final 2.0

.gitignore

@@ -8,6 +8,7 @@
 *.bbl
 *.tdo
 *.blg
+*.lof
 TeX/auto/*
 main-blx.bib
 
@@ -15,6 +16,7 @@ main-blx.bib
 *.tex~
 *#*.tex*
 *~
+*#*
 
 # no pdfs
 *.pdf
@@ -24,6 +26,7 @@ main-blx.bib
 *.png
 *.jpg
 *.xcf
+*.gif
 
 # no slurm logs
 *slurm*.out
@@ -31,3 +34,5 @@ main-blx.bib
 # no plot data
 *.csv
 *.mean
+
+*Plots_*

TF/Main.scala

@@ -0,0 +1,76 @@
+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)
+}

TF/cnn.py

@@ -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]]
+    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], [0.4,0.8], [0.4, 0.5], [0.4, 0.8],[0.4,0.8]]
+    end_ratio_list = [[0.4, 0.5], [0.4, 0.8], [0.4,0.5], [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)

TF/scratch.scala

@@ -0,0 +1,52 @@
+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

TeX/Appendix_code.tex

@@ -0,0 +1,621 @@
+\section{Implementations}
+In this section the implementations of the models used are given.
+The randomized shallow neural network used in Section~\ref{sec:conv} is
+implemented in Scala. No pre-existing frameworks were used to ensure
+the implementation was according to the definitions used in Theorem~\ref{theo:main1}.
+
+The neural networks used in Section~\ref{sec:cnn} are implemented in Python using
+the Keras framework given in TensorFlow. TensorFlow is a library
+containing highly efficient GPU implementations of a wide variety of
+tensor operations and algorithms
+for training neural networks.% (computing derivatives, updating parameters).
+
+\vspace*{-0.5cm}
+\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 Section~\ref{sec:rsnn_sim}.}
+  % 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{lstfloat}
+\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}
+  \caption{Python code used to build the network modeling the MNIST
+    handwritten digits data set.}
+  \label{lst:handwriting}
+\end{lstfloat}
+\clearpage
+\begin{lstfloat}
+\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 = 6,
+    zoom_range = 0.15,
+    width_shift_range=2,
+    height_shift_range=2,
+    shear_range = 0.15,
+    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=x_train.shape[0]//30,
+        validation_data=(x_test, y_test),
+        epochs=125, callbacks=[csv_logger],
+        shuffle=True)
+
+\end{lstlisting}
+  \caption[Python Code for fashion MNIST]{Python code
+    used to build the network modeling the fashion MNIST data set.}
+  \label{lst:fashion}
+\end{lstfloat}
+\clearpage
+\begin{lstfloat}
+\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}
+  \caption{Python code used to generate the data sets containing a
+    certain amount of random data points per class.}
+\end{lstfloat}
+
+\section{Additional Comparisons}
+\label{app:comp}
+In this section, comparisons of cross entropy loss and training
+accuracy for the models trained in Section~\ref{sec:smalldata} are given.
+\begin{figure}[h]
+  \centering
+  \small
+  \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 = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch},ylabel = {Test Loss}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}]
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_1.mean};  
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_02_1.mean};
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_1.mean}; 
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_02_1.mean};
+
+        
+        \addlegendentry{\footnotesize{Default}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G. + 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 = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch},ylabel = {Test Loss}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}]
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_00_10.mean}; 
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_02_10.mean}; 
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_00_10.mean}; 
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_02_10.mean};
+
+
+        \addlegendentry{\footnotesize{Default.}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G + D. 0.2}}
+      \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 = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch}, ylabel = {Test Loss}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}]
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_00_100.mean};
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_02_100.mean}; 
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_00_100.mean}; 
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_02_100.mean}; 
+        
+        \addlegendentry{\footnotesize{Default.}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G + D. 0.2}}
+      \end{axis}
+    \end{tikzpicture}
+    \caption{100 Samples per Class}
+    \vspace{.25cm}
+  \end{subfigure}
+  \caption[Mean Test Loss for Subsets of MNIST Handwritten
+  Digits]{Mean test cross entropy loss of the models fitting the
+    sampled subsets of MNIST 
+    handwritten digits over the 125 epochs of training.}
+\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,
+                     /pgf/number format/precision=3},tick style =
+                   {draw = none}, width = \textwidth,
+        height = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch},ylabel = {Test Loss}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}]
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_0_1.mean};  
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_2_1.mean};
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_0_1.mean}; 
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_2_1.mean};
+
+        
+        \addlegendentry{\footnotesize{Default}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G. + D. 0.2}}
+        \addlegendentry{\footnotesize{D. 0.4}}
+      \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 = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch},ylabel = {Test Loss}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}, ymin = {0.62}]
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_0_10.mean};  
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_2_10.mean};
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_0_10.mean}; 
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_2_10.mean};
+
+        
+        \addlegendentry{\footnotesize{Default.}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G + D. 0.2}}
+      \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 = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch}, ylabel = {Test Loss}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}]
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_0_100.mean};  
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_2_100.mean};
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_0_100.mean}; 
+        \addplot table
+        [x=epoch, y=val_loss, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_2_100.mean};
+        
+        \addlegendentry{\footnotesize{Default.}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G + D. 0.2}}
+      \end{axis}
+    \end{tikzpicture}
+    \caption{100 Samples per Class}
+    \vspace{.25cm}
+  \end{subfigure}
+  \caption[Mean Test Accuracies for Subsets of Fashion MNIST]{Mean
+    test cross entropy loss of the models fitting the sampled subsets
+    of fashion MNIST
+    over the 125 epochs of training.}
+\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,
+                     /pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
+        height = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch},ylabel = {Training Accuracy}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}]
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_1.mean};  
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_02_1.mean};
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_1.mean}; 
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_02_1.mean};
+
+        
+        \addlegendentry{\footnotesize{Default}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G. + 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 = 0.4\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}]
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_00_10.mean}; 
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_02_10.mean}; 
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_00_10.mean}; 
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_02_10.mean};
+
+
+        \addlegendentry{\footnotesize{Default.}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G + D. 0.2}}
+      \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 = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch}, ylabel = {Training Accuracy}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}, ymin = {0.92}]
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_00_100.mean};
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_dropout_02_100.mean}; 
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_00_100.mean}; 
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/adam_datagen_dropout_02_100.mean}; 
+        
+        \addlegendentry{\footnotesize{Default.}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G + D. 0.2}}
+      \end{axis}
+    \end{tikzpicture}
+    \caption{100 Samples per Class}
+    \vspace{.25cm}
+  \end{subfigure}
+  \caption[Mean Training Accuracies for Subsets of MNIST Handwritten
+  Digits]{Mean training accuracies of the models fitting the sampled
+    subsets of MNIST 
+    handwritten digits over the 125 epochs of training.}
+\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,
+                     /pgf/number format/precision=3},tick style =
+                   {draw = none}, width = \textwidth,
+        height = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch},ylabel = {Training Accuracy}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}]
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_0_1.mean};  
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_2_1.mean};
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_0_1.mean}; 
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_2_1.mean};
+
+        
+        \addlegendentry{\footnotesize{Default}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G. + D. 0.2}}
+        \addlegendentry{\footnotesize{D. 0.4}}
+      \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 = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch},ylabel = {Training Accuracy}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}, ymin = {0.62}]
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_0_10.mean};  
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_2_10.mean};
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_0_10.mean}; 
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_2_10.mean};
+
+        
+        \addlegendentry{\footnotesize{Default.}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G + D. 0.2}}
+      \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 = 0.4\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+        xlabel = {Epoch}, ylabel = {Training Accuracy}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt}]
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_0_100.mean};  
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_dropout_2_100.mean};
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_0_100.mean}; 
+        \addplot table
+        [x=epoch, y=accuracy, col sep=comma, mark = none]
+        {Figures/Data/fashion_datagen_dropout_2_100.mean};
+        
+        \addlegendentry{\footnotesize{Default.}}
+        \addlegendentry{\footnotesize{D. 0.2}}
+        \addlegendentry{\footnotesize{G.}}
+        \addlegendentry{\footnotesize{G + D. 0.2}}
+      \end{axis}
+    \end{tikzpicture}
+    \caption{100 Samples per Class}
+    \vspace{.25cm}
+  \end{subfigure}
+  \caption[Mean Training Accuracies for Subsets of Fashion MNIST]{Mean
+    training accuracies of the models fitting the sampled subsets of fashion MNIST 
+    over the 125 epochs of training.}
+\end{figure}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "main"
+%%% End:

TeX/Figures/RN_vs_RS.tex

@@ -10,13 +10,14 @@ plot coordinates {
 }
 }
 \begin{figure}
-  \begin{subfigure}[b]{0.5\textwidth}
+  \begin{subfigure}[b]{0.48\textwidth}
     \begin{subfigure}[b]{\textwidth}
       \begin{adjustbox}{width=\textwidth, height=0.25\textheight}
         \begin{tikzpicture}
           \begin{axis}[
         ytick = {-1, 0, 1, 2},
-        yticklabels = {$-1$, $\phantom{-0.}0$, $1$, $2$},]
+        yticklabels = {$-1$, $\phantom{-0.}0$, $1$, $2$},
+        restrict x to domain=-4:4, enlarge x limits = {0.1}]
             \addplot table [x=x, y=y, col sep=comma, only marks,
             forget plot] {Figures/Data/sin_6.csv};
             \addplot [black, line width=2pt] table [x=x, y=y, col
@@ -33,7 +34,7 @@ plot coordinates {
     \begin{subfigure}[b]{\textwidth}
       \begin{adjustbox}{width=\textwidth, height=0.25\textheight}
         \begin{tikzpicture}
-          \begin{axis}
+          \begin{axis}[restrict x to domain=-4:4, enlarge x limits = {0.1}]
             \addplot table [x=x, y=y, col sep=comma, only marks,
             forget plot] {Figures/Data/sin_6.csv};
             \addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Figures/Data/matlab_1.csv};
@@ -49,7 +50,7 @@ plot coordinates {
     \begin{subfigure}[b]{\textwidth}
       \begin{adjustbox}{width=\textwidth, height=0.25\textheight}
         \begin{tikzpicture}
-          \begin{axis}
+          \begin{axis}[restrict x to domain=-4:4, enlarge x limits = {0.1}]
             \addplot table [x=x, y=y, col sep=comma, only marks,
             forget plot] {Figures/Data/sin_6.csv};
             \addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Figures/Data/matlab_3.csv};
@@ -63,13 +64,14 @@ plot coordinates {
       \caption{$\lambda = 3.0$}
     \end{subfigure}
   \end{subfigure}
-  \begin{subfigure}[b]{0.5\textwidth}
+  \begin{subfigure}[b]{0.48\textwidth}
     \begin{subfigure}[b]{\textwidth}
       \begin{adjustbox}{width=\textwidth, height=0.245\textheight}
         \begin{tikzpicture}
           \begin{axis}[
         ytick = {-2,-1, 0, 1, 2},
-        yticklabels = {$-2$,$-1$, $\phantom{-0.}0$, $1$, $2$},]
+        yticklabels = {$-2$,$-1$, $\phantom{-0.}0$, $1$, $2$},
+        restrict x to domain=-4:4, enlarge x limits = {0.1}]
             \addplot table [x=x, y=y, col sep=comma, only marks,
             forget plot] {Figures/Data/data_sin_d_t.csv};
             \addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Figures/Data/matlab_sin_d_01.csv};
@@ -85,7 +87,7 @@ plot coordinates {
     \begin{subfigure}[b]{\textwidth}
       \begin{adjustbox}{width=\textwidth, height=0.25\textheight}
         \begin{tikzpicture}
-          \begin{axis}
+          \begin{axis}[restrict x to domain=-4:4, enlarge x limits = {0.1}]
             \addplot table [x=x, y=y, col sep=comma, only marks,
             forget plot] {Figures/Data/data_sin_d_t.csv};
             \addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Figures/Data/matlab_sin_d_1.csv};
@@ -101,7 +103,7 @@ plot coordinates {
     \begin{subfigure}[b]{\textwidth}
       \begin{adjustbox}{width=\textwidth, height=0.25\textheight}
         \begin{tikzpicture}
-          \begin{axis}
+          \begin{axis}[restrict x to domain=-4:4, enlarge x limits = {0.1}]
             \addplot table [x=x, y=y, col sep=comma, only marks,
             forget plot] {Figures/Data/data_sin_d_t.csv};
             \addplot [black, line width=2pt] table [x=x, y=y, col sep=comma, mark=none] {Figures/Data/matlab_sin_d_3.csv};
@@ -115,8 +117,8 @@ plot coordinates {
       \caption{$\lambda = 3.0$}
     \end{subfigure}
   \end{subfigure}
-  \caption[Comparison of shallow neural networks and regression
+  \caption[Comparison of Shallow Neural Networks and Regression
-  splines]{% In these Figures the behaviour stated in ... is
+  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
@@ -131,11 +133,12 @@ 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:

TeX/Figures/SGD_vs_GD.tex

@@ -4,28 +4,32 @@ legend image code/.code={
 \draw[mark repeat=2,mark phase=2]
 plot coordinates {
 (0cm,0cm)
-(0.0cm,0cm)        %% default is (0.3cm,0cm)
+(0.15cm,0cm)        %% default is (0.3cm,0cm)
-(0.0cm,0cm)         %% default is (0.6cm,0cm)
+(0.3cm,0cm)         %% default is (0.6cm,0cm)
 };%
 }
 }
 \begin{figure}
   \begin{subfigure}[h!]{\textwidth}
     \begin{tikzpicture}
-      \begin{axis}[tick style = {draw = none}, width = \textwidth,
+      \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
-        height = 0.6\textwidth,
+                     /pgf/number format/precision=3},tick style = {draw = none}, width = 0.975\textwidth,
+        height = 0.6\textwidth, legend
+        style={at={(0.0125,0.7)},anchor=north west},
+        xlabel = {Epoch}, ylabel = {Test Accuracy}, cycle
+        list/Dark2, every axis plot/.append style={line width
+          =1.25pt, mark = *, mark size=1pt},
         xtick = {1, 3, 5,7,9,11,13,15,17,19},
         xticklabels = {$2$, $4$, $6$, $8$,
-          $10$,$12$,$14$,$16$,$18$,$20$},
+          $10$,$12$,$14$,$16$,$18$,$20$}]
-        xlabel = {training epoch}, ylabel = {classification accuracy}]
         \addplot table
         [x=epoch, y=val_accuracy, col sep=comma] {Figures/Data/GD_01.log};
         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma] {Figures/Data/GD_05.log};
+        [x=epoch, y=val_accuracy, col sep=comma, mark = *] {Figures/Data/GD_05.log};
         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma] {Figures/Data/GD_1.log};
+        [x=epoch, y=val_accuracy, col sep=comma, mark = *] {Figures/Data/GD_1.log};
         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma]
+        [x=epoch, y=val_accuracy, col sep=comma, mark = *]
         {Figures/Data/SGD_01_b32.log};
         
          \addlegendentry{GD$_{0.01}$}
@@ -34,59 +38,65 @@ plot coordinates {
         \addlegendentry{SGD$_{0.01}$}
       \end{axis}
     \end{tikzpicture}
-    %\caption{Classification accuracy}
+    \caption{Test accuracy during training.}
   \end{subfigure}
-  \begin{subfigure}[b]{\textwidth}
+  % \begin{subfigure}[b]{\textwidth}
-    \begin{tikzpicture}
+  %   \begin{tikzpicture}
-      \begin{axis}[tick style = {draw = none}, width = \textwidth,
+  %     \begin{axis}[tick style = {draw = none}, width = \textwidth,
-        height = 0.6\textwidth,
+  %       height = 0.6\textwidth,
-        ytick = {0, 1, 2, 3, 4},
+  %       ytick = {0, 1, 2, 3, 4},
-        yticklabels = {$0$, $1$, $\phantom{0.}2$, $3$, $4$},
+  %       yticklabels = {$0$, $1$, $\phantom{0.}2$, $3$, $4$},
-        xtick = {1, 3, 5,7,9,11,13,15,17,19},
+  %       xtick = {1, 3, 5,7,9,11,13,15,17,19},
-        xticklabels = {$2$, $4$, $6$, $8$,
+  %       xticklabels = {$2$, $4$, $6$, $8$,
-          $10$,$12$,$14$,$16$,$18$,$20$},
+  %         $10$,$12$,$14$,$16$,$18$,$20$},
-        xlabel = {training epoch}, ylabel = {error measure\vphantom{fy}}]
+  %       xlabel = {training epoch}, ylabel = {error measure\vphantom{fy}}]
-        \addplot table
+  %       \addplot table
-        [x=epoch, y=val_loss, col sep=comma] {Figures/Data/GD_01.log};
+  %       [x=epoch, y=val_loss, col sep=comma] {Figures/Data/GD_01.log};
-        \addplot table
+  %       \addplot table
-        [x=epoch, y=val_loss, col sep=comma] {Figures/Data/GD_05.log};
+  %       [x=epoch, y=val_loss, col sep=comma] {Figures/Data/GD_05.log};
-        \addplot table
+  %       \addplot table
-        [x=epoch, y=val_loss, col sep=comma] {Figures/Data/GD_1.log};
+  %       [x=epoch, y=val_loss, col sep=comma] {Figures/Data/GD_1.log};
-        \addplot table
+  %       \addplot table
-        [x=epoch, y=val_loss, col sep=comma] {Figures/Data/SGD_01_b32.log};
+  %       [x=epoch, y=val_loss, col sep=comma] {Figures/Data/SGD_01_b32.log};
         
-        \addlegendentry{GD$_{0.01}$}
+  %       \addlegendentry{GD$_{0.01}$}
-        \addlegendentry{GD$_{0.05}$}
+  %       \addlegendentry{GD$_{0.05}$}
-        \addlegendentry{GD$_{0.1}$}
+  %       \addlegendentry{GD$_{0.1}$}
-        \addlegendentry{SGD$_{0.01}$}
+  %       \addlegendentry{SGD$_{0.01}$}
         
-      \end{axis}
+  %     \end{axis}
-    \end{tikzpicture}
+  %   \end{tikzpicture}
-    \caption{Performance metrics during training}
+  %   \caption{Performance metrics during training}
-  \end{subfigure}
+  % \end{subfigure}
   % \\~\\
-  \caption[Performance comparison of SDG and GD]{The neural network given in ?? trained with different
+  
+  \begin{subfigure}[b]{1.0\linewidth}
+    \begin{tabu} to \textwidth {@{} *4{X[c]}c*4{X[c]} @{}}
+      \multicolumn{4}{c}{Test Accuracy}
+      &~&\multicolumn{4}{c}{Test Loss}
+      \\\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}
+      0.265&0.633&0.203&0.989&&2.267&1.947&3.911&0.032 \\
+      \multicolumn{4}{c}{Training Accuracy}
+      &~&\multicolumn{4}{c}{Training Loss}
+      \\\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}
+      0.250&0.599&0.685&0.996&&2.271&1.995&1.089&0.012 \\
+    \end{tabu}
+    \caption{Performance metrics after 20 training epochs.}
+    \label{table:sgd_vs_gd}
+  \end{subfigure}
+  
+  \caption[Performance Comparison of SDG and GD]{The neural network
+    given in Figure~\ref{fig:mnist_architecture} 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
+    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"

