files changed, 148 insertions(+), 4 deletions(-) create mode 100644 .gitignore create mode 100644 TeX/introduction_nn.tex create mode 100644 TeX/main.pdf diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..9c004a3 --- /dev/null +++ b/.gitignore @@ -0,0 +1,12 @@ +# weird latex files +*.log +*.aux +*.toc +*.gz +*.xml +TeX/auto/* +main-blx.bib + +# emacs autosaves +*.tex~ + diff --git a/TeX/introduction_nn.tex b/TeX/introduction_nn.tex new file mode 100644 index 0000000..cb4c168 --- /dev/null +++ b/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) 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