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Masterarbeit
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Code in Appendix

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Tobias Arndt 4 years ago
parent 340e0017c4
commit e96331d072
39 changed files with 840 additions and 9861 deletions
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3
.gitignore vendored
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@ -8,6 +8,7 @@
*.bbl
*.tdo
*.blg
*.lof
TeX/auto/*
main-blx.bib
@ -31,3 +32,5 @@ main-blx.bib
# no plot data
*.csv
*.mean
*Plots_*

76
TF/Main.scala
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import breeze.stats.distributions.Uniform
import breeze.stats.distributions.Gaussian
import scala.language.postfixOps
object Activation {
def apply(x: Double): Double = math.max(0, x)
def d(x: Double): Double = if (x > 0) 1 else 0
}
class RSNN(val n: Int, val gamma: Double = 0.001) {
val g = Uniform(-10, 10)
val g_1 = Uniform(-5, 5)//scala.math.exp(1))
val g_3 = Gaussian(0, 5)
val xis = g.sample(n)
val vs = g_3.sample(n)
val bs = xis zip vs map {case(xi, v) => xi * v}
//val vs = g_1.sample(n)
//val bs = g.sample(n)
def computeL1(x: Double) = (bs zip vs) map { case (b, v) => Activation(b + v * x) }
def computeL2(l1: Seq[Double], ws: Seq[Double]): Double =
(l1 zip ws) map { case (l, w) => w * l } sum
def output(ws: Seq[Double])(x: Double): Double = computeL2(computeL1(x), ws)
def learn(data: Seq[(Double, Double)], ws: Seq[Double], lambda: Double, gamma: Double): Seq[Double] = {
// data: N \times 2
// ws: n \times 1
lazy val deltas = data.map {
case (x, y) =>
val l1 = computeL1(x) // n
val out = computeL2(l1, ws) // 1
(l1 zip ws) map {case (l1, w) => (l1 * 2 * (out - y) + lambda * 2 * w) * gamma * -1} // n
}
// deltas: N × n
deltas.foldRight(ws)(
(delta, ws) => // delta: n
ws zip (delta) map { case (w, d) => w + d } // n
)// map (w => w - lambda * gamma * 2 * w)
}
def train(data: Seq[(Double, Double)], iter: Int, lambda: Double, gamma: Double = gamma): (Seq[Double], Double => Double)= {
val ws = (1 to iter).foldRight((1 to n).map(_ => 0.0) :Seq[Double])((i, w) => {
println(s"Training iteration $i")
println(w.sum/w.length)
learn(data, w, lambda, gamma / 10)
})
(ws, output(ws))
}
}
object Main {
def main(args: Array[String]): Unit = {
val nn = new RSNN(10, gamma = 0.0001)
val data = (1 to 100) map (_ * 0.01) map (t => (t, math.sin(t)))
val (ws, evaluate) = nn.train(data, iter = 1000, lambda = 0.8)
val results = data.map(_._1).map(evaluate(_))
data zip results foreach {
println(_)
}
}
}
object EqSeq {
def apply(left: Double, right: Double, steps: Int): Seq[Double] =
(0 to steps) map (_ * (right - left) / steps + left)
}

8
TF/cnn.py
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@ -149,8 +149,8 @@ if __name__ == '__main__':
############################
# in between layers
start_ratio_list = [[0.4, 0.5], [0.4, 0.8], [0.4,0.8], [0.4, 0.5], [0.4, 0.8],[0.4,0.8]]
end_ratio_list = [[0.4, 0.5], [0.4, 0.8], [0.4,0.8], [0.4, 0.5], [0.4, 0.8],[0.4,0.8]]
start_ratio_list = [[0.4, 0.5], [0.4, 0.8], [0.4,0.5], [0.4, 0.8], [0.4, 0.5],[0.4,0.8]]
end_ratio_list = [[0.4, 0.5], [0.4, 0.8], [0.4,0.5], [0.4, 0.8], [0.4, 0.5],[0.4,0.8]]
patch_size_list = [(3, 3), (3, 3), (2, 2), (3,3), (3, 3), (2, 2)]
ind_bgn_list = range(len(patch_size_list))
text_list = ['Conv.', 'Conv.', 'Max-pool.', 'Conv.', 'Conv.', 'Max-pool.']
@ -211,7 +211,7 @@ if __name__ == '__main__':
# plt.show()
fig.set_size_inches(8, 2.5)
fig_dir = '/home/tobi/Masterarbeit/TeX/Plots/Data/'
fig_dir = '/home/tobi/Masterarbeit/TeX/Figures/Data/'
fig_ext = '.pdf'
fig.savefig(os.path.join(fig_dir, 'cnn_fashion_fig' + fig_ext),
fig.savefig(os.path.join(fig_dir, 'cnn_fashion_fig1' + fig_ext),
bbox_inches='tight', pad_inches=0)

52
TF/scratch.scala
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import breeze.plot._
import breeze.plot.DomainFunction._
import breeze.linalg._
import breeze.stats.distributions.Gaussian
val nn = new RSNN(5000, 0.0000001)
val g = Gaussian(0, 0.3)
//val data = EqSeq(-math.Pi, math.Pi, 15) map (t => (t, math.sin(t)+ g.sample(1).last))
val (ws, evaluate) = nn.train(data, iter = 100000, lambda = (1.0/20) / 5 * (nn.n * 8) * 1)
val f = Figure()
val p = f.subplot(0)
val x = linspace(-5, 5)
val y = x.map(evaluate)
//print_data(nn, x, y, 3)
p += plot(x, y)
p += scatter(data.map(_._1), data.map(_._2), x => 0.1)
f.saveas("lines.png")
val x_i = data map {case (x,y) => x}
val y_i = data map {case (x,y) => y}
def print_data(nn: RSNN, x: DenseVector[Double], y: DenseVector[Double], tlambda: Double): Unit = {
val n = nn.n
reflect.io.File("C:/Users/tobia/Documents/Studium/Masterarbeit/Outputs/scala_out_d_1.csv").appendAll(s"x_n_$n"+s"_tl_$tlambda;" + x.toArray.mkString(";") + "\n")
reflect.io.File("C:/Users/tobia/Documents/Studium/Masterarbeit/Outputs/scala_out_d_1.csv").appendAll(s"y_n_$n"+s"_tl_$tlambda;" + y.toArray.mkString(";") + "\n")
}
reflect.io.File("C:/Users/tobia/Documents/Studium/Masterarbeit/Outputs/data_sin_d.csv").appendAll(x_i.mkString(";") + "\n")
reflect.io.File("C:/Users/tobia/Documents/Studium/Masterarbeit/Outputs/data_sin_d.csv").appendAll(y_i.mkString(";") + "\n")
reflect.io.File("C:/Users/tobia/Documents/Studium/Masterarbeit/Outputs/vals1.csv").appendAll(x.toArray.mkString(";") + "\n")
reflect.io.File("C:/Users/tobia/Documents/Studium/Masterarbeit/Outputs/vals1.csv").appendAll(y.toArray.mkString(";") + "\n")
for(j <- List(0.1, 1, 3)) {
for (i <- 3 until 4) {
val nn = new RSNN((5 * math.pow(10, i)).asInstanceOf[Int], 0.0000001)
val (ws, evaluate) = nn.train(data, iter = 100000, lambda = (1.0 / 20) / 5 * (nn.n * 8) * j)
val x = linspace(-5, 5)
val y = x.map(evaluate)
print_data(nn, x, y, j)
}
}
val x_i = Seq(-3.141592653589793,-2.722713633111154,-2.303834612632515,-1.8849555921538759,-1.4660765716752369,-1.0471975511965979,-0.6283185307179586,-0.2094395102393194,0.2094395102393194,0.6283185307179586,1.0471975511965974,1.4660765716752362,1.8849555921538759,2.3038346126325155,2.7227136331111543,3.1415926535897922)
val y_i = Seq(0.0802212608585366,-0.3759376368887911,-1.3264180339054117,-0.8971334213504949,-0.7724344034354425,-0.9501497164520739,-0.6224628757084738,-0.35622668982623207,-0.18377660088356823,0.7836770998126841,0.5874762732054489,1.0696991264956026,1.1297065441952743,0.7587275382323738,-0.030547103790458163,0.044327111895927106)
val data = x_i zip y_i

189
TeX/Appendix_code.tex
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\section{Code...}
\begin{itemize}
\item Code for randomized shallow neural network
\item Code for keras
\end{itemize}
\clearpage
\begin{lstfloat}
\begin{lstlisting}[language=iPython]
import breeze.stats.distributions.Uniform
import breeze.stats.distributions.Gaussian
import scala.language.postfixOps
object Activation {
def apply(x: Double): Double = math.max(0, x)
def d(x: Double): Double = if (x > 0) 1 else 0
}
class RSNN(val n: Int, val gamma: Double = 0.001) {
val g_unif = Uniform(-10, 10)
val g_gauss = Gaussian(0, 5)
val xis = g_unif.sample(n)
val vs = g_gauss.sample(n)
val bs = xis zip vs map {case(xi, v) => xi * v}
def computeL1(x: Double) = (bs zip vs) map {
case (b, v) => Activation(b + v * x) }
def computeL2(l1: Seq[Double], ws: Seq[Double]): Double =
(l1 zip ws) map { case (l, w) => w * l } sum
def output(ws: Seq[Double])(x: Double): Double =
computeL2(computeL1(x), ws)
def learn(data: Seq[(Double, Double)], ws: Seq[Double],
lamb: Double, gamma: Double): Seq[Double] = {
lazy val deltas = data.map {
case (x, y) =>
val l1 = computeL1(x)
val out = computeL2(l1, ws)
(l1 zip ws) map {case (l1, w) => (l1 * 2 * (out - y) +
lam * 2 * w) * gamma * -1}
}
deltas.foldRight(ws)(
(delta, ws) => ws zip (delta) map { case (w, d) => w + d })
}
def train(data: Seq[(Double, Double)], iter: Int, lam: Double,
gamma: Double = gamma): (Seq[Double], Double => Double)= {
val ws = (1 to iter).foldRight((1 to n).map(
_ => 0.0) :Seq[Double])((i, w) => {
println(s"Training iteration $i")
println(w.sum/w.length)
learn(data, w, lam, gamma / 10)
})
(ws, output(ws))
}
}
\end{lstlisting}
\caption{Scala code used to build and train the ridge penalized
randomized shallow neural network in .... The parameter \textit{lam}
in the train function represents the $\lambda$ parameter in the error
function. The parameters \textit{n} and \textit{gamma} set the number
of hidden nodes and the stepsize for training.}
\label{lst:rsnn}
\end{lstfloat}
\clearpage
\begin{lstlisting}[language=iPython]
import tensorflow as tf
import numpy as np
from tensorflow.keras.callbacks import CSVLogger
from tensorflow.keras.preprocessing.image import ImageDataGenerator
mnist = tf.keras.datasets.mnist
(x_train, y_train), (x_test, y_test) = mnist.load_data()
x_train = x_train.reshape(x_train.shape[0], 28, 28, 1)
x_train = x_train / 255.0
x_test = x_test.reshape(x_test.shape[0], 28, 28, 1)
x_test = x_test / 255.0
y_train = tf.keras.utils.to_categorical(y_train)
y_test = tf.keras.utils.to_categorical(y_test)
model = tf.keras.models.Sequential()
model.add(tf.keras.layers.Conv2D(24,kernel_size=5,padding='same',
activation='relu',input_shape=(28,28,1)))
model.add(tf.keras.layers.MaxPool2D())
model.add(tf.keras.layers.Conv2D(64,kernel_size=5,padding='same',
activation='relu'))
model.add(tf.keras.layers.MaxPool2D(padding='same'))
model.add(tf.keras.layers.Flatten())
model.add(tf.keras.layers.Dense(256, activation='relu'))
model.add(tf.keras.layers.Dropout(0.2))
model.add(tf.keras.layers.Dense(10, activation='softmax'))
model.compile(optimizer='adam', loss="categorical_crossentropy",
metrics=["accuracy"])
datagen = ImageDataGenerator(
rotation_range = 30,
zoom_range = 0.15,
width_shift_range=2,
height_shift_range=2,
shear_range = 1)
csv_logger = CSVLogger(<Target File>)
history = model.fit(datagen.flow(x_train, y_train, batch_size=50),
validation_data=(x_test, y_test),
epochs=125, callbacks=[csv_logger],
steps_per_epoch = x_train.shape[0]//50)
\end{lstlisting}
\clearpage
\begin{lstlisting}[language=iPython]
import tensorflow as tf
import numpy as np
from tensorflow.keras.callbacks import CSVLogger
from tensorflow.keras.preprocessing.image import ImageDataGenerator
mnist = tf.keras.datasets.fashion_mnist
(x_train, y_train), (x_test, y_test) = mnist.load_data()
x_train = x_train.reshape(x_train.shape[0], 28, 28, 1)
x_test = x_test.reshape(x_test.shape[0], 28, 28, 1)
x_train, x_test = x_train / 255.0, x_test / 255.0
y_train = tf.keras.utils.to_categorical(y_train)
y_test = tf.keras.utils.to_categorical(y_test)
model = tf.keras.Sequential()
model.add(tf.keras.layers.Conv2D(filters = 32, kernel_size = (3, 3),
activation='relu', input_shape = (28, 28, 1), padding='same'))
model.add(tf.keras.layers.Conv2D(filters = 32, kernel_size = (2, 2), activation='relu', padding = 'same'))
model.add(tf.keras.layers.MaxPool2D(strides=(2,2)))
model.add(tf.keras.layers.Conv2D(filters = 64, kernel_size = (3, 3), activation='relu', padding='same'))
model.add(tf.keras.layers.Conv2D(filters = 64, kernel_size = (3, 3), activation='relu', padding='same'))
model.add(tf.keras.layers.MaxPool2D(strides=(2,2)))
model.add(tf.keras.layers.Flatten())
model.add(tf.keras.layers.Dense(256, activation='relu'))
model.add(tf.keras.layers.Dropout(0.2))
model.add(tf.keras.layers.Dense(10, activation='softmax'))
model.compile(optimizer=tf.keras.optimizers.Adam(lr = 1e-3), loss="categorical_crossentropy", metrics=["accuracy"])
datagen = ImageDataGenerator(
rotation_range = 15,
zoom_range = 0.1,
width_shift_range=2,
height_shift_range=2,
shear_range = 0.5,
fill_mode = 'constant',
cval = 0)
csv_logger = CSVLogger(<Target File>)
history = model.fit(datagen.flow(x_train, y_train, batch_size=30),
steps_per_epoch=2000,
validation_data=(x_test, y_test),
epochs=125, callbacks=[csv_logger],
shuffle=True)
\end{lstlisting}
\clearpage
\begin{lstlisting}[language=iPython]
def get_random_sample(a, b, number_of_samples=10):
x = []
y = []
for category_number in range(0,10):
# get all samples of a category
train_data_category = a[b==category_number]
# pick a number of random samples from the category
train_data_category = train_data_category[np.random.randint(
train_data_category.shape[0], size=number_of_samples), :]
x.extend(train_data_category)
y.append([category_number]*number_of_samples)
return (np.asarray(x).reshape(-1, 28, 28, 1),
np.asarray(y).reshape(10*number_of_samples,1))
\end{lstlisting}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End:

5
TeX/Figures/RN_vs_RS.tex
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@ -131,11 +131,12 @@ plot coordinates {
Ridge Penalized Neural Network compared to Regression Spline,
with them being trained on $\text{data}_A$ in a), b), c) and on
$\text{data}_B$ in d), e), f).
The Parameters of each are given above.
The Parameters of each are given above. The implementation of the
network in Scala is given in Listing~\ref{lst:rsnn}
}
\label{fig:rn_vs_rs}
\end{figure}
%%% Local Variables:
%%% mode: latex
%%% TeX-master:
%%% TeX-master: "main"
%%% End:

20
TeX/Figures/mnist.tex
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@ -1,34 +1,34 @@
\begin{figure}[h]
\centering
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist0.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist0.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist1.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist1.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist2.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist2.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist3.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist3.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist4.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist4.pdf}
\end{subfigure}\\
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist5.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist5.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist6.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist6.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist7.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist7.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist8.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist8.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist9.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist9.pdf}
\end{subfigure}
\caption[MNIST data set]{The MNIST data set contains 70.000 images of preprocessed handwritten
digits. Of these images 60.000 are used as training images, while

10
TeX/Figures/pfg_test.tex
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@ -177,12 +177,12 @@ plot coordinates {
\begin{center}
\begin{figure}[h]
\centering
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist0.pdf}
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{Data/cnn_fashion_fig.pdf}
\caption{original\\image}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist_gen_zoom.pdf}
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{Data/cnn_fashion_fig1.pdf}
\caption{random\\zoom}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
@ -196,7 +196,7 @@ plot coordinates {
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist_gen_shift.pdf}
\caption{random\\positional shift}
\end{subfigure}\\
\end{subfigure}\\
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist5.pdf}
\end{subfigure}

146
TeX/Figures/test.tex
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@ -11,9 +11,15 @@
\definecolor{ipython_red}{RGB}{186, 33, 33}
\definecolor{ipython_green}{RGB}{0, 128, 0}
\definecolor{ipython_cyan}{RGB}{64, 128, 128}
\definecolor{ipython_purple}{RGB}{170, 34, 255}
\definecolor{ipython_purple}{RGB}{110, 64, 130}
\usepackage{listings}
\usepackage{float}
\newfloat{lstfloat}{htbp}{lop}
\floatname{lstfloat}{Listing}
\def\lstfloatautorefname{Listing}
\lstset{
breaklines=true,
%
@ -38,10 +44,11 @@
%% modified by me (should not have empty lines)
%%
\lstdefinelanguage{iPython}{
morekeywords={access,and,break,class,continue,def,del,elif,else,except,exec,finally,for,from,global,if,import,in,is,lambda,not,or,pass,print,raise,return,try,while},%
morekeywords={access,and,break,class,continue,def,del,elif,else,except,exec,finally,for,from,global,if,import,
in,is,lambda,not,or,pass,print,raise,return,try,while},%
%
% Built-ins
morekeywords=[2]{abs,all,any,basestring,bin,bool,bytearray,callable,chr,classmethod,cmp,compile,complex,delattr,dict,dir,divmod,enumerate,eval,execfile,file,filter,float,format,frozenset,getattr,globals,hasattr,hash,help,hex,id,input,int,isinstance,issubclass,iter,len,list,locals,long,map,max,memoryview,min,next,object,oct,open,ord,pow,property,range,raw_input,reduce,reload,repr,reversed,round,set,setattr,slice,sorted,staticmethod,str,sum,super,tuple,type,unichr,unicode,vars,xrange,zip,apply,buffer,coerce,intern},%
morekeywords=[2]{abs,all,any,basestring,bin,bool,bytearray,callable,chr,classmethod,cmp,compile,complex,delattr,dict,dir,divmod,enumerate,eval,execfile,file,filter,float,format,frozenset,getattr,globals,hasattr,hash,help,hex,id,input,int,isinstance,issubclass,iter,len,list,locals,long,map,max,memoryview,min,next,object,oct,open,ord,pow,property,range,raw_input,reduce,reload,repr,reversed,round,set,setattr,slice,sorted,staticmethod,str,sum,super,tuple,type,unichr,unicode,vars,xrange,zip,apply,buffer,coerce,intern,val},%
%
sensitive=true,%
morecomment=[l]\#,%
@ -91,7 +98,7 @@
{?}{{{\color{ipython_purple}?}}}1,
%
identifierstyle=\color{black}\ttfamily,
commentstyle=\color{ipython_cyan}\ttfamily,
commentstyle=\color{ipython_red}\ttfamily,
stringstyle=\color{ipython_red}\ttfamily,
keepspaces=true,
showspaces=false,
@ -109,9 +116,80 @@
% extendedchars=true,
basicstyle=\scriptsize,
keywordstyle=\color{ipython_green}\ttfamily,
morekeywords = [3]{Int, Double},
morekeywords = [2]{foldRight, case},
keywordstyle = [3]{\color{ipython_purple}\ttfamily},
keywordstyle = [2]{\color{ipython_cyan}\ttfamily},
}
\begin{document}
\begin{lstfloat}
\begin{lstlisting}[language=iPython]
import breeze.stats.distributions.Uniform
import breeze.stats.distributions.Gaussian
import scala.language.postfixOps
object Activation {
def apply(x: Double): Double = math.max(0, x)
def d(x: Double): Double = if (x > 0) 1 else 0
}
class RSNN(val n: Int, val gamma: Double = 0.001) {
val g_unif = Uniform(-10, 10)
val g_gauss = Gaussian(0, 5)
val xis = g_unif.sample(n)
val vs = g_gauss.sample(n)
val bs = xis zip vs map {case(xi, v) => xi * v}
def computeL1(x: Double) = (bs zip vs) map {
case (b, v) => Activation(b + v * x) }
def computeL2(l1: Seq[Double], ws: Seq[Double]): Double =
(l1 zip ws) map { case (l, w) => w * l } sum
def output(ws: Seq[Double])(x: Double): Double =
computeL2(computeL1(x), ws)
def learn(data: Seq[(Double, Double)], ws: Seq[Double],
lamb: Double, gamma: Double): Seq[Double] = {
lazy val deltas = data.map {
case (x, y) =>
val l1 = computeL1(x) // n
val out = computeL2(l1, ws) // 1
(l1 zip ws) map {case (l1, w) => (l1 * 2 * (out - y) +
lam * 2 * w) * gamma * -1}
}
deltas.foldRight(ws)(
(delta, ws) => ws zip (delta) map { case (w, d) => w + d })
}
def train(data: Seq[(Double, Double)], iter: Int, lam: Double,
gamma: Double = gamma): (Seq[Double], Double => Double)= {
val ws = (1 to iter).foldRight((1 to n).map(
_ => 0.0) :Seq[Double])((i, w) => {
println(s"Training iteration $i")
println(w.sum/w.length)
learn(data, w, lam, gamma / 10)
})
(ws, output(ws))
}
}
\end{lstlisting}
\caption{Scala code used to build and train the ridge penalized
randomized shallow neural network in .... The parameter \textit{lam}
in the train function represents the $\lambda$ parameter in the error
function. The parameters \textit{n} and \textit{gamma} set the number
of hidden nodes and the stepsize for training.}
\end{lstfloat}
\clearpage
\begin{lstlisting}[language=iPython]
import tensorflow as tf
import numpy as np
@ -136,7 +214,7 @@ model.add(tf.keras.layers.Conv2D(64,kernel_size=5,padding='same',activation='rel
model.add(tf.keras.layers.MaxPool2D(padding='same'))
model.add(tf.keras.layers.Flatten())
model.add(tf.keras.layers.Dense(256, activation='relu'))
model.add(tf.keras.layers.Dropout(j))
model.add(tf.keras.layers.Dropout(0.2))
model.add(tf.keras.layers.Dense(10, activation='softmax'))
model.compile(optimizer='adam', loss="categorical_crossentropy",
metrics=["accuracy"])
@ -150,10 +228,59 @@ datagen = ImageDataGenerator(
csv_logger = CSVLogger(<Target File>)
history = model.fit(datagen.flow(x_train_, y_train_, batch_size=50),
validation_data=(x_test, y_test), epochs=125,
callbacks=[csv_logger],
steps_per_epoch = x_train_.shape[0]//50)
history = model.fit(datagen.flow(x_train, y_train, batch_size=50),
validation_data=(x_test, y_test),
epochs=125, callbacks=[csv_logger],
steps_per_epoch = x_train.shape[0]//50)
\end{lstlisting}
\clearpage
\begin{lstlisting}[language=iPython]
import tensorflow as tf
import numpy as np
from tensorflow.keras.callbacks import CSVLogger
from tensorflow.keras.preprocessing.image import ImageDataGenerator
mnist = tf.keras.datasets.fashion_mnist
(x_train, y_train), (x_test, y_test) = mnist.load_data()
x_train = x_train.reshape(x_train.shape[0], 28, 28, 1)
x_test = x_test.reshape(x_test.shape[0], 28, 28, 1)
x_train, x_test = x_train / 255.0, x_test / 255.0
y_train = tf.keras.utils.to_categorical(y_train)
y_test = tf.keras.utils.to_categorical(y_test)
model = tf.keras.Sequential()
model.add(tf.keras.layers.Conv2D(filters = 32, kernel_size = (3, 3), activation='relu',
input_shape = (28, 28, 1), padding='same'))
model.add(tf.keras.layers.Conv2D(filters = 32, kernel_size = (2, 2), activation='relu', padding = 'same'))
model.add(tf.keras.layers.MaxPool2D(strides=(2,2)))
model.add(tf.keras.layers.Conv2D(filters = 64, kernel_size = (3, 3), activation='relu', padding='same'))
model.add(tf.keras.layers.Conv2D(filters = 64, kernel_size = (3, 3), activation='relu', padding='same'))
model.add(tf.keras.layers.MaxPool2D(strides=(2,2)))
model.add(tf.keras.layers.Flatten())
model.add(tf.keras.layers.Dense(256, activation='relu'))
model.add(tf.keras.layers.Dropout(0.2))
model.add(tf.keras.layers.Dense(10, activation='softmax'))
model.compile(optimizer=tf.keras.optimizers.Adam(lr = 1e-3), loss="categorical_crossentropy", metrics=["accuracy"])
datagen = ImageDataGenerator(
rotation_range = 15,
zoom_range = 0.1,
width_shift_range=2,
height_shift_range=2,
shear_range = 0.5,
fill_mode = 'constant',
cval = 0)
csv_logger = CSVLogger(<Target File>)
history = model.fit(datagen.flow(x_train, y_train, batch_size=30),
steps_per_epoch=2000,
validation_data=(x_test, y_test),
epochs=125, callbacks=[csv_logger],
shuffle=True)
\end{lstlisting}
\begin{lstlisting}[language=iPython]
@ -172,4 +299,5 @@ def get_random_sample(a, b, number_of_samples=10):
return (np.asarray(x).reshape(-1, 28, 28, 1),
np.asarray(y).reshape(10*number_of_samples,1))
\end{lstlisting}
\end{document}