TeX/Figures/fashion_mnist.tex

@@ -40,11 +40,11 @@
     \includegraphics[width=\textwidth]{Figures/Data/fashion_mnist9.pdf}
     \caption{Ankle boot}
   \end{subfigure}
-  \caption[Fashion MNIST data set]{The fashtion MNIST data set contains 70.000 images of
+  \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.}
+    the rest is used to validate the models trained.}
   \label{fig:fashionMNIST}
 \end{figure}
 %%% Local Variables:

TeX/Figures/gen_dropout.tex

@@ -16,7 +16,7 @@ plot coordinates {
       \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
                      /pgf/number format/precision=3},tick style = {draw = none}, width = 0.975\textwidth,
         height = 0.6\textwidth, ymin = 0.988, legend style={at={(0.9825,0.0175)},anchor=south east},
-        xlabel = {epoch}, ylabel = {Classification Accuracy}, cycle
+        xlabel = {Epoch}, 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]
@@ -45,18 +45,18 @@ plot coordinates {
         \addlegendentry{\footnotesize{Default}}
       \end{axis}
     \end{tikzpicture}
-    \caption{Classification accuracy}
+    \caption{Test 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 \\
+      \Tstrut \Bstrut & Default & 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 \\
+      mean     \Tstrut   & 0.9914 & 0.9923 & 0.9930 & 0.9937 & 0.9943 & 0.9944 \\
-      max                & 0.9926 & 0.9930 & 0.9934 & 0.9946 & 0.9955 & 0.9956 \\
+      max                & 0.9926 & 0.9930 & 0.9934 & 0.9946 & 0.9957 & 0.9956 \\
-      min                & 0.9887 & 0.9909 & 0.9922 & 0.9929 & 0.9929 & 0.9934 \\
+      min                & 0.9887 & 0.9909 & 0.9922 & 0.9929 & 0.9930 & 0.9934 \\
       \hline
       \multicolumn{7}{c}{Training Accuracy}\Bstrut                             \\
       \cline{2-7}
@@ -64,15 +64,16 @@ plot coordinates {
       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.}
+    \caption{Mean, maximum and minimum accuracy after 50 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.),
+  \caption[Performance Comparison of Overfitting Measures]{Accuracy
+    for the net given in Figure~\ref{fig:mnist_architecture} 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
+    model was trained five 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}

TeX/Figures/mnist.tex

@@ -1,38 +1,39 @@
 \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
+  \caption[MNIST Database of Handwritten Digits]{The MNIST database of handwritten
+    digits 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.}
+    the rest is used to validate the models trained.}
   \label{fig:MNIST}
 \end{figure}
 %%% Local Variables:

TeX/Figures/pfg_test.tex

@@ -8,286 +8,363 @@
 \usepackage{showframe}
 \usepackage{graphicx}
 \usepackage{titlecaps}
+\usepackage{amssymb}
+\usepackage{mathtools}%add-on and patches to amsmath
 \usetikzlibrary{calc, 3d}
 \usepgfplotslibrary{colorbrewer}
 
 \newcommand\Tstrut{\rule{0pt}{2.6ex}}         % = `top' strut
 \newcommand\Bstrut{\rule[-0.9ex]{0pt}{0pt}}   % = `bottom' strut
 
+\DeclareMathOperator*{\plim}{\mathbb{P}\text{-}\lim}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator*{\argmin}{arg\,min}
+\DeclareMathOperator*{\po}{\mathbb{P}\text{-}\mathcal{O}}
+\DeclareMathOperator*{\equals}{=}
 \begin{document}
-\pgfplotsset{
+\newcommand{\plimn}[0]{\plim\limits_{n \to \infty}}
-compat=1.11,
+\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
-legend image code/.code={
+% \pgfplotsset{
-\draw[mark repeat=2,mark phase=2]
+% compat=1.11,
-plot coordinates {
+% legend image code/.code={
-(0cm,0cm)
+% \draw[mark repeat=2,mark phase=2]
-(0.3cm,0cm)        %% default is (0.3cm,0cm)
+% plot coordinates {
-(0.6cm,0cm)         %% default is (0.6cm,0cm)
+% (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{figure}
-      \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
+%   \begin{subfigure}[h]{\textwidth}
-                     /pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
+%     \begin{tikzpicture}
-        height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
+%       \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
-        ylabel = {Test Accuracy}, cycle
+%                      /pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
-        list/Dark2, every axis plot/.append style={line width
+%         height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
-          =1.25pt}]
+%         ylabel = {Test Accuracy}, cycle
-        % \addplot [dashed] table
+%         list/Dark2, every axis plot/.append style={line width
-        % [x=epoch, y=accuracy, col sep=comma, mark = none]
+%           =1.25pt}]
-        % {Data/adam_datagen_full.log};
+%         % \addplot [dashed] table
-        \addplot table
+%         % [x=epoch, y=accuracy, col sep=comma, mark = none]
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         % {Data/adam_datagen_full.log};
-        {Data/adam_1.mean};  
+%         \addplot table
-        % \addplot [dashed] table
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        % [x=epoch, y=accuracy, col sep=comma, mark = none]
+%         {Data/adam_1.mean};  
-        % {Data/adam_datagen_dropout_02_full.log}; 
+%         % \addplot [dashed] table
-        \addplot table
+%         % [x=epoch, y=accuracy, col sep=comma, mark = none]
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         % {Data/adam_datagen_dropout_02_full.log}; 
-        {Data/adam_datagen_1.mean}; 
+%         \addplot table
-        \addplot table
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         {Data/adam_datagen_1.mean}; 
-        {Data/adam_datagen_dropout_02_1.mean};
+%         \addplot table
-        \addplot table
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         {Data/adam_datagen_dropout_02_1.mean};
-        {Data/adam_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.}}
-        \addlegendentry{\footnotesize{G. + D. 0.2}}
+%         \addlegendentry{\footnotesize{G. + D. 0.2}}
-        \addlegendentry{\footnotesize{G. + D. 0.4}}
+%         \addlegendentry{\footnotesize{G. + D. 0.4}}
-        \addlegendentry{\footnotesize{D. 0.2}}
+%         \addlegendentry{\footnotesize{D. 0.2}}
-        \addlegendentry{\footnotesize{D. 0.4}}
+%         \addlegendentry{\footnotesize{D. 0.4}}
-        \addlegendentry{\footnotesize{Default}}
+%         \addlegendentry{\footnotesize{Default}}
-      \end{axis}
+%       \end{axis}
-    \end{tikzpicture}
+%     \end{tikzpicture}
-    \caption{1 sample per class}
+%     \caption{1 sample per class}
-    \vspace{0.25cm}
+%     \vspace{0.25cm}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}[h]{\textwidth}
+%   \begin{subfigure}[h]{\textwidth}
-    \begin{tikzpicture}
+%     \begin{tikzpicture}
-      \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
+%       \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
-                     /pgf/number format/precision=3},tick style = {draw = none}, width = \textwidth,
+%                      /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},
+%         height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
-        ylabel = {Test Accuracy}, cycle
+%         ylabel = {Test Accuracy}, cycle
-        list/Dark2, every axis plot/.append style={line width
+%         list/Dark2, every axis plot/.append style={line width
-          =1.25pt}]
+%           =1.25pt}]
-        \addplot table
+%         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Data/adam_dropout_00_10.mean}; 
+%         {Data/adam_dropout_00_10.mean}; 
-        \addplot table
+%         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Data/adam_dropout_02_10.mean}; 
+%         {Data/adam_dropout_02_10.mean}; 
-        \addplot table
+%         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Data/adam_datagen_dropout_00_10.mean}; 
+%         {Data/adam_datagen_dropout_00_10.mean}; 
-        \addplot table
+%         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Data/adam_datagen_dropout_02_10.mean};
+%         {Data/adam_datagen_dropout_02_10.mean};
 
         
-        \addlegendentry{\footnotesize{G.}}
+%         \addlegendentry{\footnotesize{G.}}
-        \addlegendentry{\footnotesize{G. + D. 0.2}}
+%         \addlegendentry{\footnotesize{G. + D. 0.2}}
-        \addlegendentry{\footnotesize{G. + D. 0.4}}
+%         \addlegendentry{\footnotesize{G. + D. 0.4}}
-        \addlegendentry{\footnotesize{D. 0.2}}
+%         \addlegendentry{\footnotesize{D. 0.2}}
-        \addlegendentry{\footnotesize{D. 0.4}}
+%         \addlegendentry{\footnotesize{D. 0.4}}
-        \addlegendentry{\footnotesize{Default}}
+%         \addlegendentry{\footnotesize{Default}}
-      \end{axis}
+%       \end{axis}
-    \end{tikzpicture}
+%     \end{tikzpicture}
-    \caption{10 samples per class}
+%     \caption{10 samples per class}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}[h]{\textwidth}
+%   \begin{subfigure}[h]{\textwidth}
-    \begin{tikzpicture}
+%     \begin{tikzpicture}
-      \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
+%       \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,
+%                      /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},
+%         height = 0.35\textwidth, legend style={at={(0.9825,0.0175)},anchor=south east},
-        xlabel = {epoch}, ylabel = {Test Accuracy}, cycle
+%         xlabel = {epoch}, ylabel = {Test Accuracy}, cycle
-        list/Dark2, every axis plot/.append style={line width
+%         list/Dark2, every axis plot/.append style={line width
-          =1.25pt}, ymin = {0.92}]
+%           =1.25pt}, ymin = {0.92}]
-        \addplot table
+%         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Data/adam_dropout_00_100.mean};
+%         {Data/adam_dropout_00_100.mean};
-        \addplot table
+%         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Data/adam_dropout_02_100.mean}; 
+%         {Data/adam_dropout_02_100.mean}; 
-        \addplot table
+%         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Data/adam_datagen_dropout_00_100.mean}; 
+%         {Data/adam_datagen_dropout_00_100.mean}; 
-        \addplot table
+%         \addplot table
-        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+%         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Data/adam_datagen_dropout_02_100.mean}; 
+%         {Data/adam_datagen_dropout_02_100.mean}; 
         
-        \addlegendentry{\footnotesize{G.}}
+%         \addlegendentry{\footnotesize{G.}}
-        \addlegendentry{\footnotesize{G. + D. 0.2}}
+%         \addlegendentry{\footnotesize{G. + D. 0.2}}
-        \addlegendentry{\footnotesize{G. + D. 0.4}}
+%         \addlegendentry{\footnotesize{G. + D. 0.4}}
-        \addlegendentry{\footnotesize{D. 0.2}}
+%         \addlegendentry{\footnotesize{D. 0.2}}
-        \addlegendentry{\footnotesize{D. 0.4}}
+%         \addlegendentry{\footnotesize{D. 0.4}}
-        \addlegendentry{\footnotesize{Default}}
+%         \addlegendentry{\footnotesize{Default}}
-      \end{axis}
+%       \end{axis}
-    \end{tikzpicture}
+%     \end{tikzpicture}
-    \caption{100 samples per class}
+%     \caption{100 samples per class}
-    \vspace{.25cm}
+%     \vspace{.25cm}
-  \end{subfigure}
+%   \end{subfigure}
-  \caption{Accuracy for the net given in ... with Dropout (D.),
+%   \caption{Accuracy for the net given in ... with Dropout (D.),
-    data generation (G.), a combination, or neither (Default) implemented and trained
+%     data generation (G.), a combination, or neither (Default) implemented and trained
-    with \textsc{Adam}. For each epoch the 60.000 training samples     
+%     with \textsc{Adam}. For each epoch the 60.000 training samples     
-    were used, or for data generation 10.000 steps with each using
+%     were used, or for data generation 10.000 steps with each using
-    batches of 60 generated data points. For each configuration the
+%     batches of 60 generated data points. For each configuration the
-    model was trained 5 times and the average accuracies at each epoch
+%     model was trained 5 times and the average accuracies at each epoch
-    are given in (a). Mean, maximum and minimum values of accuracy on
+%     are given in (a). Mean, maximum and minimum values of accuracy on
-    the test and training set are given in (b).}
+%     the test and training set are given in (b).}
-\end{figure}
+% \end{figure}
-\begin{table}
+% \begin{table}
-  \centering
+%   \centering
-  \begin{tabu} to \textwidth {@{}l*4{X[c]}@{}}
+%   \begin{tabu} to \textwidth {@{}l*4{X[c]}@{}}
-      \Tstrut \Bstrut    & \textsc{Adam} & D. 0.2 & Gen    & Gen.+D. 0.2  \\
+%       \Tstrut \Bstrut    & \textsc{Adam} & D. 0.2 & Gen    & Gen.+D. 0.2  \\
-      \hline           
+%       \hline           
-                         & 
+%                          & 
-      \multicolumn{4}{c}{\titlecap{test accuracy for 1 sample}}\Bstrut    \\
+%       \multicolumn{4}{c}{\titlecap{test accuracy for 1 sample}}\Bstrut    \\
-      \cline{2-5}
+%       \cline{2-5}
-       max     \Tstrut   & 0.5633        & 0.5312 & 0.6704 & 0.6604       \\
+%        max     \Tstrut   & 0.5633        & 0.5312 & 0.6704 & 0.6604       \\
-       min               & 0.3230        & 0.4224 & 0.4878 & 0.5175       \\
+%        min               & 0.3230        & 0.4224 & 0.4878 & 0.5175       \\
-       mean              & 0.4570        & 0.4714 & 0.5862 & 0.6014       \\
+%        mean              & 0.4570        & 0.4714 & 0.5862 & 0.6014       \\
-       var               & 0.0040        & 0.0012 & 0.0036 & 0.0023       \\
+%        var               & 0.0040        & 0.0012 & 0.0036 & 0.0023       \\
-      \hline
+%       \hline
-                         & 
+%                          & 
-      \multicolumn{4}{c}{\titlecap{test accuracy for 10 samples}}\Bstrut  \\
+%       \multicolumn{4}{c}{\titlecap{test accuracy for 10 samples}}\Bstrut  \\
-      \cline{2-5}
+%       \cline{2-5}
-       max       \Tstrut & 0.8585        & 0.9423 & 0.9310 & 0.9441       \\
+%        max       \Tstrut & 0.8585        & 0.9423 & 0.9310 & 0.9441       \\
-       min               & 0.8148        & 0.9081 & 0.9018 & 0.9061       \\
+%        min               & 0.8148        & 0.9081 & 0.9018 & 0.9061       \\
-       mean              & 0.8377        & 0.9270 & 0.9185 & 0.9232       \\
+%        mean              & 0.8377        & 0.9270 & 0.9185 & 0.9232       \\
-       var               & 2.7e-4        & 1.3e-4 & 6e-05  & 1.5e-4       \\
+%        var               & 2.7e-4        & 1.3e-4 & 6e-05  & 1.5e-4       \\
-      \hline
+%       \hline
-                         & 
+%                          & 
-      \multicolumn{4}{c}{\titlecap{test accuracy for 100 samples}}\Bstrut \\
+%       \multicolumn{4}{c}{\titlecap{test accuracy for 100 samples}}\Bstrut \\
-      \cline{2-5}
+%       \cline{2-5}
-       max               & 0.9637        & 0.9796 & 0.9810 & 0.9805       \\
+%        max               & 0.9637        & 0.9796 & 0.9810 & 0.9805       \\
-       min               & 0.9506        & 0.9719 & 0.9702 & 0.9727       \\
+%        min               & 0.9506        & 0.9719 & 0.9702 & 0.9727       \\
-       mean              & 0.9582        & 0.9770 & 0.9769 & 0.9783       \\
+%        mean              & 0.9582        & 0.9770 & 0.9769 & 0.9783       \\
-       var               & 2e-05         & 1e-05  & 1e-05  & 0            \\
+%        var               & 2e-05         & 1e-05  & 1e-05  & 0            \\
-      \hline
+%       \hline
-    \end{tabu}
+%     \end{tabu}
-    \caption{Values of the test accuracy of the model trained 10 times
+%     \caption{Values of the test accuracy of the model trained 10 times
-      of random training sets containing 1, 10 and 100 data points per
+%       of random training sets containing 1, 10 and 100 data points per
-      class.}
+%       class.}
-\end{table}
+% \end{table}
 
-\begin{center}
+% \begin{center}
-  \begin{figure}[h]
+%   \begin{figure}[h]
-    \centering
+%     \centering
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist0.pdf}
+%     \includegraphics[width=\textwidth]{Data/cnn_fashion_fig.pdf}
-    \caption{original\\image}
+%     \caption{original\\image}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist_gen_zoom.pdf}
+%     \includegraphics[width=\textwidth]{Data/cnn_fashion_fig1.pdf}
-    \caption{random\\zoom}
+%     \caption{random\\zoom}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{0.19\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist_gen_shear.pdf}
+%     \includegraphics[width=\textwidth]{Data/mnist_gen_shear.pdf}
-    \caption{random\\shear}
+%     \caption{random\\shear}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{0.19\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist_gen_rotation.pdf}
+%     \includegraphics[width=\textwidth]{Data/mnist_gen_rotation.pdf}
-    \caption{random\\rotation}
+%     \caption{random\\rotation}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{0.19\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist_gen_shift.pdf}
+%     \includegraphics[width=\textwidth]{Data/mnist_gen_shift.pdf}
-  \caption{random\\positional shift}
+%   \caption{random\\positional shift}
-  \end{subfigure}\\
+%   \end{subfigure}\\ 
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{0.19\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist5.pdf}
+%     \includegraphics[width=\textwidth]{Data/mnist5.pdf}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{0.19\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist6.pdf}
+%     \includegraphics[width=\textwidth]{Data/mnist6.pdf}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{0.19\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist7.pdf}
+%     \includegraphics[width=\textwidth]{Data/mnist7.pdf}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{0.19\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist8.pdf}
+%     \includegraphics[width=\textwidth]{Data/mnist8.pdf}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{0.19\textwidth}
+%   \begin{subfigure}{0.19\textwidth}
-    \includegraphics[width=\textwidth]{Data/mnist9.pdf}
+%     \includegraphics[width=\textwidth]{Data/mnist9.pdf}
-  \end{subfigure}
+%   \end{subfigure}
-  \caption{The MNIST data set contains 70.000 images of preprocessed handwritten
+%   \caption{The MNIST data set contains 70.000 images of preprocessed handwritten
-    digits. Of these images 60.000 are used as training images, while
+%     digits. Of these images 60.000 are used as training images, while
-    the rest are used to validate the models trained.}
+%     the rest are used to validate the models trained.}
-\end{figure}
+% \end{figure}
-\end{center}
+% \end{center}
 