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datagen_dropout_02_1
test
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datagen_dropout_00_1
test
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dropout_02_1
test
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default_1
test
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datagen_dropout_02_10
test
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train
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datagen_dropout_00_10
test
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train
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dropout_02_10
test
0.9423& 0.9081& 0.92696& 0.00013
train
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default_10
test
0.8585& 0.8148& 0.83771& 0.00027
train
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datagen_dropout_02_100
test
0.9805& 0.9727& 0.97826& 0
train
datagen_dropout_00_100
test
0.981& 0.9702& 0.9769& 1e-05
train
dropout_02_100
test
0.9796& 0.9719& 0.97703& 1e-05
train
default_100
test
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"56",5.5,-0.705540325570392,5.30834840605735,-0.721772537926303,5.28807996342596,-0.766498807502665
"57",5.6,-0.631266637872321,5.53199687756185,-0.583133415115471,5.40779902870202,-0.688843253413245
"58",5.7,-0.550685542597638,5.9238064899769,-0.541063721566544,5.59865656961444,-0.627040990301198
"59",5.8,-0.464602179413757,5.8067999294844,-0.43156566524513,5.68077207716296,-0.552246304884294
"60",5.9,-0.373876664830236,5.93089453525347,-0.604056792592816,5.80084302534748,-0.550733954237757
"61",6,-0.279415498198926,6.02965160059402,-0.234452930170458,5.91786841211583,-0.434812265604247
"62",6.1,-0.182162504272095,5.88697419016579,-0.135764844759742,5.91990685000071,-0.323660336266941
"63",6.2,-0.0830894028174964,5.91445270773648,-0.0073552500992853,5.92798052258888,-0.205537962618181
1 x_i y_i x_d y_d x y
2 1 0 0 -0.251688505259414 -0.109203329280437 -0.0838961684198045 -0.0364011097601456
3 2 0.1 0.0998334166468282 0.216143831477992 0.112557051753147 0.00912581751114394 0.0102181849309398
4 3 0.2 0.198669330795061 0.351879533708722 0.52138915851383 0.120991434720523 0.180094983253476
5 4 0.3 0.29552020666134 -0.0169121548298757 0.0870956013269369 0.0836131805695847 0.163690012207993
6 5 0.4 0.389418342308651 0.278503661037003 0.464752686490904 0.182421968363305 0.294268636359638
7 6 0.5 0.479425538604203 0.241783494554983 0.521480762031938 0.216291763003623 0.399960258238722
8 7 0.6 0.564642473395035 0.67288177436767 0.617435509386938 0.35521581484916 0.469717955748659
9 8 0.7 0.644217687237691 0.692239292735764 0.395366561077235 0.492895242512842 0.472257444593698
10 9 0.8 0.717356090899523 0.779946606884677 0.830045203984444 0.621840812496715 0.609161571471379
11 10 0.9 0.783326909627483 0.796987424421658 0.801263132114778 0.723333122197902 0.682652280249237
12 11 1 0.841470984807897 1.06821012817873 0.869642838589798 0.860323524382936 0.752971972337735
13 12 1.1 0.891207360061435 1.50128637982775 0.899079529605641 1.09148187598916 0.835465707990221
14 13 1.2 0.932039085967226 1.1194263347154 0.906626360727432 1.13393429991233 0.875953352580199
15 14 1.3 0.963558185417193 1.24675170552299 1.07848030956084 1.2135821540696 0.950969562327306
16 15 1.4 0.98544972998846 1.32784804980202 0.76685418220594 1.2818141129714 0.899892140468108
17 16 1.5 0.997494986604054 1.23565831982523 1.07310713979952 1.2548338349408 0.961170357331681
18 17 1.6 0.999573603041505 1.90289281875567 0.88003153305018 1.47254506382487 0.94006950203764
19 18 1.7 0.991664810452469 1.68871194985252 1.01829329437246 1.56940444551462 0.955793455192302
20 19 1.8 0.973847630878195 1.72179983981017 1.02268013575533 1.64902528694529 0.988666907865147
21 20 1.9 0.946300087687414 2.0758716236832 0.805032560816536 1.83908127693465 0.928000158917177
22 21 2 0.909297426825682 2.11118945422405 1.0134691646089 1.94365432453739 0.957334347939419
23 22 2.1 0.863209366648874 2.00475777514698 0.86568986134637 1.9826265174693 0.924298444442167
24 23 2.2 0.80849640381959 2.40773948766051 0.667018023975934 2.15807575978944 0.826761739840873
25 24 2.3 0.74570521217672 2.14892522112975 0.872704236332415 2.17485332420928 0.839957045849706
26 25 2.4 0.675463180551151 2.41696701330131 0.253955021611832 2.26412064248401 0.631186439537074
27 26 2.5 0.598472144103957 2.4087686184711 0.49450592290142 2.33847747374241 0.557319074033222
28 27 2.6 0.515501371821464 2.55312145187913 0.343944677655963 2.4151672191424 0.467867318187242
29 28 2.7 0.42737988023383 2.6585492172135 0.528990826178838 2.51649125567521 0.447178678139147
30 29 2.8 0.334988150155905 2.86281283456189 0.311400289332401 2.65184232661008 0.399952143417531
31 30 2.9 0.239249329213982 2.74379162744449 0.501282616227342 2.70796893413474 0.432791852065713
32 31 3 0.141120008059867 2.95951338295806 0.241385538727577 2.81576254355573 0.373424929745113
33 32 3.1 0.0415806624332905 2.87268165585702 0.0764217470113609 2.85626015646841 0.264426413128825
34 33 3.2 -0.0583741434275801 3.29898326143096 -0.272500742891131 3.0101734240017 0.0756660807058224
35 34 3.3 -0.157745694143249 3.64473302259565 -0.24394459655987 3.24463496592626 -0.0688606479078372
36 35 3.4 -0.255541102026832 3.46698556586598 -0.184272732807665 3.35339770834784 -0.15210430721581
37 36 3.5 -0.35078322768962 3.67208160089566 -0.119933071489115 3.51318482264886 -0.176430496141549
38 37 3.6 -0.442520443294852 3.73738883546162 -0.486197268315415 3.62961845872181 -0.283186040443485
39 38 3.7 -0.529836140908493 3.77209072631297 -0.70275845349803 3.68619468325631 -0.422698101171958
40 39 3.8 -0.611857890942719 3.66424718733509 -0.482410535792735 3.69727905622484 -0.462935060857071
41 40 3.9 -0.687766159183974 3.72257849834575 -0.58477261395861 3.71784166083333 -0.543108060927685
42 41 4 -0.756802495307928 3.85906293918747 -0.703015362823377 3.76539960460785 -0.618449987254768
43 42 4.1 -0.818277111064411 4.0131961543859 -0.900410257326814 3.84632588679948 -0.708384794580195
44 43 4.2 -0.871575772413588 4.0263131749378 -0.906044808231391 3.92085812717095 -0.789303202089581
45 44 4.3 -0.916165936749455 4.77220075671212 -0.530827398816399 4.22925719163087 -0.729943577630504
46 45 4.4 -0.951602073889516 4.4795636311648 -1.26672674728111 4.35331987391088 -0.921377204806384
47 46 4.5 -0.977530117665097 4.5088210845027 -0.886168448505782 4.44898342417679 -0.914264630323723
48 47 4.6 -0.993691003633465 4.70645816063034 -1.1082213336257 4.58861983576766 -0.97806804633887
49 48 4.7 -0.999923257564101 4.48408312008838 -0.98352521226689 4.55827710678399 -1.01979325501755
50 49 4.8 -0.996164608835841 4.97817348334347 -1.03043977928678 4.69715193557134 -1.02203657500247
51 50 4.9 -0.982452612624332 5.09171179984929 -0.948912592308037 4.8484480091335 -0.999631162740658
52 51 5 -0.958924274663138 4.87710566000798 -0.825224506141761 4.87693462801326 -0.937722874707385
53 52 5.1 -0.925814682327732 5.04139294635392 -0.718936957124138 4.97198282698482 -0.856650521199568
54 53 5.2 -0.883454655720153 4.94893136398377 -0.992753696742329 4.98294046406006 -0.885371127105841
55 54 5.3 -0.832267442223901 5.38128555915899 -0.717434652733088 5.10670981664685 -0.816103747160468
56 55 5.4 -0.772764487555987 5.46192736637355 -0.724060934669406 5.2398375587704 -0.780347098915984
57 56 5.5 -0.705540325570392 5.30834840605735 -0.721772537926303 5.28807996342596 -0.766498807502665
58 57 5.6 -0.631266637872321 5.53199687756185 -0.583133415115471 5.40779902870202 -0.688843253413245
59 58 5.7 -0.550685542597638 5.9238064899769 -0.541063721566544 5.59865656961444 -0.627040990301198
60 59 5.8 -0.464602179413757 5.8067999294844 -0.43156566524513 5.68077207716296 -0.552246304884294
61 60 5.9 -0.373876664830236 5.93089453525347 -0.604056792592816 5.80084302534748 -0.550733954237757
62 61 6 -0.279415498198926 6.02965160059402 -0.234452930170458 5.91786841211583 -0.434812265604247
63 62 6.1 -0.182162504272095 5.88697419016579 -0.135764844759742 5.91990685000071 -0.323660336266941
64 63 6.2 -0.0830894028174964 5.91445270773648 -0.0073552500992853 5.92798052258888 -0.205537962618181