-\begin{figure}
+% \begin{figure}
-  \begin{adjustbox}{width=\textwidth}
+%   \begin{adjustbox}{width=\textwidth}
-    \begin{tikzpicture}  
+%     \begin{tikzpicture}  
-      \begin{scope}[x = (0:1cm), y=(90:1cm), z=(15:-0.5cm)]
+%       \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)
+%         \node[canvas is xy plane at z=0, transform shape] at (0,0)
-        {\includegraphics[width=5cm]{Data/klammern_r.jpg}};
+%         {\includegraphics[width=5cm]{Data/klammern_r.jpg}};
-        \node[canvas is xy plane at z=2, transform shape] at (0,-0.2)
+%         \node[canvas is xy plane at z=2, transform shape] at (0,-0.2)
-        {\includegraphics[width=5cm]{Data/klammern_g.jpg}};
+%         {\includegraphics[width=5cm]{Data/klammern_g.jpg}};
-        \node[canvas is xy plane at z=4, transform shape] at (0,-0.4)
+%         \node[canvas is xy plane at z=4, transform shape] at (0,-0.4)
-        {\includegraphics[width=5cm]{Data/klammern_b.jpg}};
+%         {\includegraphics[width=5cm]{Data/klammern_b.jpg}};
-        \node[canvas is xy plane at z=4, transform shape] at (-8,-0.2)
+%         \node[canvas is xy plane at z=4, transform shape] at (-8,-0.2)
-        {\includegraphics[width=5.3cm]{Data/klammern_rgb.jpg}};
+%         {\includegraphics[width=5.3cm]{Data/klammern_rgb.jpg}};
-      \end{scope}
+%       \end{scope}
-    \end{tikzpicture}
+%     \end{tikzpicture}
-  \end{adjustbox}
+%   \end{adjustbox}
-  \caption{On the right the red, green and blue chanels of the picture
+%   \caption{On the right the red, green and blue chanels of the picture
-    are displayed. In order to better visualize the color channes the
+%     are displayed. In order to better visualize the color channes the
-    black and white picture of each channel has been colored in the
+%     black and white picture of each channel has been colored in the
-    respective color. Combining the layers results in the image on the
+%     respective color. Combining the layers results in the image on the
-    left}
+%     left}
-\end{figure}
+% \end{figure}
 
-\begin{figure}
+% \begin{figure}
-  \centering
+%   \centering
-  \begin{subfigure}{\linewidth}
+%   \begin{subfigure}{\linewidth}
-    \centering
+%     \centering
-    \includegraphics[width=\textwidth]{Data/convnet_fig.pdf}
+%     \includegraphics[width=\textwidth]{Data/convnet_fig.pdf}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{.45\linewidth}
+%   \begin{subfigure}{.45\linewidth}
-    \centering
+%     \centering
-    \begin{tikzpicture}
+%     \begin{tikzpicture}
-      \begin{axis}[enlargelimits=false, width=\textwidth]
+%       \begin{axis}[enlargelimits=false, width=\textwidth]
-        \addplot[domain=-5:5, samples=100]{tanh(x)};
+%         \addplot[domain=-5:5, samples=100]{tanh(x)};
-      \end{axis}
+%       \end{axis}
-    \end{tikzpicture}
+%     \end{tikzpicture}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{.45\linewidth}
+%   \begin{subfigure}{.45\linewidth}
-    \centering
+%     \centering
-    \begin{tikzpicture}
+%     \begin{tikzpicture}
-      \begin{axis}[enlargelimits=false, width=\textwidth,
+%       \begin{axis}[enlargelimits=false, width=\textwidth,
-        ytick={0,2,4},yticklabels={\hphantom{4.}0,2,4}, ymin=-1]
+%         ytick={0,2,4},yticklabels={\hphantom{4.}0,2,4}, ymin=-1]
-        \addplot[domain=-5:5, samples=100]{max(0,x)};
+%         \addplot[domain=-5:5, samples=100]{max(0,x)};
-      \end{axis}
+%       \end{axis}
-    \end{tikzpicture}
+%     \end{tikzpicture}
-  \end{subfigure}
+%   \end{subfigure}
-  \begin{subfigure}{.45\linewidth}
+%   \begin{subfigure}{.45\linewidth}
-    \centering
+%     \centering
-    \begin{tikzpicture}
+%     \begin{tikzpicture}
-      \begin{axis}[enlargelimits=false, width=\textwidth, ymin=-1,
+%       \begin{axis}[enlargelimits=false, width=\textwidth, ymin=-1,
-        ytick={0,2,4},yticklabels={$\hphantom{-5.}0$,2,4}]
+%         ytick={0,2,4},yticklabels={$\hphantom{-5.}0$,2,4}]
-        \addplot[domain=-5:5, samples=100]{max(0,x)+ 0.1*min(0,x)};
+%         \addplot[domain=-5:5, samples=100]{max(0,x)+ 0.1*min(0,x)};
-      \end{axis}
+%       \end{axis}
-    \end{tikzpicture}
+%     \end{tikzpicture}
-  \end{subfigure}
+%   \end{subfigure}
-\end{figure}
+% \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}
+\newcommand{\abs}[1]{\ensuremath{\left\vert#1\right\vert}}
+
+\[
+  \sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
+  h_{k,n} = \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
+      (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
+  \left(\sum_{\substack{k \in \kappa \\ \xi_k \in
+        [\delta l , \delta(l+1))}} \varphi(\xi_k, v_k) 
+    h_{k,n}\right) \approx
+\]
+\[
+  \approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
+      (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
+  \left(\sum_{\substack{k \in \kappa \\ \xi_k \in
+        [\delta l , \delta(l+1))}} \left(\varphi(\delta l, v_k)
+      \frac{1}{n g_\xi (\delta l)} \pm \frac{\varepsilon}{n}\right) 
+    \frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
+          \delta(l+1))\right\}}}{\abs{\left\{m \in \kappa : \xi_m
+          \in [\delta l, \delta(l+1))\right\}}}\right)
+\]
+\[
+  \approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
+      (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
+  \left(\frac{\sum_{\substack{k \in \kappa \\ \xi_k \in
+          [\delta l , \delta(l+1))}}\varphi(\delta l,
+      v_k)}{\abs{\left\{m \in \kappa : \xi_m
+          \in [\delta l, \delta(l+1))\right\}}} 
+    \frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
+          \delta(l+1))\right\}}}{n g_\xi (\delta l)}\right) \pm \varepsilon
+\]
+The amount of kinks in a given interval of length $\delta$ follows a
+binomial distribution,
+\[
+  \mathbb{E} \left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
+        \delta(l+1))\right\}}\right] = n \int_{\delta
+    l}^{\delta(l+1)}g_\xi (x) dx \approx n (\delta g_\xi(\delta l)
+  \pm \delta \tilde{\varepsilon}),
+\]
+for any $\delta \leq \delta(\varepsilon, \tilde{\varepsilon})$, since $g_\xi$ is uniformly continuous on its
+support by Assumption..
+As the distribution of $v$ is continuous as well we get
+$\mathcal{L}(v_k) = \mathcal{L} v| \xi = \delta l) \forall k \in
+\kappa : \xi_k \in [\delta l, \delta(l+1))$ for $delta \leq
+\delta(\varepsilon, \tilde{\varepsilon})$. Thus we get with the law of
+large numbers
+\begin{align*}
+  &\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
+  h_{k,n} \approx\\  
+  &\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
+  (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
+  \}]}}\left(\mathbb{E}[\phi(\xi, v)|\xi=\delta l]
+  \stackrel{\mathbb{P}}{\pm}\right) \delta \left(1 \pm
+  \frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
+  \\
+  &\approx \left(\sum_{\substack{l \in \mathbb{Z} \\ [\delta
+  l, \delta 
+  (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
+  \}]}}\mathbb{E}[\phi(\xi, v)|\xi=\delta l] \delta
+  \stackrel{\mathbb{P}}{\pm}\tilde{\tilde{\varepsilon}}
+  \abs{C_{g_\xi}^u - C_{g_\xi}^l}
+  \right)\\
+  &\phantom{\approx}\cdot \left(1 \pm
+  \frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
+\end{align*}
+\newpage
 
 
-\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}
 

TeX/Figures/sdg_comparison.tex

@@ -4,34 +4,56 @@ legend image code/.code={
 \draw[mark repeat=2,mark phase=2]
 plot coordinates {
 (0cm,0cm)
-(0.0cm,0cm)        %% default is (0.3cm,0cm)
+(0.15cm,0cm)        %% default is (0.3cm,0cm)
-(0.0cm,0cm)         %% default is (0.6cm,0cm)
+(0.3cm,0cm)         %% default is (0.6cm,0cm)
 };%
 }
 }
 \begin{figure}
   \begin{subfigure}[h]{\textwidth}
     \begin{tikzpicture}
-      \begin{axis}[tick style = {draw = none}, width = \textwidth,
+      \begin{axis}[legend cell align={left},yticklabel style={/pgf/number format/fixed,
-        height = 0.6\textwidth, ymin = 0.92, legend style={at={(0.9825,0.75)},anchor=north east},
+                     /pgf/number format/precision=3},tick style = {draw = none}, width = 0.975\textwidth,
-        xlabel = {epoch}, ylabel = {Classification Accuracy}]
+        height = 0.6\textwidth, ymin = 0.885, 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}]
+        % [tick style = {draw = none}, width = \textwidth,
+        % height = 0.6\textwidth, ymin = 0.905, 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]
+        % {Figures/Data/adagrad.log}; 
+        % \addplot table
+        % [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+        % {Figures/Data/adadelta.log}; 
+        % \addplot table
+        % [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+        % {Figures/Data/adam.log}; 
         \addplot table
         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Figures/Data/adagrad.log}; 
+        {Figures/Data/Adagrad.mean}; 
         \addplot table
         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Figures/Data/adadelta.log}; 
+        {Figures/Data/Adadelta.mean}; 
         \addplot table
         [x=epoch, y=val_accuracy, col sep=comma, mark = none]
-        {Figures/Data/adam.log}; 
+        {Figures/Data/Adam.mean}; 
+        \addplot table
+        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+        {Figures/Data/SGD_00.mean}; 
+        \addplot table
+        [x=epoch, y=val_accuracy, col sep=comma, mark = none]
+        {Figures/Data/SGD_09.mean}; 
         
-         \addlegendentry{\footnotesize{ADAGRAD}}
+         \addlegendentry{\footnotesize{\textsc{AdaGrad}}}
-        \addlegendentry{\footnotesize{ADADELTA}}
+        \addlegendentry{\footnotesize{\textsc{Adadelta}}}
-        \addlegendentry{\footnotesize{ADAM}}
+        \addlegendentry{\footnotesize{\textsc{Adam}}}
-        \addlegendentry{SGD$_{0.01}$}
+        \addlegendentry{\footnotesize{\textsc{Sgd}}}
+        \addlegendentry{\footnotesize{Momentum}}
       \end{axis}
     \end{tikzpicture}
-    %\caption{Classification accuracy}
+    \caption{Test accuracies during training}
     \vspace{.25cm}
   \end{subfigure}
   % \begin{subfigure}[b]{\textwidth}
@@ -58,18 +80,27 @@ plot coordinates {
   %   \vspace{.25cm}
   % \end{subfigure}
   \begin{subfigure}[b]{1.0\linewidth}
-    \begin{tabu} to \textwidth {@{}  *3{X[c]}c*3{X[c]} @{}}
+    \begin{tabu} to \textwidth {@{}l*5{X[c]}@{}}
-      \multicolumn{3}{c}{Classification Accuracy}
+      \Tstrut \Bstrut &\textsc{AdaGrad}& \textsc{AdaDelta}&
-      &~&\multicolumn{3}{c}{Error Measure}
+      \textsc{Adam} & \textsc{Sgd} & Momentum \\
-      \\\cline{1-3}\cline{5-7}
+      \hline           
-      ADAGRAD&ADADELTA&ADAM&&ADAGRAD&ADADELTA&ADAM
+      \Tstrut  Accuracy &0.9870 & 0.9562 & 0.9925 & 0.9866 & 0.9923 \\
-      \\\cline{1-3}\cline{5-7}
+      \Tstrut  Loss &0.0404 & 0.1447 & 0.0999 & 0.0403 & 0.0246 \\     
-      1&1&1&&1&1&1
     \end{tabu}
-    \caption{Performace metrics after 20 epochs}
+    % \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}
+    %   \textsc{AdaGad}&\textsc{AdaDelta}&\textsc{Adam}&&\textsc{AdaGrad}&\textsc{AdaDelta}&\textsc{Adam}
+    %   \\\cline{1-3}\cline{5-7}
+    %   1&1&1&&1&1&1
+    % \end{tabu}
+    \caption{Performace metrics after 50 epochs}
   \end{subfigure}
-  \caption[Performance comparison of training algorithms]{Classification accuracy on the test set and ...Performance metrics of the network given in ... trained
+  \caption[Performance Comparison of Training Algorithms]{
-    with different optimization algorithms}
+    Average performance metrics of the neural network given in
+   Figure~\ref{fig:mnist_architecture} trained 5 times for 50 epochs
+   using different optimization algorithms.}
   \label{fig:comp_alg}
 \end{figure}
 %%% Local Variables:

TeX/Figures/sin_conv.tex

@@ -14,6 +14,7 @@
     \end{adjustbox}
     \caption{True position (\textcolor{red}{red}), distorted position data (black)}
   \end{subfigure}
+  \hfill
   \begin{subfigure}[b]{0.49\textwidth}
     \centering
     \begin{adjustbox}{width=\textwidth, height=0.25\textheight}
@@ -28,11 +29,11 @@
     \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
+  \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 $g$
+      data separately. The convolution of both signals with $g$
       improves the MSE of the positions from 0.196 to 0.170 and
       visibly smoothes the data.
   }

TeX/Figures/test.tex

@@ -11,165 +11,306 @@
 \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}
-\lstset{
+\usepackage{float}
-    breaklines=true,
-    %
-    extendedchars=true,
-    literate=
-    {á}{{\'a}}1 {é}{{\'e}}1 {í}{{\'i}}1 {ó}{{\'o}}1 {ú}{{\'u}}1
-    {Á}{{\'A}}1 {É}{{\'E}}1 {Í}{{\'I}}1 {Ó}{{\'O}}1 {Ú}{{\'U}}1
-    {à}{{\`a}}1 {è}{{\`e}}1 {ì}{{\`i}}1 {ò}{{\`o}}1 {ù}{{\`u}}1
-    {À}{{\`A}}1 {È}{{\'E}}1 {Ì}{{\`I}}1 {Ò}{{\`O}}1 {Ù}{{\`U}}1
-    {ä}{{\"a}}1 {ë}{{\"e}}1 {ï}{{\"i}}1 {ö}{{\"o}}1 {ü}{{\"u}}1
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-    {€}{{\EUR}}1 {£}{{\pounds}}1
-}
-
-%%
-%% Python definition (c) 1998 Michael Weber
-%% Additional definitions (2013) Alexis Dimitriadis
-%% 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},%
-    %
-    % 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},%
-    %
-    sensitive=true,%
-    morecomment=[l]\#,%
-    morestring=[b]',%
-    morestring=[b]",%
-    %
-    morestring=[s]{'''}{'''},% used for documentation text (mulitiline strings)
-    morestring=[s]{"""}{"""},% added by Philipp Matthias Hahn
-    %
-    morestring=[s]{r'}{'},% `raw' strings
-    morestring=[s]{r"}{"},%
-    morestring=[s]{r'''}{'''},%
-    morestring=[s]{r"""}{"""},%
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-    %
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-    % *{-}{-}{1} will not replace in comments and so on
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-    {€}{{\EUR}}1 {£}{{\pounds}}1
-    %
-    {^}{{{\color{ipython_purple}\^{}}}}1
-    {=}{{{\color{ipython_purple}=}}}1
-    %
-    {+}{{{\color{ipython_purple}+}}}1
-    {*}{{{\color{ipython_purple}$^\ast$}}}1
-    {/}{{{\color{ipython_purple}/}}}1
-    %
-    {+=}{{{+=}}}1
-    {-=}{{{-=}}}1
-    {*=}{{{$^\ast$=}}}1
-    {/=}{{{/=}}}1,
-    literate=
-    *{-}{{{\color{ipython_purple}-}}}1
-     {?}{{{\color{ipython_purple}?}}}1,
-    %
-    identifierstyle=\color{black}\ttfamily,
-    commentstyle=\color{ipython_cyan}\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,
-    numberstyle=\tiny\color{halfgray},
-    %
-    %
-    backgroundcolor=\color{ipython_bg},
-    %   extendedchars=true,
-    basicstyle=\scriptsize,
-    keywordstyle=\color{ipython_green}\ttfamily,
-}
 