141
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@ -1,141 +0,0 @@
\pgfplotsset{
compat=1.11,
legend image code/.code={
\draw[mark repeat=2,mark phase=2]
plot coordinates {
(0cm,0cm)
(0.075cm,0cm) %% default is (0.3cm,0cm)
(0.15cm,0cm) %% default is (0.6cm,0cm)
};%
}
}
\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\begin{subfigure}[b]{\textwidth}
\begin{adjustbox}{width=\textwidth, height=0.25\textheight}
\begin{tikzpicture}
\begin{axis}[
ytick = {-1, 0, 1, 2},
yticklabels = {$-1$, $\phantom{-0.}0$, $1$, $2$},]
\addplot table [x=x, y=y, col sep=comma, only marks,
forget plot] {Plots/Data/sin_6.csv};
\addplot [black, line width=2pt] table [x=x, y=y, col
sep=comma, mark=none] {Plots/Data/matlab_0.csv};
\addplot [red, line width = 1.5pt, dashed] table [x=x_n_5000_tl_0.0,
y=y_n_5000_tl_0.0, col sep=comma, mark=none] {Plots/Data/scala_out_sin.csv};
\addlegendentry{$f_1^{*, 0.1}$};
\addlegendentry{$\mathcal{RN}_w^{\tilde{\lambda}}$};
\end{axis}
\end{tikzpicture}
\end{adjustbox}
\caption{$\lambda = 0.1$}
\end{subfigure}\\
\begin{subfigure}[b]{\textwidth}
\begin{adjustbox}{width=\textwidth, height=0.25\textheight}
\begin{tikzpicture}
\begin{axis}
\addplot table [x=x, y=y, col sep=comma, only marks,
forget plot] {Plots/Data/sin_6.csv};
\addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Plots/Data/matlab_1.csv};
\addplot [red, line width = 1.5pt, dashed] table [x=x_n_5000_tl_1.0,
y=y_n_5000_tl_1.0, col sep=comma, mark=none] {Plots/Data/scala_out_sin.csv};
\addlegendentry{$f_1^{*, 1.0}$};
\addlegendentry{$\mathcal{RN}_w^{\tilde{\lambda}}$};
\end{axis}
\end{tikzpicture}
\end{adjustbox}
\caption{$\lambda = 1.0$}
\end{subfigure}\\
\begin{subfigure}[b]{\textwidth}
\begin{adjustbox}{width=\textwidth, height=0.25\textheight}
\begin{tikzpicture}
\begin{axis}
\addplot table [x=x, y=y, col sep=comma, only marks,
forget plot] {Plots/Data/sin_6.csv};
\addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Plots/Data/matlab_3.csv};
\addplot [red, line width = 1.5pt, dashed] table [x=x_n_5000_tl_3.0,
y=y_n_5000_tl_3.0, col sep=comma, mark=none] {Plots/Data/scala_out_sin.csv};
\addlegendentry{$f_1^{*, 3.0}$};
\addlegendentry{$\mathcal{RN}_w^{\tilde{\lambda}}$};
\end{axis}
\end{tikzpicture}
\end{adjustbox}
\caption{$\lambda = 3.0$}
\end{subfigure}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\begin{subfigure}[b]{\textwidth}
\begin{adjustbox}{width=\textwidth, height=0.245\textheight}
\begin{tikzpicture}
\begin{axis}[
ytick = {-2,-1, 0, 1, 2},
yticklabels = {$-2$,$-1$, $\phantom{-0.}0$, $1$, $2$},]
\addplot table [x=x, y=y, col sep=comma, only marks,
forget plot] {Plots/Data/data_sin_d_t.csv};
\addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Plots/Data/matlab_sin_d_01.csv};
\addplot [red, line width = 1.5pt, dashed] table [x=x_n_5000_tl_0.1,
y=y_n_5000_tl_0.1, col sep=comma, mark=none] {Plots/Data/scala_out_d_1_t.csv};
\addlegendentry{$f_1^{*, 0.1}$};
\addlegendentry{$\mathcal{RN}_w^{\tilde{\lambda}}$};
\end{axis}
\end{tikzpicture}
\end{adjustbox}
\caption{$\lambda = 0.1$}
\end{subfigure}\\
\begin{subfigure}[b]{\textwidth}
\begin{adjustbox}{width=\textwidth, height=0.25\textheight}
\begin{tikzpicture}
\begin{axis}
\addplot table [x=x, y=y, col sep=comma, only marks,
forget plot] {Plots/Data/data_sin_d_t.csv};
\addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Plots/Data/matlab_sin_d_1.csv};
\addplot [red, line width = 1.5pt, dashed] table [x=x_n_5000_tl_1.0,
y=y_n_5000_tl_1.0, col sep=comma, mark=none] {Plots/Data/scala_out_d_1_t.csv};
\addlegendentry{$f_1^{*, 1.0}$};
\addlegendentry{$\mathcal{RN}_w^{\tilde{\lambda},*}$};
\end{axis}
\end{tikzpicture}
\end{adjustbox}
\caption{$\lambda = 1.0$}
\end{subfigure}\\
\begin{subfigure}[b]{\textwidth}
\begin{adjustbox}{width=\textwidth, height=0.25\textheight}
\begin{tikzpicture}
\begin{axis}
\addplot table [x=x, y=y, col sep=comma, only marks,
forget plot] {Plots/Data/data_sin_d_t.csv};
\addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Plots/Data/matlab_sin_d_3.csv};
\addplot [red, line width = 1.5pt, dashed] table [x=x_n_5000_tl_3.0,
y=y_n_5000_tl_3.0, col sep=comma, mark=none] {Plots/Data/scala_out_d_1_t.csv};
\addlegendentry{$f_1^{*, 3.0}$};
\addlegendentry{$\mathcal{RN}_w^{\tilde{\lambda}}$};
\end{axis}
\end{tikzpicture}
\end{adjustbox}
\caption{$\lambda = 3.0$}
\end{subfigure}
\end{subfigure}
\caption[Comparison of shallow neural networks and regression
splines]{% In these Figures the behaviour stated in ... is
% visualized
% in two exaples. For $(a), (b), (c)$ six values of sinus equidistantly
% spaced on $[-\pi, \pi]$ have been used as training data. For
% $(d),(e),(f)$ 15 equidistand values have been used, where
% $y_i^{train} = \sin(x_i^{train}) + \varepsilon_i$ and
% $\varepsilon_i \sim \mathcal{N}(0, 0.3)$. For
% $\mathcal{RN}_w^{\tilde{\lambda, *}}$ the random weights are
% distributed as follows
% \begin{align*}
% \xi_k &\sim
% \end{align*}
Ridge Penalized Neural Network compared to Regression Spline,
with them being trained on $\text{data}_A$ in a), b), c) and on
$\text{data}_B$ in d), e), f).
The Parameters of each are given above.
}
\label{fig:rn_vs_rs}
\end{figure}
%%% Local Variables:
%%% mode: latex
%%% TeX-master:
%%% End:

93
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@ -1,93 +0,0 @@
\pgfplotsset{
compat=1.11,
legend image code/.code={
\draw[mark repeat=2,mark phase=2]
plot coordinates {
(0cm,0cm)
(0.0cm,0cm) %% default is (0.3cm,0cm)
(0.0cm,0cm) %% default is (0.6cm,0cm)
};%
}
}
\begin{figure}
\begin{subfigure}[h!]{\textwidth}
\begin{tikzpicture}
\begin{axis}[tick style = {draw = none}, width = \textwidth,
height = 0.6\textwidth,
xtick = {1, 3, 5,7,9,11,13,15,17,19},
xticklabels = {$2$, $4$, $6$, $8$,
$10$,$12$,$14$,$16$,$18$,$20$},
xlabel = {training epoch}, ylabel = {classification accuracy}]
\addplot table
[x=epoch, y=val_accuracy, col sep=comma] {Plots/Data/GD_01.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma] {Plots/Data/GD_05.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma] {Plots/Data/GD_1.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma]
{Plots/Data/SGD_01_b32.log};
\addlegendentry{GD$_{0.01}$}
\addlegendentry{GD$_{0.05}$}
\addlegendentry{GD$_{0.1}$}
\addlegendentry{SGD$_{0.01}$}
\end{axis}
\end{tikzpicture}
%\caption{Classification accuracy}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\begin{tikzpicture}
\begin{axis}[tick style = {draw = none}, width = \textwidth,
height = 0.6\textwidth,
ytick = {0, 1, 2, 3, 4},
yticklabels = {$0$, $1$, $\phantom{0.}2$, $3$, $4$},
xtick = {1, 3, 5,7,9,11,13,15,17,19},
xticklabels = {$2$, $4$, $6$, $8$,
$10$,$12$,$14$,$16$,$18$,$20$},
xlabel = {training epoch}, ylabel = {error measure\vphantom{fy}}]
\addplot table
[x=epoch, y=val_loss, col sep=comma] {Plots/Data/GD_01.log};
\addplot table
[x=epoch, y=val_loss, col sep=comma] {Plots/Data/GD_05.log};
\addplot table
[x=epoch, y=val_loss, col sep=comma] {Plots/Data/GD_1.log};
\addplot table
[x=epoch, y=val_loss, col sep=comma] {Plots/Data/SGD_01_b32.log};
\addlegendentry{GD$_{0.01}$}
\addlegendentry{GD$_{0.05}$}
\addlegendentry{GD$_{0.1}$}
\addlegendentry{SGD$_{0.01}$}
\end{axis}
\end{tikzpicture}
\caption{Performance metrics during training}
\end{subfigure}
% \\~\\
\caption[Performance comparison of SDG and GD]{The neural network given in ?? trained with different
algorithms on the MNIST handwritten digits data set. For gradient
descent the learning rated 0.01, 0.05 and 0.1 are (GD$_{\cdot}$). For
stochastic gradient descend a batch size of 32 and learning rate
of 0.01 is used (SDG$_{0.01}$).}
\label{fig:sgd_vs_gd}
\end{figure}
\begin{table}[h]
\begin{tabu} to \textwidth {@{} *4{X[c]}c*4{X[c]} @{}}
\multicolumn{4}{c}{Classification Accuracy}
&~&\multicolumn{4}{c}{Error Measure}
\\\cline{1-4}\cline{6-9}
GD$_{0.01}$&GD$_{0.05}$&GD$_{0.1}$&SGD$_{0.01}$&&GD$_{0.01}$&GD$_{0.05}$&GD$_{0.1}$&SGD$_{0.01}$
\\\cline{1-4}\cline{6-9}
\multicolumn{9}{c}{test}\\
0.265&0.633&0.203&0.989&&2.267&1.947&3.91&0.032
\end{tabu}
\caption{Performance metrics of the networks trained in
Figure~\ref{fig:sgd_vs_gd} after 20 training epochs.}
\label{table:sgd_vs_gd}
\end{table}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "../main"
%%% End:

71
TeX/Plots/_region_.tex
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@ -1,71 +0,0 @@
\message{ !name(pfg_test.tex)}\documentclass{article}
\usepackage{pgfplots}
\usepackage{filecontents}
\usepackage{subcaption}
\usepackage{adjustbox}
\usepackage{xcolor}
\usepackage{graphicx}
\usetikzlibrary{calc, 3d}
\begin{document}
\message{ !name(pfg_test.tex) !offset(6) }
\end{axis}
\end{tikzpicture}
\end{adjustbox}
\caption{True position (\textcolor{red}{red}), distorted data (black)}
\end{figure}
\begin{center}
\begin{figure}[h]
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{Data/klammern.jpg}
\caption{Original Picure}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{Data/image_conv4.png}
\caption{test}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{Data/image_conv5.png}
\caption{test}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\textwidth]{Data/image_conv6.png}
\caption{test}
\end{subfigure}
\end{figure}
\end{center}
\begin{figure}
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\begin{scope}[x = (0:1cm), y=(90:1cm), z=(15:-0.5cm)]
\node[canvas is xy plane at z=0, transform shape] at (0,0)
{\includegraphics[width=5cm]{Data/klammern_r.jpg}};
\node[canvas is xy plane at z=2, transform shape] at (0,-0.2)
{\includegraphics[width=5cm]{Data/klammern_g.jpg}};
\node[canvas is xy plane at z=4, transform shape] at (0,-0.4)
{\includegraphics[width=5cm]{Data/klammern_b.jpg}};
\node[canvas is xy plane at z=4, transform shape] at (-8,-0.2)
{\includegraphics[width=5.3cm]{Data/klammern_rgb.jpg}};
\end{scope}
\end{tikzpicture}
\end{adjustbox}
\caption{On the right the red, green and blue chanels of the picture
are displayed. In order to better visualize the color channes the
black and white picture of each channel has been colored in the
respective color. Combining the layers results in the image on the
left}
\end{figure}
\message{ !name(pfg_test.tex) !offset(3) }
\end{document}
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\begin{figure}[h]
\centering
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist0.pdf}
\caption{T-shirt/top}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist1.pdf}
\caption{Trousers}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist2.pdf}
\caption{Pullover}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist3.pdf}
\caption{Dress}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist4.pdf}
\caption{Coat}
\end{subfigure}\\
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist5.pdf}
\caption{Sandal}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist6.pdf}
\caption{Shirt}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist7.pdf}
\caption{Sneaker}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist8.pdf}
\caption{Bag}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/fashion_mnist9.pdf}
\caption{Ankle boot}
\end{subfigure}
\caption[Fashion MNIST data set]{The fashtion MNIST data set contains 70.000 images of
preprocessed product images from Zalando, which are categorized as
T-shirt/top, Trouser, Pullover, Dress, Coat, Sandal, Shirt,
Sneaker, Bag, Ankle boot. Of these images 60.000 are used as training images, while
the rest are used to validate the models trained.}
\label{fig:MNIST}
\end{figure}
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\pgfplotsset{
compat=1.11,
legend image code/.code={
\draw[mark repeat=2,mark phase=2]
plot coordinates {
(0cm,0cm)
(0.15cm,0cm) %% default is (0.3cm,0cm)
(0.3cm,0cm) %% default is (0.6cm,0cm)
};%
}
}
\begin{figure}
\begin{subfigure}[h]{\textwidth}
\begin{tikzpicture}
\begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
/pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
height = 0.6\textwidth, ymin = 0.988, legend style={at={(0.9825,0.0175)},anchor=south east},
xlabel = {epoch}, ylabel = {Classification Accuracy}, cycle
list/Dark2, every axis plot/.append style={line width =1.25pt}]
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_datagen_full_mean.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_datagen_dropout_02_full_mean.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_datagen_dropout_04_full_mean.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_dropout_02_full_mean.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_dropout_04_full_mean.log};
\addplot [dashed] table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_full_mean.log};
\addlegendentry{\footnotesize{G.}}
\addlegendentry{\footnotesize{G. + D. 0.2}}
\addlegendentry{\footnotesize{G. + D. 0.4}}
\addlegendentry{\footnotesize{D. 0.2}}
\addlegendentry{\footnotesize{D. 0.4}}
\addlegendentry{\footnotesize{Default}}
\end{axis}
\end{tikzpicture}
\caption{Classification accuracy}
\vspace{.25cm}
\end{subfigure}
\begin{subfigure}[h]{1.0\linewidth}
\begin{tabu} to \textwidth {@{}lc*5{X[c]}@{}}
\Tstrut \Bstrut & \textsc{\,Adam\,} & D. 0.2 & D. 0.4 & G. &G.+D.\,0.2 & G.+D.\,0.4 \\
\hline
\multicolumn{7}{c}{Test Accuracy}\Bstrut \\
\cline{2-7}
mean \Tstrut & 0.9914 & 0.9923 & 0.9930 & 0.9937 & 0.9938 & 0.9943 \\
max & 0.9926 & 0.9930 & 0.9934 & 0.9946 & 0.9955 & 0.9956 \\
min & 0.9887 & 0.9909 & 0.9922 & 0.9929 & 0.9929 & 0.9934 \\
\hline
\multicolumn{7}{c}{Training Accuracy}\Bstrut \\
\cline{2-7}
mean \Tstrut & 0.9994 & 0.9991 & 0.9989 & 0.9967 & 0.9954 & 0.9926 \\
max & 0.9996 & 0.9996 & 0.9992 & 0.9979 & 0.9971 & 0.9937 \\
min & 0.9992 & 0.9990 & 0.9984 & 0.9947 & 0.9926 & 0.9908 \\
\end{tabu}
\caption{Mean and maximum accuracy after 48 epochs of training.}
\label{fig:gen_dropout_b}
\end{subfigure}
\caption[Performance comparison of overfitting measures]{Accuracy for the net given in ... with Dropout (D.),
data generation (G.), a combination, or neither (Default) implemented and trained
with \textsc{Adam}. For each epoch the 60.000 training samples
were used, or for data generation 10.000 steps with each using
batches of 60 generated data points. For each configuration the
model was trained 5 times and the average accuracies at each epoch
are given in (a). Mean, maximum and minimum values of accuracy on
the test and training set are given in (b).}
\label{fig:gen_dropout}
\end{figure}
%%% Local Variables:
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%%% TeX-master: "../main"
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\begin{figure}[h]
\centering
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist0.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist1.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist2.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist3.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist4.pdf}
\end{subfigure}\\
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist5.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist6.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist7.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist8.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist9.pdf}
\end{subfigure}
\caption[MNIST data set]{The MNIST data set contains 70.000 images of preprocessed handwritten
digits. Of these images 60.000 are used as training images, while
the rest are used to validate the models trained.}
\label{fig:MNIST}
\end{figure}
%%% Local Variables:
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%%% TeX-master: "../main"
%%% End:

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\documentclass[a4paper, 12pt, draft=true]{article}
\usepackage{pgfplots}
\usepackage{filecontents}
\usepackage{subcaption}
\usepackage{adjustbox}
\usepackage{xcolor}
\usepackage{tabu}
\usepackage{showframe}
\usepackage{graphicx}
\usepackage{titlecaps}
\usetikzlibrary{calc, 3d}
\usepgfplotslibrary{colorbrewer}
\newcommand\Tstrut{\rule{0pt}{2.6ex}} % = `top' strut
\newcommand\Bstrut{\rule[-0.9ex]{0pt}{0pt}} % = `bottom' strut
\begin{document}
\pgfplotsset{
compat=1.11,
legend image code/.code={
\draw[mark repeat=2,mark phase=2]
plot coordinates {
(0cm,0cm)
(0.3cm,0cm) %% default is (0.3cm,0cm)
(0.6cm,0cm) %% default is (0.6cm,0cm)
};%
}
}
\begin{figure}
\begin{subfigure}[h]{\textwidth}
\begin{tikzpicture}
\begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
/pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
ylabel = {Test Accuracy}, cycle
list/Dark2, every axis plot/.append style={line width
=1.25pt}]
% \addplot [dashed] table
% [x=epoch, y=accuracy, col sep=comma, mark = none]
% {Data/adam_datagen_full.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_1.mean};
% \addplot [dashed] table
% [x=epoch, y=accuracy, col sep=comma, mark = none]
% {Data/adam_datagen_dropout_02_full.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_datagen_1.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_datagen_dropout_02_1.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_dropout_02_1.mean};
\addlegendentry{\footnotesize{G.}}
\addlegendentry{\footnotesize{G. + D. 0.2}}
\addlegendentry{\footnotesize{G. + D. 0.4}}
\addlegendentry{\footnotesize{D. 0.2}}
\addlegendentry{\footnotesize{D. 0.4}}
\addlegendentry{\footnotesize{Default}}
\end{axis}
\end{tikzpicture}
\caption{1 sample per class}
\vspace{0.25cm}
\end{subfigure}
\begin{subfigure}[h]{\textwidth}
\begin{tikzpicture}
\begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
/pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
ylabel = {Test Accuracy}, cycle
list/Dark2, every axis plot/.append style={line width
=1.25pt}]
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_dropout_00_10.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_dropout_02_10.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_datagen_dropout_00_10.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_datagen_dropout_02_10.mean};
\addlegendentry{\footnotesize{G.}}
\addlegendentry{\footnotesize{G. + D. 0.2}}
\addlegendentry{\footnotesize{G. + D. 0.4}}
\addlegendentry{\footnotesize{D. 0.2}}
\addlegendentry{\footnotesize{D. 0.4}}
\addlegendentry{\footnotesize{Default}}
\end{axis}
\end{tikzpicture}
\caption{10 samples per class}
\end{subfigure}
\begin{subfigure}[h]{\textwidth}
\begin{tikzpicture}
\begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
/pgf/number format/precision=3},tick style = {draw = none}, width = 0.9875\textwidth,
height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
xlabel = {epoch}, ylabel = {Test Accuracy}, cycle
list/Dark2, every axis plot/.append style={line width
=1.25pt}, ymin = {0.92}]
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_dropout_00_100.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_dropout_02_100.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_datagen_dropout_00_100.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Data/adam_datagen_dropout_02_100.mean};
\addlegendentry{\footnotesize{G.}}
\addlegendentry{\footnotesize{G. + D. 0.2}}
\addlegendentry{\footnotesize{G. + D. 0.4}}
\addlegendentry{\footnotesize{D. 0.2}}
\addlegendentry{\footnotesize{D. 0.4}}
\addlegendentry{\footnotesize{Default}}
\end{axis}
\end{tikzpicture}
\caption{100 samples per class}
\vspace{.25cm}
\end{subfigure}
\caption{Accuracy for the net given in ... with Dropout (D.),
data generation (G.), a combination, or neither (Default) implemented and trained
with \textsc{Adam}. For each epoch the 60.000 training samples
were used, or for data generation 10.000 steps with each using
batches of 60 generated data points. For each configuration the
model was trained 5 times and the average accuracies at each epoch
are given in (a). Mean, maximum and minimum values of accuracy on
the test and training set are given in (b).}
\end{figure}
\begin{table}
\centering
\begin{tabu} to \textwidth {@{}l*4{X[c]}@{}}
\Tstrut \Bstrut & \textsc{Adam} & D. 0.2 & Gen & Gen.+D. 0.2 \\
\hline
&
\multicolumn{4}{c}{\titlecap{test accuracy for 1 sample}}\Bstrut \\
\cline{2-5}
max \Tstrut & 0.5633 & 0.5312 & 0.6704 & 0.6604 \\
min & 0.3230 & 0.4224 & 0.4878 & 0.5175 \\
mean & 0.4570 & 0.4714 & 0.5862 & 0.6014 \\
var & 0.0040 & 0.0012 & 0.0036 & 0.0023 \\
\hline
&
\multicolumn{4}{c}{\titlecap{test accuracy for 10 samples}}\Bstrut \\
\cline{2-5}
max \Tstrut & 0.8585 & 0.9423 & 0.9310 & 0.9441 \\
min & 0.8148 & 0.9081 & 0.9018 & 0.9061 \\
mean & 0.8377 & 0.9270 & 0.9185 & 0.9232 \\
var & 2.7e-4 & 1.3e-4 & 6e-05 & 1.5e-4 \\
\hline
&
\multicolumn{4}{c}{\titlecap{test accuracy for 100 samples}}\Bstrut \\
\cline{2-5}
max & 0.9637 & 0.9796 & 0.9810 & 0.9805 \\
min & 0.9506 & 0.9719 & 0.9702 & 0.9727 \\
mean & 0.9582 & 0.9770 & 0.9769 & 0.9783 \\
var & 2e-05 & 1e-05 & 1e-05 & 0 \\
\hline
\end{tabu}
\caption{Values of the test accuracy of the model trained 10 times
of random training sets containing 1, 10 and 100 data points per
class.}
\end{table}
\begin{center}
\begin{figure}[h]
\centering
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist0.pdf}
\caption{original\\image}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist_gen_zoom.pdf}
\caption{random\\zoom}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist_gen_shear.pdf}
\caption{random\\shear}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist_gen_rotation.pdf}
\caption{random\\rotation}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist_gen_shift.pdf}
\caption{random\\positional shift}
\end{subfigure}\\
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist5.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist6.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist7.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist8.pdf}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Data/mnist9.pdf}
\end{subfigure}
\caption{The MNIST data set contains 70.000 images of preprocessed handwritten
digits. Of these images 60.000 are used as training images, while
the rest are used to validate the models trained.}
\end{figure}
\end{center}
\begin{figure}
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}
\begin{scope}[x = (0:1cm), y=(90:1cm), z=(15:-0.5cm)]
\node[canvas is xy plane at z=0, transform shape] at (0,0)
{\includegraphics[width=5cm]{Data/klammern_r.jpg}};
\node[canvas is xy plane at z=2, transform shape] at (0,-0.2)
{\includegraphics[width=5cm]{Data/klammern_g.jpg}};
\node[canvas is xy plane at z=4, transform shape] at (0,-0.4)
{\includegraphics[width=5cm]{Data/klammern_b.jpg}};
\node[canvas is xy plane at z=4, transform shape] at (-8,-0.2)
{\includegraphics[width=5.3cm]{Data/klammern_rgb.jpg}};
\end{scope}
\end{tikzpicture}
\end{adjustbox}
\caption{On the right the red, green and blue chanels of the picture
are displayed. In order to better visualize the color channes the
black and white picture of each channel has been colored in the
respective color. Combining the layers results in the image on the
left}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{.45\linewidth}
\centering
\begin{tikzpicture}
\begin{axis}[enlargelimits=false, ymin=0, ymax = 1, width=\textwidth]
\addplot [domain=-5:5, samples=101,unbounded coords=jump]{1/(1+exp(-x)};
\end{axis}
\end{tikzpicture}
\end{subfigure}
\begin{subfigure}{.45\linewidth}
\centering
\begin{tikzpicture}
\begin{axis}[enlargelimits=false, width=\textwidth]
\addplot[domain=-5:5, samples=100]{tanh(x)};
\end{axis}
\end{tikzpicture}
\end{subfigure}
\begin{subfigure}{.45\linewidth}
\centering
\begin{tikzpicture}
\begin{axis}[enlargelimits=false, width=\textwidth,
ytick={0,2,4},yticklabels={\hphantom{4.}0,2,4}, ymin=-1]
\addplot[domain=-5:5, samples=100]{max(0,x)};
\end{axis}
\end{tikzpicture}
\end{subfigure}
\begin{subfigure}{.45\linewidth}
\centering
\begin{tikzpicture}
\begin{axis}[enlargelimits=false, width=\textwidth, ymin=-1,
ytick={0,2,4},yticklabels={$\hphantom{-5.}0$,2,4}]
\addplot[domain=-5:5, samples=100]{max(0,x)+ 0.1*min(0,x)};
\end{axis}
\end{tikzpicture}
\end{subfigure}
\end{figure}
\begin{tikzpicture}
\begin{axis}[enlargelimits=false]
\addplot [domain=-5:5, samples=101,unbounded coords=jump]{1/(1+exp(-x)};
\addplot[domain=-5:5, samples=100]{tanh(x)};
\addplot[domain=-5:5, samples=100]{max(0,x)};
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}
\begin{axis}[enlargelimits=false]
\addplot[domain=-2*pi:2*pi, samples=100]{cos(deg(x))};
\end{axis}
\end{tikzpicture}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:

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@ -1,78 +0,0 @@
\pgfplotsset{
compat=1.11,
legend image code/.code={
\draw[mark repeat=2,mark phase=2]
plot coordinates {
(0cm,0cm)
(0.0cm,0cm) %% default is (0.3cm,0cm)
(0.0cm,0cm) %% default is (0.6cm,0cm)
};%
}
}
\begin{figure}
\begin{subfigure}[h]{\textwidth}
\begin{tikzpicture}
\begin{axis}[tick style = {draw = none}, width = \textwidth,
height = 0.6\textwidth, ymin = 0.92, legend style={at={(0.9825,0.75)},anchor=north east},
xlabel = {epoch}, ylabel = {Classification Accuracy}]
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adagrad.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adadelta.log};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam.log};
\addlegendentry{\footnotesize{ADAGRAD}}
\addlegendentry{\footnotesize{ADADELTA}}
\addlegendentry{\footnotesize{ADAM}}
\addlegendentry{SGD$_{0.01}$}
\end{axis}
\end{tikzpicture}
%\caption{Classification accuracy}
\vspace{.25cm}
\end{subfigure}
% \begin{subfigure}[b]{\textwidth}
% \begin{tikzpicture}
% \begin{axis}[tick style = {draw = none}, width = \textwidth,
% height = 0.6\textwidth, ymax = 0.5,
% xlabel = {epoch}, ylabel = {Error Measure\vphantom{y}},ytick ={0,0.1,0.2,0.3,0.4,0.45,0.5}, yticklabels =
% {0,0.1,0.2,0.3,0.4,\phantom{0.94},0.5}]
% \addplot table
% [x=epoch, y=val_loss, col sep=comma, mark = none] {Plots/Data/adagrad.log};
% \addplot table
% [x=epoch, y=val_loss, col sep=comma, mark = none] {Plots/Data/adadelta.log};
% \addplot table
% [x=epoch, y=val_loss, col sep=comma, mark = none] {Plots/Data/adam.log};
% \addlegendentry{\footnotesize{ADAGRAD}}
% \addlegendentry{\footnotesize{ADADELTA}}
% \addlegendentry{\footnotesize{ADAM}}
% \addlegendentry{SGD$_{0.01}$}
% \end{axis}
% \end{tikzpicture}
% \caption{Performance metrics during training}
% \vspace{.25cm}
% \end{subfigure}
\begin{subfigure}[b]{1.0\linewidth}
\begin{tabu} to \textwidth {@{} *3{X[c]}c*3{X[c]} @{}}
\multicolumn{3}{c}{Classification Accuracy}
&~&\multicolumn{3}{c}{Error Measure}
\\\cline{1-3}\cline{5-7}
ADAGRAD&ADADELTA&ADAM&&ADAGRAD&ADADELTA&ADAM
\\\cline{1-3}\cline{5-7}
1&1&1&&1&1&1
\end{tabu}
\caption{Performace metrics after 20 epochs}
\end{subfigure}
\caption[Performance comparison of training algorithms]{Classification accuracy on the test set and ...Performance metrics of the network given in ... trained
with different optimization algorithms}
\label{fig:comp_alg}
\end{figure}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "../main"
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@ -1,64 +0,0 @@
"","x_i","y_i","x_d","y_d","x","y"
"1",0,0,0.0815633019993375,0.095134925029757,0.0815633019993375,0.095134925029757
"2",0.1,0.0998334166468282,-0.137539012603596,0.503920419784276,-0.137539012603596,0.503920419784276
"3",0.2,0.198669330795061,0.219868163218743,0.32022289024623,0.219868163218743,0.32022289024623
"4",0.3,0.29552020666134,0.378332723534869,0.474906286765401,0.378332723534869,0.474906286765401
"5",0.4,0.389418342308651,0.286034335293811,0.422891394375764,0.215056588291437,0.412478430748051
"6",0.5,0.479425538604203,-0.109871707385461,0.229661026779107,0.122574532557623,0.353221043330047
"7",0.6,0.564642473395035,0.91036951450573,0.56079130435097,0.451160317716352,0.452893574072324
"8",0.7,0.644217687237691,0.899001194675409,0.714355793051917,0.491731451724399,0.514477919331008
"9",0.8,0.717356090899523,0.733791390723896,0.694085383523086,0.488943974889845,0.530054084580656
"10",0.9,0.783326909627483,0.893642943873427,0.739792642916928,0.599785378272423,0.575149967162231
"11",1,0.841470984807897,0.895913227983752,0.658288213778898,0.650886140047209,0.577618711891772
"12",1.1,0.891207360061435,1.01252219752013,0.808981437684505,0.726263244907525,0.643161394030218
"13",1.2,0.932039085967226,1.30930912337975,1.04111824066026,0.872590842152803,0.745714536528734
"14",1.3,0.963558185417193,1.0448292335495,0.741250429230841,0.850147062957694,0.687171673021914
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"17",1.6,0.999573603041505,1.34409615749122,0.976992098566069,1.13543598207093,0.889434319996364
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"56",5.5,-0.705540325570392,5.33298025214155,-0.343702005257262,5.0497327607228,-0.711573964373115
"57",5.6,-0.631266637872321,5.49935694796791,-0.828968673188174,5.15036520204232,-0.816467931201244
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"60",5.9,-0.373876664830236,5.48166674139637,-0.597796931577294,5.3357436834558,-0.649913835818738
"61",6,-0.279415498198926,5.77474590863769,-0.280234463056808,5.46956415981143,-0.524503219480344
"62",6.1,-0.182162504272095,6.36764321572312,-0.0996286988755344,5.7169871104113,-0.422854073705143
"63",6.2,-0.0830894028174964,6.46175133910451,-0.025702847911482,5.83540227044819,-0.355719019286555
1 x_i y_i x_d y_d x y
2 1 0 0 0.0815633019993375 0.095134925029757 0.0815633019993375 0.095134925029757
3 2 0.1 0.0998334166468282 -0.137539012603596 0.503920419784276 -0.137539012603596 0.503920419784276
4 3 0.2 0.198669330795061 0.219868163218743 0.32022289024623 0.219868163218743 0.32022289024623
5 4 0.3 0.29552020666134 0.378332723534869 0.474906286765401 0.378332723534869 0.474906286765401
6 5 0.4 0.389418342308651 0.286034335293811 0.422891394375764 0.215056588291437 0.412478430748051
7 6 0.5 0.479425538604203 -0.109871707385461 0.229661026779107 0.122574532557623 0.353221043330047
8 7 0.6 0.564642473395035 0.91036951450573 0.56079130435097 0.451160317716352 0.452893574072324
9 8 0.7 0.644217687237691 0.899001194675409 0.714355793051917 0.491731451724399 0.514477919331008
10 9 0.8 0.717356090899523 0.733791390723896 0.694085383523086 0.488943974889845 0.530054084580656
11 10 0.9 0.783326909627483 0.893642943873427 0.739792642916928 0.599785378272423 0.575149967162231
12 11 1 0.841470984807897 0.895913227983752 0.658288213778898 0.650886140047209 0.577618711891772
13 12 1.1 0.891207360061435 1.01252219752013 0.808981437684505 0.726263244907525 0.643161394030218
14 13 1.2 0.932039085967226 1.30930912337975 1.04111824066026 0.872590842152803 0.745714536528734
15 14 1.3 0.963558185417193 1.0448292335495 0.741250429230841 0.850147062957694 0.687171673021914
16 15 1.4 0.98544972998846 1.57369086195552 1.17277927321094 1.06520673597544 0.847936751231165
17 16 1.5 0.997494986604054 1.61427415976939 1.3908361301708 1.15616745244604 0.969474391592075
18 17 1.6 0.999573603041505 1.34409615749122 0.976992098566069 1.13543598207093 0.889434319996364
19 18 1.7 0.991664810452469 1.79278028030419 1.02939764179765 1.33272772191879 0.935067381106346
20 19 1.8 0.973847630878195 1.50721559744085 0.903076361857071 1.30862923824728 0.91665506605512
21 20 1.9 0.946300087687414 1.835014641556 0.830477479204284 1.45242210409837 0.889715842048808
22 21 2 0.909297426825682 1.98589997236352 0.887302138185342 1.56569111721857 0.901843632635883
23 22 2.1 0.863209366648874 2.31436634488224 0.890096618924313 1.73810390755555 0.899632162941341
24 23 2.2 0.80849640381959 2.14663445612581 0.697012453130415 1.77071083163663 0.831732978616874
25 24 2.3 0.74570521217672 2.17162372560288 0.614243640399509 1.84774268936257 0.787400621584077
26 25 2.4 0.675463180551151 2.2488591417345 0.447664288915269 1.93366609303299 0.707449056213168
27 26 2.5 0.598472144103957 2.56271588872389 0.553368843490625 2.08922735802261 0.702402440783529
28 27 2.6 0.515501371821464 2.60986205081511 0.503762006272682 2.17548673152621 0.657831176057599
29 28 2.7 0.42737988023383 2.47840649766003 0.215060732402894 2.20251747034638 0.533903400086802
30 29 2.8 0.334988150155905 2.99861119922542 0.28503285049582 2.43015164462239 0.512492561673074
31 30 2.9 0.239249329213982 3.09513467852082 0.245355736487949 2.54679545455398 0.461447717313721
32 31 3 0.141120008059867 2.86247369846558 0.0960140633436418 2.55274767368554 0.371740588261606
33 32 3.1 0.0415806624332905 2.79458017090243 -0.187923650913249 2.59422388058738 0.234694070506915
34 33 3.2 -0.0583741434275801 3.6498183243501 -0.186738431858275 2.9216851043241 0.173308072295566
35 34 3.3 -0.157745694143249 3.19424275971809 -0.221908035274934 2.86681135711315 0.101325637659584
36 35 3.4 -0.255541102026832 3.53166785156005 -0.295496842654793 3.03827050777863 0.0191967841533109
37 36 3.5 -0.35078322768962 3.53250700922714 -0.364585027403596 3.12709094619305 -0.0558446366563474
38 37 3.6 -0.442520443294852 3.52114271616751 -0.363845774016092 3.18702722489489 -0.10585071711408
39 38 3.7 -0.529836140908493 3.72033580551176 -0.386489608468821 3.31200591645168 -0.158195730190865
40 39 3.8 -0.611857890942719 4.0803717995796 -0.64779795182054 3.49862620703954 -0.284999326812438
41 40 3.9 -0.687766159183974 3.88351729419721 -0.604406622894426 3.51908925124143 -0.324791870057922
42 41 4 -0.756802495307928 3.9941257036697 -0.8061112437715 3.62222513609486 -0.438560071688316
43 42 4.1 -0.818277111064411 3.81674488816054 -0.548538951165239 3.63032709398802 -0.41285438330036
44 43 4.2 -0.871575772413588 4.47703348424544 -0.998992385231986 3.88581748102334 -0.592305016590357
45 44 4.3 -0.916165936749455 4.46179199544059 -0.969288921090897 3.96444243944485 -0.643076376622242
46 45 4.4 -0.951602073889516 4.15184730382548 -1.11987501275525 3.93838897981045 -0.743258835859858
47 46 4.5 -0.977530117665097 4.64522916494355 -0.772872365801468 4.15504805602606 -0.691414328153313
48 47 4.6 -0.993691003633465 4.68087925098283 -0.650422764094352 4.24176417425486 -0.675107584174976
49 48 4.7 -0.999923257564101 5.00475403211142 -0.922605880059771 4.41432228408005 -0.770625346502085
50 49 4.8 -0.996164608835841 4.71428836112322 -1.14280193223997 4.41279031790692 -0.861010494025717
51 50 4.9 -0.982452612624332 5.02115518218406 -0.9819618243158 4.57449352886454 -0.843786948015608
52 51 5 -0.958924274663138 4.92057344952522 -0.872931430146499 4.61418118503201 -0.836318916150308
53 52 5.1 -0.925814682327732 5.37277893732831 -0.91444926304078 4.81555148166217 -0.864686555983682
54 53 5.2 -0.883454655720153 5.19524942845082 -1.41169784739596 4.84152902094499 -1.03768305406186
55 54 5.3 -0.832267442223901 5.4432222181271 -0.726481337519931 4.98565483155961 -0.856094353978009
56 55 5.4 -0.772764487555987 4.98285013865449 -0.692803346852181 4.90897053115903 -0.838425020062396
57 56 5.5 -0.705540325570392 5.33298025214155 -0.343702005257262 5.0497327607228 -0.711573964373115
58 57 5.6 -0.631266637872321 5.49935694796791 -0.828968673188174 5.15036520204232 -0.816467931201244
59 58 5.7 -0.550685542597638 5.69204187550805 -0.481580461165225 5.26232964126231 -0.689500817105975
60 59 5.8 -0.464602179413757 5.84391772412888 -0.20453899468884 5.38069867877875 -0.564365367144995
61 60 5.9 -0.373876664830236 5.48166674139637 -0.597796931577294 5.3357436834558 -0.649913835818738
62 61 6 -0.279415498198926 5.77474590863769 -0.280234463056808 5.46956415981143 -0.524503219480344
63 62 6.1 -0.182162504272095 6.36764321572312 -0.0996286988755344 5.7169871104113 -0.422854073705143
64 63 6.2 -0.0830894028174964 6.46175133910451 -0.025702847911482 5.83540227044819 -0.355719019286555

45
TeX/Plots/sin_conv.tex
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@ -1,45 +0,0 @@
\begin{figure}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\begin{adjustbox}{width=\textwidth, height=0.25\textheight}
\begin{tikzpicture}
\begin{axis}[tick style = {draw = none}, xticklabel = \empty,
yticklabel=\empty]
\addplot [mark options={scale = 0.7}, mark = o] table
[x=x_d,y=y_d, col sep = comma] {Plots/Data/sin_conv.csv};
\addplot [red, mark=x] table [x=x_i, y=y_i, col sep=comma, color ='black'] {Plots/Data/sin_conv.csv};
\end{axis}
\end{tikzpicture}
\end{adjustbox}
\caption{True position (\textcolor{red}{red}), distorted position data (black)}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\begin{adjustbox}{width=\textwidth, height=0.25\textheight}
\begin{tikzpicture}
\begin{axis}[tick style = {draw = none}, xticklabel = \empty,
yticklabel=\empty]
\addplot [mark options={scale = 0.7}, mark = o] table [x=x,y=y, col
sep = comma] {Plots/Data/sin_conv.csv};
\addplot [red, mark=x] table [x=x_i, y=y_i, col sep=comma, color ='black'] {Plots/Data/sin_conv.csv};
\end{axis}
\end{tikzpicture}
\end{adjustbox}
\caption{True position (\textcolor{red}{red}), filtered position data (black)}
\end{subfigure}
\caption[Signal smoothing using convolution]{Example for noise reduction using convolution with simulated
positional data. As filter
$g(i)=\left(\nicefrac{1}{3},\nicefrac{1}{4},\nicefrac{1}{5},\nicefrac{1}{6},\nicefrac{1}{20}\right)_{(i-1)}$
is chosen and applied to the $x$ and $y$ coordinate
data seperately. The convolution of both signals with $g$
improves the MSE of the positions from 0.196 to 0.170 and
visibly smoothes the data.
}
\label{fig:sin_conv}
\end{figure}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "../main"
%%% End:

5
TeX/Plots/y.tex
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@ -1,5 +0,0 @@
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "../main"
%%% End:

22
TeX/appendixA.tex
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@ -1,6 +1,7 @@
\newpage
\begin{appendices}
\counterwithin{lstfloat}{section}
\section{Proofs for sone Lemmata in ...}
In the following there will be proofs for some important Lemmata in
Section~\ref{sec:theo38}. Further proofs not discussed here can be
@ -8,17 +9,20 @@
\begin{Theorem}[Proof of Lemma~\ref{theo38}]
\end{Theorem}
\begin{Lemma}[$\frac{w^{*,\tilde{\lambda}}_k}{v_k}\approx\mathcal{O}(\frac{1}{n})$]
For any $\lambda > 0$ and training data $(x_i^{\text{train}},
y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
\left\{1,\dots,N\right\}$, we have
\[
\max_{k \in \left\{1,\dots,n\right\}} \frac{w^{*,
\tilde{\lambda}}_k}{v_k} = \po_{n\to\infty}
\]
\begin{Lemma}[$\frac{w^{*,\tilde{\lambda}}_k}{v_k}\approx\mathcal{O}(\frac{1}{n})$]
For any $\lambda > 0$ and training data $(x_i^{\text{train}},
y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
\left\{1,\dots,N\right\}$, we have
\[
\max_{k \in \left\{1,\dots,n\right\}} \frac{w^{*,
\tilde{\lambda}}_k}{v_k} = \po_{n\to\infty}
\]
\end{Lemma}
\input{Appendix_code.tex}
\end{Lemma}
\end{appendices}

45
TeX/bibliograpy.bib
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@ -201,4 +201,49 @@ url={https://openreview.net/forum?id=rkgz2aEKDr}
doi = "https://doi.org/10.1016/j.neucom.2018.09.013",
url = "http://www.sciencedirect.com/science/article/pii/S0925231218310749",
author = "Maayan Frid-Adar and Idit Diamant and Eyal Klang and Michal Amitai and Jacob Goldberger and Hayit Greenspan"
}
@online{fashionMNIST,
author = {Han Xiao and Kashif Rasul and Roland Vollgraf},
title = {Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms},
date = {2017-08-28},
year = {2017},
eprintclass = {cs.LG},
eprinttype = {arXiv},
eprint = {cs.LG/1708.07747},
}
@inproceedings{10.1145/3206098.3206111,
author = {Kowsari, Kamran and Heidarysafa, Mojtaba and Brown, Donald E. and Meimandi, Kiana Jafari and Barnes, Laura E.},
title = {RMDL: Random Multimodel Deep Learning for Classification},
year = {2018},
isbn = {9781450363549},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/3206098.3206111},
doi = {10.1145/3206098.3206111},
booktitle = {Proceedings of the 2nd International Conference on Information System and Data Mining},
pages = {19–28},
numpages = {10},
keywords = {Supervised Learning, Deep Learning, Data Mining, Text Classification, Deep Neural Networks, Image Classification},
location = {Lakeland, FL, USA},
series = {ICISDM '18}
}
@article{random_erasing,
author = {Zhun Zhong and
Liang Zheng and
Guoliang Kang and
Shaozi Li and
Yi Yang},
title = {Random Erasing Data Augmentation},
journal = {CoRR},
volume = {abs/1708.04896},
year = {2017},
url = {http://arxiv.org/abs/1708.04896},
archivePrefix = {arXiv},
eprint = {1708.04896},
timestamp = {Mon, 13 Aug 2018 16:47:52 +0200},
biburl = {https://dblp.org/rec/journals/corr/abs-1708-04896.bib},
bibsource = {dblp computer science bibliography, https://dblp.org}
}

227
TeX/further_applications_of_nn.tex
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@ -2,9 +2,13 @@
As neural networks are applied to problems of higher complexity often
resulting in higher dimensionality of the input the amount of
parameters in the network rises drastically. For example a network
with ...
A way to combat the
parameters in the network rises drastically.
For very large inputs such as high resolution image data due to the
fully connected nature of the neural network the amount of parameters
can ... exceed the amount that is feasible for training and storage.
A way to combat this is by using layers which are only sparsely
connected and share parameters between nodes. This can be implemented
using convolution.\todo{Überleitung besser schreiben}
\subsection{Convolution}
@ -18,7 +22,7 @@ functions is integrated after one has been reversed and shifted.
This operation can be described as a filter-function $g$ being applied
to $f$,
as values $f(t)$ are being replaced by an average of values of $f$
weighted by $g$ in position $t$.
weighted by a filter-function $g$ in position $t$.
The convolution operation allows plentiful manipulation of data, with
a simple example being smoothing of real-time data. Consider a sensor
measuring the location of an object (e.g. via GPS). We expect the
@ -29,7 +33,7 @@ the data to reduce the noise. Using convolution for this task, we
can control the significance we want to give each data-point. We
might want to give a larger weight to more recent measurements than
older ones. If we assume these measurements are taken on a discrete
timescale, we need to introduce discrete convolution first. Let $f$,
timescale, we need to introduce discrete convolution first. \\Let $f$,
$g: \mathbb{Z} \to \mathbb{R}$ then
\[
@ -39,7 +43,7 @@ Applying this on the data with the filter $g$ chosen accordingly we
are
able to improve the accuracy, which can be seen in
Figure~\ref{fig:sin_conv}.
\input{Plots/sin_conv.tex}
\input{Figures/sin_conv.tex}
This form of discrete convolution can also be applied to functions
with inputs of higher dimensionality. Let $f$, $g: \mathbb{Z}^d \to
\mathbb{R}$ then
@ -51,12 +55,12 @@ with inputs of higher dimensionality. Let $f$, $g: \mathbb{Z}^d \to
This will prove to be a useful framework for image manipulation but
in order to apply convolution to images we need to discuss
representation of image data first. Most often images are represented
by each pixel being a mixture of base colors these base colors define
by each pixel being a mixture of base colors. These base colors define
the color-space in which the image is encoded. Often used are
color-spaces RGB (red,
blue, green) or CMYK (cyan, magenta, yellow, black). An example of an
image split in its red, green and blue channel is given in
Figure~\ref{fig:rgb} Using this
Figure~\ref{fig:rgb}. Using this
encoding of the image we can define a corresponding discrete function
describing the image, by mapping the coordinates $(x,y)$ of an pixel
and the
@ -75,13 +79,13 @@ channel (color) $c$ to the respective value $v$
\begin{tikzpicture}
\begin{scope}[x = (0:1cm), y=(90:1cm), z=(15:-0.5cm)]
\node[canvas is xy plane at z=0, transform shape] at (0,0)
{\includegraphics[width=5cm]{Plots/Data/klammern_r.jpg}};
{\includegraphics[width=5cm]{Figures/Data/klammern_r.jpg}};
\node[canvas is xy plane at z=2, transform shape] at (0,-0.2)
{\includegraphics[width=5cm]{Plots/Data/klammern_g.jpg}};
{\includegraphics[width=5cm]{Figures/Data/klammern_g.jpg}};
\node[canvas is xy plane at z=4, transform shape] at (0,-0.4)
{\includegraphics[width=5cm]{Plots/Data/klammern_b.jpg}};
{\includegraphics[width=5cm]{Figures/Data/klammern_b.jpg}};
\node[canvas is xy plane at z=4, transform shape] at (-8,-0.2)
{\includegraphics[width=5.3cm]{Plots/Data/klammern_rgb.jpg}};
{\includegraphics[width=5.3cm]{Figures/Data/klammern_rgb.jpg}};
\end{scope}
\end{tikzpicture}
\end{adjustbox}
@ -104,6 +108,14 @@ convolution
(I * g)_{x,y,c} = \sum_{i,j,l \in \mathbb{Z}} I_{x-i,y-j,c-l} g_{i,j,l}.
\]
As images are finite in size for pixels close enough to the border
that the filter ... the convolution is not well defined. In such cases
padding can be used. With padding the image is enlarged beyond .. with
0 entries to
ensure the convolution is well defined for all pixels. If no padding
is used the size of the output is reduced to \textit{size of input -
size of kernel +1} in each dimension.
Simple examples for image manipulation using
convolution are smoothing operations or
rudimentary detection of edges in grayscale images, meaning they only
@ -143,38 +155,38 @@ wise. Examples of convolution with both kernels are given in Figure~\ref{fig:img
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Plots/Data/klammern.jpg}
\includegraphics[width=\textwidth]{Figures/Data/klammern.jpg}
\caption{Original Picture}
\label{subf:OrigPicGS}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Plots/Data/image_conv9.png}
\includegraphics[width=\textwidth]{Figures/Data/image_conv9.png}
\caption{\hspace{-2pt}Gaussian Blur $\sigma^2 = 1$}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Plots/Data/image_conv10.png}
\includegraphics[width=\textwidth]{Figures/Data/image_conv10.png}
\caption{Gaussian Blur $\sigma^2 = 4$}
\end{subfigure}\\
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Plots/Data/image_conv4.png}
\includegraphics[width=\textwidth]{Figures/Data/image_conv4.png}
\caption{Sobel Operator $x$-direction}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Plots/Data/image_conv5.png}
\includegraphics[width=\textwidth]{Figures/Data/image_conv5.png}
\caption{Sobel Operator $y$-direction}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{Plots/Data/image_conv6.png}
\includegraphics[width=\textwidth]{Figures/Data/image_conv6.png}
\caption{Sobel Operator combined}
\end{subfigure}
% \begin{subfigure}{0.24\textwidth}
% \centering
% \includegraphics[width=\textwidth]{Plots/Data/image_conv6.png}
% \includegraphics[width=\textwidth]{Figures/Data/image_conv6.png}
% \caption{test}
% \end{subfigure}
\caption[Convolution applied on image]{Convolution of original greyscale Image (a) with different
@ -344,7 +356,7 @@ In order to illustrate this behavior we modeled a convolutional neural
network to ... handwritten digits. The data set used for this is the
MNIST database of handwritten digits (\textcite{MNIST},
Figure~\ref{fig:MNIST}).
\input{Plots/mnist.tex}
\input{Figures/mnist.tex}
The network used consists of two convolution and max pooling layers
followed by one fully connected hidden layer and the output layer.
Both covolutional layers utilize square filters of size five which are
@ -359,7 +371,7 @@ The architecture of the convolutional neural network is summarized in
Figure~\ref{fig:mnist_architecture}.
\begin{figure}
\missingfigure{network architecture}
\includegraphics[width=\textwidth]{Figures/Data/convnet_fig.pdf}
\caption{architecture}
\label{fig:mnist_architecture}
\end{figure}
@ -380,7 +392,7 @@ gradient calculated on the subset it performs far better than the
network using true gradients when training for the same mount of time.
\todo{vergleich training time}
\input{Plots/SGD_vs_GD.tex}
\input{Figures/SGD_vs_GD.tex}
\clearpage
\subsection{\titlecap{modified stochastic gradient descent}}
An inherent problem of the stochastic gradient descent algorithm is
@ -631,7 +643,7 @@ Here it can be seen that the ADAM algorithm performs far better than
the other algorithms, with AdaGrad and Adelta following... bla bla
\input{Plots/sdg_comparison.tex}
\input{Figures/sdg_comparison.tex}
% \subsubsubsection{Stochastic Gradient Descent}
\clearpage
@ -741,23 +753,23 @@ mirroring.
\begin{figure}[h]
\centering
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist0.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist0.pdf}
\caption{original\\image}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist_gen_zoom.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist_gen_zoom.pdf}
\caption{random\\zoom}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist_gen_shear.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist_gen_shear.pdf}
\caption{random\\shear}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist_gen_rotation.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist_gen_rotation.pdf}
\caption{random\\rotation}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{Plots/Data/mnist_gen_shift.pdf}
\includegraphics[width=\textwidth]{Figures/Data/mnist_gen_shift.pdf}
\caption{random\\positional shift}
\end{subfigure}
\caption[Image data generation]{Example for the manipuations used in ... As all images are
@ -781,9 +793,9 @@ reduction in overfitting can be seen in
accuracy decreases with test accuracy increasing. However utlitizing
data generation as well as dropout with a probability of 0.4 seems to
be a too aggressive approach as the training accuracy drops below the
test accuracy.
test accuracy\todo{kleine begründung}.
\input{Plots/gen_dropout.tex}
\input{Figures/gen_dropout.tex}
\todo{Vergleich verschiedene dropout größen auf MNSIT o.ä., subset als
training set?}
@ -796,24 +808,56 @@ the available data can be highly limited.
In these problems the networks are highly ... for overfitting the
data. In order to get a understanding of accuracys achievable and the
impact of the measures to prevent overfitting discussed above we and train
the network on datasets of varying sizes.
First we use the mnist handwriting dataset and then a slightly harder
problem given by the mnist fashion dataset which contains PREEDITED
pictures of clothes from 10 different categories.
\input{Plots/fashion_mnist.tex}
For training for each class a certain number of random datapoints are
chosen for training the network. The sizes chosen are:
full dataset: ... per class\\
1000 per class
100 per class
10 per class
the results for training .. are given in ... Here can be seen... that
for small training sets data generation has a large impact on the accuracy.
\begin{table}
the network on datasets of varying sizes with different measures implemented.
For training we use the mnist handwriting dataset as well as the fashion
mnist dataset. The fashion mnist dataset is a benchmark set build by
\textcite{fashionMNIST} in order to provide a harder set, as state of
the art models are able to achive accuracies of 99.88\%
(\textcite{10.1145/3206098.3206111}) on the handwriting set.
The dataset contains 70.000 preprocessed images of clothes from
zalando, a overview is given in Figure~\ref{fig:fashionMNIST}.
\input{Figures/fashion_mnist.tex}
\afterpage{
\noindent
\begin{minipage}{\textwidth}
\small
\begin{tabu} to \textwidth {@{}l*4{X[c]}@{}}
\Tstrut \Bstrut & \textsc{Adam} & D. 0.2 & Gen & Gen.+D. 0.2 \\
\hline
&
\multicolumn{4}{c}{\titlecap{test accuracy for 1 sample}}\Bstrut \\
\cline{2-5}
max \Tstrut & 0.5633 & 0.5312 & 0.6704 & 0.6604 \\
min & 0.3230 & 0.4224 & 0.4878 & 0.5175 \\
mean & 0.4570 & 0.4714 & 0.5862 & 0.6014 \\
var & 0.0040 & 0.0012 & 0.0036 & 0.0023 \\
\hline
&
\multicolumn{4}{c}{\titlecap{test accuracy for 10 samples}}\Bstrut \\
\cline{2-5}
max \Tstrut & 0.8585 & 0.9423 & 0.9310 & 0.9441 \\
min & 0.8148 & 0.9081 & 0.9018 & 0.9061 \\
mean & 0.8377 & 0.9270 & 0.9185 & 0.9232 \\
var & 2.7e-4 & 1.3e-4 & 6e-05 & 1.5e-4 \\
\hline
&
\multicolumn{4}{c}{\titlecap{test accuracy for 100 samples}}\Bstrut \\
\cline{2-5}
max & 0.9637 & 0.9796 & 0.9810 & 0.9805 \\
min & 0.9506 & 0.9719 & 0.9702 & 0.9727 \\
mean & 0.9582 & 0.9770 & 0.9769 & 0.9783 \\
var & 2e-05 & 1e-05 & 1e-05 & 0 \\
\hline
\end{tabu}
\normalsize
\captionof{table}{Values of the test accuracy of the model trained
10 times
on random MNIST handwriting training sets containing 1, 10 and 100
data points per class after 125 epochs. The mean achieved accuracy
for the full set employing both overfitting measures is }
\small
\centering
\begin{tabu} to \textwidth {@{}l*4{X[c]}@{}}
\Tstrut \Bstrut & \textsc{Adam} & D. 0.2 & Gen & Gen.+D. 0.2 \\
@ -843,14 +887,51 @@ for small training sets data generation has a large impact on the accuracy.
var & 2e-05 & 1e-05 & 1e-05 & 0 \\
\hline
\end{tabu}
\caption{Values of the test accuracy of the model trained 10 times
of random training sets containing 1, 10 and 100 data points per
class.}
\end{table}
\normalsize
\captionof{table}{Values of the test accuracy of the model trained 10 times
on random fashion MNIST training sets containing 1, 10 and 100 data points per
class. The mean achieved accuracy for the full dataset is: ....}
\end{minipage}
\clearpage % if needed/desired
}
The random datasets chosen for training are made up of a certain
number of datapoints for each class, which are chosen at random. The
sizes chosen for the comparisons are the full dataset, 100, 10 and 1
data points
per class.
For the task of classifying the fashion data a slightly altered model
is used. The convolutional layers with filters of size 5 are replaced
by two consecutive convolutional layers with filters of size 3.
This is done in order to have more ... in order to better ... the data
in the model. A diagram of the architecture is given in
Figure~\ref{fig:fashion_MNIST}.
For both scenarios the model are trained 10 times on randomly
... training sets. Additionally models of the same architecture where
a dropout layer with a ... 20\% is implemented and/or datageneration
is used to augment the data during training. The values for the
datageneration are given in CODE APPENDIX.
The models are trained for 125 epoch to ensure enough random
augmentations of the input images are considered to ensure
convergence. The test accuracies of the models after training for 125
epoch are given in Figure~\ref{...} for the handwriting
and in Figure~\ref{...} for the fashion scenario. Additionally the
average test accuracies of the models are given for each epoch in
Figure ... and Figure...
\begin{figure}
\includegraphics[width=\textwidth]{Figures/Data/cnn_fashion_fig.pdf}
\caption{Convolutional neural network architecture used to model the
fashion MNIST dataset.}
\label{fig:mnist_architecture}
\end{figure}
\begin{figure}[h]
\centering
\small
\begin{subfigure}[h]{\textwidth}
\begin{tikzpicture}
\begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
@ -861,16 +942,16 @@ for small training sets data generation has a large impact on the accuracy.
=1.25pt}]
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_1.mean};
{Figures/Data/adam_1.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_dropout_02_1.mean};
{Figures/Data/adam_dropout_02_1.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_datagen_1.mean};
{Figures/Data/adam_datagen_1.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_datagen_dropout_02_1.mean};
{Figures/Data/adam_datagen_dropout_02_1.mean};
\addlegendentry{\footnotesize{Default}}
@ -894,16 +975,16 @@ for small training sets data generation has a large impact on the accuracy.
=1.25pt}]
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_dropout_00_10.mean};
{Figures/Data/adam_dropout_00_10.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_dropout_02_10.mean};
{Figures/Data/adam_dropout_02_10.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_datagen_dropout_00_10.mean};
{Figures/Data/adam_datagen_dropout_00_10.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_datagen_dropout_02_10.mean};
{Figures/Data/adam_datagen_dropout_02_10.mean};
\addlegendentry{\footnotesize{Default.}}
@ -924,16 +1005,16 @@ for small training sets data generation has a large impact on the accuracy.
=1.25pt}, ymin = {0.92}]
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_dropout_00_100.mean};
{Figures/Data/adam_dropout_00_100.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_dropout_02_100.mean};
{Figures/Data/adam_dropout_02_100.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_datagen_dropout_00_100.mean};
{Figures/Data/adam_datagen_dropout_00_100.mean};
\addplot table
[x=epoch, y=val_accuracy, col sep=comma, mark = none]
{Plots/Data/adam_datagen_dropout_02_100.mean};
{Figures/Data/adam_datagen_dropout_02_100.mean};
\addlegendentry{\footnotesize{Default.}}
\addlegendentry{\footnotesize{D. 0.2}}
@ -945,27 +1026,29 @@ for small training sets data generation has a large impact on the accuracy.
\vspace{.25cm}
\end{subfigure}
\caption{}
\label{mnist fashion}
\label{fig:MNISTfashion}
\end{figure}
\begin{figure}[h]
\centering
\missingfigure{datagen fashion}
\caption{Sample pictures of the mnist fashioyn dataset, one per
\caption{Sample pictures of the mnist fashion dataset, one per
class.}
\label{mnist fashion}
\end{figure}
\clearpage
\section{Bla}
\section{Schluss}
\begin{itemize}
\item generate more data, GAN etc
\item generate more data, GAN etc \textcite{gan}
\item Transfer learning, use network trained on different task and
repurpose it / train it with the training data
repurpose it / train it with the training data \textcite{transfer_learning}
\item random erasing fashion mnist 96.35\% accuracy \textcite{random_erasing}
\end{itemize}
\textcite{transfer_learning}
\textcite{gan}
%%% Local Variables:
%%% mode: latex

150
TeX/main.tex
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@ -34,10 +34,13 @@
\usepackage{todonotes}
\usepackage{lipsum}
\usepackage[ruled,vlined]{algorithm2e}
\usepackage{showframe}
%\usepackage{showframe}
\usepackage[protrusion=true, expansion=true, kerning=true, letterspace
= 150]{microtype}
\usepackage{titlecaps}
\usepackage{afterpage}
\usepackage{xcolor}
\usepackage{chngcntr}
\captionsetup[sub]{justification=centering}
@ -52,7 +55,123 @@
\pgfplotsset{compat = 1.16}
\usepackage[export]{adjustbox}
\definecolor{maroon}{cmyk}{0, 0.87, 0.68, 0.32}
\definecolor{halfgray}{gray}{0.55}
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\usepackage{listings}
\usepackage{float}
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\def\lstfloatautorefname{Listing}
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identifierstyle=\color{black}\ttfamily,
commentstyle=\color{ipython_red}\ttfamily,
stringstyle=\color{ipython_red}\ttfamily,
keepspaces=true,
showspaces=false,
showstringspaces=false,
%
rulecolor=\color{ipython_frame},
frame=single,
frameround={t}{t}{t}{t},
framexleftmargin=6mm,
numbers=left,
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keywordstyle=\color{ipython_green}\ttfamily,
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keywordstyle = [3]{\color{ipython_purple}\ttfamily},
keywordstyle = [2]{\color{ipython_cyan}\ttfamily},
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\usepackage[style=authoryear, backend=bibtex]{biblatex}
\urlstyle{same}
@ -103,14 +222,31 @@
%\textbf{Seminar Machine--Learning: Unsupervised %Learning} \newline
%Institut für Mathematik der Universität %Augsburg\\
%Lehrstuhl für Rechnerorientierte Statistik und %Datenanalyse\\
\smallskip\hrule\bigskip
\begin{center}
{\huge{Electricity Price Forecasting based on Regression Tree Models}}
\end{center}
\hrulefill
\huge \textbf{Master Thesis}\\
\vspace{1cm}
\Large \textbf{University Augsburg\\Department of Mathematics\\Chair of
Computational Statistics and Data Analysis}
\vspace{1cm}
\end{center}
\begin{figure}[h]
\centering
\includegraphics[scale=1.3]{Figures/Uni_Aug_Siegel_32Grad_schwarz.png}
\end{figure}
\begin{center}
\vspace{1cm}
\huge \textbf{TITLE Neural Network bla blub langer Titel}\\
\vspace{1cm}
\huge \textbf{Tim Tobias Arndt}\\
\vspace{1cm}
\Large \textbf{October 2020}
\end{center}
\pagenumbering{gobble}
\newpage
\clearpage
%\setcounter{tocdepth}{4}
\tableofcontents
\clearpage

10
TeX/theo_3_8.tex
Unescape Escape View File

@ -195,13 +195,13 @@ plot coordinates {
height = 0.6\textwidth]
\addplot table
[x=x, y=y, col sep=comma, only marks,mark options={scale =
0.7}] {Plots/Data/overfit.csv};
0.7}] {Figures/Data/overfit.csv};
\addplot [red, line width=0.8pt] table [x=x_n, y=s_n, col
sep=comma, forget plot] {Plots/Data/overfit.csv};
sep=comma, forget plot] {Figures/Data/overfit.csv};
\addplot [black, line width=0.8pt] table [x=x_n, y=y_n, col
sep=comma] {Plots/Data/overfit.csv};
sep=comma] {Figures/Data/overfit.csv};
\addplot [black, line width=0.8pt, dashed] table [x=x, y=y, col
sep=comma] {Plots/Data/overfit_spline.csv};
sep=comma] {Figures/Data/overfit_spline.csv};
\addlegendentry{\footnotesize{data}};
\addlegendentry{\footnotesize{$\mathcal{NN}_{\vartheta^*}$}};
@ -950,7 +950,7 @@ results are given in Figure~\ref{fig:rs_vs_rs}, here it can be seen that in
the intervall of the traing data $[-\pi, \pi]$ the neural network and
smoothing spline are nearly identical, coinciding with the proposition.
\input{Plots/RN_vs_RS}
\input{Figures/RN_vs_RS}
%%% Local Variables:

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