+\usepackage{amsmath,amssymb}
+
+% \newfloat{lstfloat}{htbp}{lop}
+% \floatname{lstfloat}{Listing}
+% \def\lstfloatautorefname{Listing}
+
+% \lstset{
+%     breaklines=true,
+%     %
+%     extendedchars=true,
+%     literate=
+%     {á}{{\'a}}1 {é}{{\'e}}1 {í}{{\'i}}1 {ó}{{\'o}}1 {ú}{{\'u}}1
+%     {Á}{{\'A}}1 {É}{{\'E}}1 {Í}{{\'I}}1 {Ó}{{\'O}}1 {Ú}{{\'U}}1
+%     {à}{{\`a}}1 {è}{{\`e}}1 {ì}{{\`i}}1 {ò}{{\`o}}1 {ù}{{\`u}}1
+%     {À}{{\`A}}1 {È}{{\'E}}1 {Ì}{{\`I}}1 {Ò}{{\`O}}1 {Ù}{{\`U}}1
+%     {ä}{{\"a}}1 {ë}{{\"e}}1 {ï}{{\"i}}1 {ö}{{\"o}}1 {ü}{{\"u}}1
+%     {Ä}{{\"A}}1 {Ë}{{\"E}}1 {Ï}{{\"I}}1 {Ö}{{\"O}}1 {Ü}{{\"U}}1
+%     {â}{{\^a}}1 {ê}{{\^e}}1 {î}{{\^i}}1 {ô}{{\^o}}1 {û}{{\^u}}1
+%     {Â}{{\^A}}1 {Ê}{{\^E}}1 {Î}{{\^I}}1 {Ô}{{\^O}}1 {Û}{{\^U}}1
+%     {œ}{{\oe}}1 {Œ}{{\OE}}1 {æ}{{\ae}}1 {Æ}{{\AE}}1 {ß}{{\ss}}1
+%     {ç}{{\c c}}1 {Ç}{{\c C}}1 {ø}{{\o}}1 {å}{{\r a}}1 {Å}{{\r A}}1
+%     {€}{{\EUR}}1 {£}{{\pounds}}1
+% }
+
+% %%
+% %% Python definition (c) 1998 Michael Weber
+% %% Additional definitions (2013) Alexis Dimitriadis
+% %% 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},%
+%     %
+%     % 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,val},%
+%     %
+%     sensitive=true,%
+%     morecomment=[l]\#,%
+%     morestring=[b]',%
+%     morestring=[b]",%
+%     %
+%     morestring=[s]{'''}{'''},% used for documentation text (mulitiline strings)
+%     morestring=[s]{"""}{"""},% added by Philipp Matthias Hahn
+%     %
+%     morestring=[s]{r'}{'},% `raw' strings
+%     morestring=[s]{r"}{"},%
+%     morestring=[s]{r'''}{'''},%
+%     morestring=[s]{r"""}{"""},%
+%     morestring=[s]{u'}{'},% unicode strings
+%     morestring=[s]{u"}{"},%
+%     morestring=[s]{u'''}{'''},%
+%     morestring=[s]{u"""}{"""},%
+%     %
+%     % {replace}{replacement}{lenght of replace}
+%     % *{-}{-}{1} will not replace in comments and so on
+%     literate=
+%     {á}{{\'a}}1 {é}{{\'e}}1 {í}{{\'i}}1 {ó}{{\'o}}1 {ú}{{\'u}}1
+%     {Á}{{\'A}}1 {É}{{\'E}}1 {Í}{{\'I}}1 {Ó}{{\'O}}1 {Ú}{{\'U}}1
+%     {à}{{\`a}}1 {è}{{\`e}}1 {ì}{{\`i}}1 {ò}{{\`o}}1 {ù}{{\`u}}1
+%     {À}{{\`A}}1 {È}{{\'E}}1 {Ì}{{\`I}}1 {Ò}{{\`O}}1 {Ù}{{\`U}}1
+%     {ä}{{\"a}}1 {ë}{{\"e}}1 {ï}{{\"i}}1 {ö}{{\"o}}1 {ü}{{\"u}}1
+%     {Ä}{{\"A}}1 {Ë}{{\"E}}1 {Ï}{{\"I}}1 {Ö}{{\"O}}1 {Ü}{{\"U}}1
+%     {â}{{\^a}}1 {ê}{{\^e}}1 {î}{{\^i}}1 {ô}{{\^o}}1 {û}{{\^u}}1
+%     {Â}{{\^A}}1 {Ê}{{\^E}}1 {Î}{{\^I}}1 {Ô}{{\^O}}1 {Û}{{\^U}}1
+%     {œ}{{\oe}}1 {Œ}{{\OE}}1 {æ}{{\ae}}1 {Æ}{{\AE}}1 {ß}{{\ss}}1
+%     {ç}{{\c c}}1 {Ç}{{\c C}}1 {ø}{{\o}}1 {å}{{\r a}}1 {Å}{{\r A}}1
+%     {€}{{\EUR}}1 {£}{{\pounds}}1
+%     %
+%     {^}{{{\color{ipython_purple}\^{}}}}1
+%     {=}{{{\color{ipython_purple}=}}}1
+%     %
+%     {+}{{{\color{ipython_purple}+}}}1
+%     {*}{{{\color{ipython_purple}$^\ast$}}}1
+%     {/}{{{\color{ipython_purple}/}}}1
+%     %
+%     {+=}{{{+=}}}1
+%     {-=}{{{-=}}}1
+%     {*=}{{{$^\ast$=}}}1
+%     {/=}{{{/=}}}1,
+%     literate=
+%     *{-}{{{\color{ipython_purple}-}}}1
+%      {?}{{{\color{ipython_purple}?}}}1,
+%     %
+%     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,
+%     numberstyle=\tiny\color{halfgray},
+%     %
+%     %
+%     backgroundcolor=\color{ipython_bg},
+%     %   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
+% 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}
+% \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}
 \begin{document}
-\begin{lstlisting}[language=iPython]
+
-import tensorflow as tf
+\begin{align}
-import numpy as np
+  \makebox[2cm][c]{$\overset{\text{Lem. A.6}}{\underset{\delta \text{
-from tensorflow.keras.callbacks import CSVLogger
+  small enough}}{=}} $} 
-from tensorflow.keras.preprocessing.image import ImageDataGenerator
+\end{align}
-
+
-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(j))
-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}
-\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}
 \end{document} 
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:

TeX/Plots/Data/data_sin_d_t.csv

@@ -1,17 +0,0 @@
-x,y
--3.141592653589793,0.0802212608585366
--2.722713633111154,-0.3759376368887911
--2.303834612632515,-1.3264180339054117
--1.8849555921538759,-0.8971334213504949
--1.4660765716752369,-0.7724344034354425
--1.0471975511965979,-0.9501497164520739
--0.6283185307179586,-0.6224628757084738
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TeX/Plots/Data/min_max.txt

@@ -1,58 +0,0 @@
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-default_100
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TeX/Plots/Data/scala_out_d_1_t.csv

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TeX/Plots/Data/scala_out_sin.csv

@@ -1,101 +0,0 @@
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TeX/Plots/Data/sin_6.csv

@@ -1,7 +0,0 @@
-x,y
--3.14159265358979 , -1.22464679914735e-16
--1.88495559215388 , -0.951056516295154
--0.628318530717959 , -0.587785252292473
-0.628318530717959 , 0.587785252292473
-1.88495559215388 , 0.951056516295154
-3.14159265358979 , 1.22464679914735e-16

TeX/Plots/Data/sin_conv.csv

@@ -1,64 +0,0 @@
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TeX/Plots/RN_vs_RS.tex

@@ -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:

TeX/Plots/SGD_vs_GD.tex

@@ -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:

TeX/Plots/_region_.tex

@@ -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}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:

TeX/Plots/fashion_mnist.tex

@@ -1,53 +0,0 @@
-\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}
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "../main"
-%%% End:

TeX/Plots/gen_dropout.tex

@@ -1,82 +0,0 @@
-\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:
-%%% mode: latex
-%%% TeX-master: "../main"
-%%% End:

TeX/Plots/mnist.tex

@@ -1,41 +0,0 @@
-\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:
-%%% mode: latex
-%%% TeX-master: "../main"
-%%% End:

TeX/Plots/pfg_test.tex

@@ -1,301 +0,0 @@
-\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:

TeX/Plots/sdg_comparison.tex

@@ -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"
-%%% End:

TeX/Plots/sin_conv.csv

@@ -1,64 +0,0 @@
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TeX/Plots/sin_conv.tex

@@ -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:

TeX/Plots/y.tex

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

TeX/appendixA.tex

@@ -1,24 +1,333 @@
 
 \newpage
 \begin{appendices}
-  \section{Proofs for sone Lemmata in ...}
+  \counterwithin{lstfloat}{section}
-  In the following there will be proofs for some important Lemmata in
+  \section{Notes on Proofs of Lemmata in Section~\ref{sec:conv}}
-  Section~\ref{sec:theo38}. Further proofs not discussed here can be
+  \label{appendix:proofs}
-  found in \textcite{heiss2019}
+  Contrary to \textcite{heiss2019}  we do not make the distinction between $f_+$ and
-  \begin{Theorem}[Proof of Lemma~\ref{theo38}]
+  $f_-$.
-  \end{Theorem}
+  This results in some alterations in the proofs being necessary. In
-  
+  the following the affected proofs and the required changes are given.
-\begin{Lemma}[$\frac{w^{*,\tilde{\lambda}}_k}{v_k}\approx\mathcal{O}(\frac{1}{n})$]
+  % Because of that slight alterations are needed in the proofs of
-  For any $\lambda > 0$ and training data $(x_i^{\text{train}},
+  % .. auxiliary lemmata.
-  y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
+  % Alterations that go beyond substituting $F_{+-}^{}$
-  \left\{1,\dots,N\right\}$, we have
+  % As the proofs are ... for the most part only
+  % the alterations needed are specified.
+
+  
+
+  
+  % In the following there will be proofs for some important Lemmata in
+  % Section~\ref{sec:theo38}. Further proofs not discussed here can be
+  % found in \textcite{heiss2019}
+  % The proves in this section are based on \textcite{heiss2019}. Slight
+  % alterations have been made to accommodate for not splitting $f$ into
+  % $f_+$ and $f_-$.
+  % \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
+  %   \[
+  %     \max_{k \in \left\{1,\dots,n\right\}} \frac{w^{*,
+  %         \tilde{\lambda}}_k}{v_k} = \po_{n\to\infty}
+  %   \]
+  
+    
+  % \end{Lemma}
+
+  \begin{Proof}[Heiss, Teichmann, and Wutte (2019, Lemma A.9)]~\\\noindent
+    \label{proof:lem9}
+    With $\tilde{\lambda} \coloneqq \lambda n g(0)$ Lemma~\ref{lem:cnvh} follows
+    analogously when considering $\tilde{w}$, $f_g^{*, \lambda}$, and $h_k$
+    instead of $\tilde{w}^+$, $f_{g,+}^{*, \lambda}$, and $\bar{h}_k$.
+    Consider $\kappa = \left\{1, \dots, n \right\}$ for $n$ nodes
+    instead of $\kappa^+$. With $h_k = \frac{1}{n g_\xi(\xi_n)}$
+    instead of $\bar{h}_k$
+    and \[
+      \mathbb{E} \left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
+        \delta(l+1))\right\}}\right] = n \int_{\delta
+    l}^{\delta(l+1)}g_\xi (x) dx \approx n (\delta g_\xi(\delta l)
+  \pm \delta \tilde{\varepsilon}).
+    \]
+   %  \[
+%   \sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
+%   h_{k,n} = \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
+%       (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
+%   \left(\sum_{\substack{k \in \kappa \\ \xi_k \in
+%         [\delta l , \delta(l+1))}} \varphi(\xi_k, v_k) 
+%     h_{k,n}\right) \approx
+% \]
+% \[
+%   \approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
+%       (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
+%   \left(\sum_{\substack{k \in \kappa \\ \xi_k \in
+%         [\delta l , \delta(l+1))}} \left(\varphi(\delta l, v_k)
+%       \frac{1}{n g_\xi (\delta l)} \pm \frac{\varepsilon}{n}\right) 
+%     \frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
+%           \delta(l+1))\right\}}}{\abs{\left\{m \in \kappa : \xi_m
+%           \in [\delta l, \delta(l+1))\right\}}}\right)
+% \]
+% \[
+%   \approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
+%       (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}
+%   \left(\frac{\sum_{\substack{k \in \kappa \\ \xi_k \in
+%           [\delta l , \delta(l+1))}}\varphi(\delta l,
+%       v_k)}{\abs{\left\{m \in \kappa : \xi_m
+%           \in [\delta l, \delta(l+1))\right\}}} 
+%     \frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
+%           \delta(l+1))\right\}}}{n g_\xi (\delta l)}\right) \pm \varepsilon
+% \]
+% The amount of kinks in a given interval of length $\delta$ follows a
+% binomial distribution,
+% \[
+%   \mathbb{E} \left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
+%         \delta(l+1))\right\}}\right] = n \int_{\delta
+%     l}^{\delta(l+1)}g_\xi (x) dx \approx n (\delta g_\xi(\delta l)
+%   \pm \delta \tilde{\varepsilon}),
+% \]
+% for any $\delta \leq \delta(\varepsilon, \tilde{\varepsilon})$, since $g_\xi$ is uniformly continuous on its
+% support by Assumption..
+% As the distribution of $v$ is continuous as well we get that
+% $\mathcal{L}(v_k) = \mathcal{L} v| \xi = \delta l) \forall k \in
+% \kappa : \xi_k \in [\delta l, \delta(l+1))$ for $\delta \leq
+% \delta(\varepsilon, \tilde{\varepsilon})$. Thus we get with the law of
+% large numbers
+% \begin{align*}
+%   &\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
+%     h_{k,n} \approx\\  
+%   &\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta
+%   (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
+%   \}]}}\left(\mathbb{E}[\phi(\xi, v)|\xi=\delta l]
+%   \stackrel{\mathbb{P}}{\pm}\right) \delta \left(1 \pm
+%   \frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
+%   \\
+%   &\approx \left(\sum_{\substack{l \in \mathbb{Z} \\ [\delta
+%   l, \delta 
+%   (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T
+%   \}]}}\mathbb{E}[\phi(\xi, v)|\xi=\delta l] \delta
+%   \stackrel{\mathbb{P}}{\pm}\tilde{\tilde{\varepsilon}}
+%   \abs{C_{g_\xi}^u - C_{g_\xi}^l}
+%   \right)\\
+%   &\phantom{\approx}\cdot \left(1 \pm
+%     \frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon
+% \end{align*}
+\end{Proof}
+
+% \begin{Lemma}[($L(f_n) \to L(f)$), Heiss, Teichmann, and
+% Wutte (2019, Lemma A.11)]
+%   For any data $(x_i^{\text{train}}, y_i^{\text{train}}) \in
+%   \mathbb{R}^2, i \in \left\{1,\dots,N\right\}$, let $(f_n)_{n \in
+%     \mathbb{N}}$ be a sequence of functions that converges point-wise
+%   in probability to a function $f : \mathbb{R}\to\mathbb{R}$, then the
+%   loss $L$ of $f_n$ converges is probability to $L(f)$ as $n$ tends to
+%   infinity,
+%   \[
+%     \plimn L(f_n) = L(f).
+%   \]
+%   \proof Vgl. ...
+% \end{Lemma}
+
+\begin{Proof}[Heiss, Teichmann, and Wutte (2019, Lemma A.12)]~\\\noindent
+    \label{proof:lem12}
+   With $\tilde{\lambda} \coloneqq \lambda n g(0)$ Lemma~\ref{lem:s2} follows
+    analogously when considering $\tilde{w}$, $f_g^{*, \lambda}$, and $h_k$
+    instead of $\tilde{w}^+$, $f_{g,+}^{*, \lambda}$, and $\bar{h}_k$.
+  % We start by showing that
+  % \[
+  %   \plimn \tilde{\lambda} \norm{\tilde{w}}_2^2 = \lambda g(0)
+  %   \left(\int \frac{\left(f_g^{*,\lambda''}\right)^2}{g(x)} dx\right)
+  % \]
+  % With the definitions of $\tilde{w}$, $\tilde{\lambda}$ and
+  % $h$ we have
+  % \begin{align*}
+  %   \tilde{\lambda} \norm{\tilde{w}}_2^2
+  %   &= \tilde{\lambda} \sum_{k \in
+  %     \kappa}\left(f_g^{*,\lambda''}(\xi_k) \frac{h_k
+  %     v_k}{\mathbb{E}v^2|\xi = \xi_k]}\right)^2\\
+  %   &= \tilde{\lambda} \sum_{k \in
+  %     \kappa}\left(\left(f_g^{*,\lambda''}\right)^2(\xi_k) \frac{h_k
+  %     v_k^2}{\mathbb{E}v^2|\xi = \xi_k]}\right) h_k\\
+  %   & = \lambda g(0) \sum_{k \in
+  %     \kappa}\left(\left(f_g^{*,\lambda''}\right)^2(\xi_k)\frac{v_k^2}{g_\xi(\xi_k)\mathbb{E}
+  %     [v^2|\xi=\xi_k]}\right)h_k.
+  % \end{align*}
+  % By using Lemma~\ref{lem} with $\phi(x,y) =
+  % \left(f_g^{*,\lambda''}\right)^2(x)\frac{y^2}{g_\xi(\xi)\mathbb{E}[v^2|\xi=y]}$
+  % this converges to
+  % \begin{align*}
+  %   &\plimn \tilde{\lambda}\norm{\tilde{w}}_2^2 = \\
+  %   &=\lambda
+  %     g_\xi(0)\mathbb{E}[v^2|\xi=0]\int_{\supp{g_\xi}}\mathbb{E}\left[
+  %     \left(f_g^{*,\lambda''}\right)^2(\xi)\frac{v^2}{
+  %     g_\xi(\xi)\mathbb{E}[v^2|\xi=x]^2}\Big{|} \xi = x\right]dx\\
+  %   &=\lambda g_\xi(0) \mathbb{E}[v^2|\xi=0] \int_{\supp{g_xi}}
+  %     \frac{\left(f_g^{*,\lambda''}\right)^2 (x)}{g_\xi(x)
+  %     \mathbb{E}[v^2|\xi=x]} dx \\
+  %   &=\lambda g(0) \int_{\supp{g_\xi}} \frac{\left(f_g^{*,\lambda''}\right)^2}{g(x)}dx.
+  % \end{align*}
+\end{Proof}
+
+\begin{Proof}[Heiss, Teichmann, and Wutte (2019, Lemma A.14)]~\\\noindent
+    \label{proof:lem14}
+  Substitute $F_{+-}^{\lambda, g}\left(f_{g,+}^{*,\lambda},
+    f_{g,-}^{*,\lambda}\right)$ with $F^{\lambda,g}\left(f_g^{*,\lambda}\right)$.
+\end{Proof}
+% \begin{Lemma}[Heiss, Teichmann, and
+%   Wutte (2019, Lemma A.13)]
+%   Using the notation of Definition .. and ... the following statement
+%   holds:
+%   $\forall \varepsilon \in \mathbb{R}_{>0} : \exists \delta \in
+%   \mathbb{R}_{>0} : \forall \omega \in \Omega : \forall l, l' \in
+%   \left\{1,\dots,N\right\} : \forall n \in \mathbb{N}$
+%   \[
+%     \left(\abs{\xi_l(\omega) - \xi_{l'}(\omega)} < \delta \wedge
+%       \text{sign}(v_l(\omega)) = \text{sign}(v_{l'}(\omega))\right)
+%     \implies \abs{\frac{w_l^{*, \tilde{\lambda}}(\omega)}{v_l(\omega)}
+%       - \frac{w_{l'}^{*, \tilde{\lambda}}(\omega)}{v_{l'}(\omega)}} <
+%     \frac{\varepsilon}{n},
+%   \]
+%   if we assume that $v_k$ is never zero.
+%   \proof given in ..
+% \end{Lemma}
+
+% \begin{Lemma}[$\frac{w^{*,\tilde{\lambda}}}{v} \approx
+%     \mathcal{O}(\frac{1}{n})$, Heiss, Teichmann, and
+%     Wutte (2019, Lemma A.14)]
+%     For any $\lambda > 0$ and data $(x_i^{\text{train}},
+%     y_i^{\text{train}}) \in \mathbb{R}^2, i\in
+%     \left\{1,\dots,\right\}$, we have
+%     \[
+%       \forall P \in (0,1) : \exists C \in \mathbb{R}_{>0} : \exists
+%       n_0 \in \mathbb{N} : \forall n > n_0 : \mathbb{P}
+%       \left[\max_{k\in \left\{1,\dots,n\right\}}
+%         \frac{w_k^{*,\tilde{\lambda}}}{v_k} < C
+%         \frac{1}{n}\right] > P
+%       % \max_{k\in \left\{1,\dots,n\right\}}
+%       % \frac{w_k^{*,\tilde{\lambda}}}{v_k} = \plimn
+%     \]
+%     \proof
+    
+    
+%     Let $k^*_+ \in \argmax_{k\in
+%       \left\{1,\dots,n\right\}}\frac{w^{*,\tilde{\lambda}}}{v_k} : v_k
+%     > 0$ and $k^*_- \in \argmax_{k\in
+%       \left\{1,\dots,n\right\}}\frac{w^{*,\tilde{\lambda}}}{v_k} : v_k
+%     < 0$. W.l.o.g. assume $\frac{w_{k_+^*}^2}{v_{k_+^*}^2} \geq
+%     \frac{w_{k_-^*}^2}{v_{k_-^*}^2}$
+%     \begin{align*}
+%       \frac{F^{\lambda,
+%       g}\left(f^{*,\lambda}_g\right)}{\tilde{\lambda}}
+%       \makebox[2cm][c]{$\stackrel{\mathbb{P}}{\geq}$}
+%       & \frac{1}{2 \tilde{\lambda}}
+%         F_n^{\tilde{\lambda}}\left(\mathcal{RN}^{*,\tilde{\lambda}}\right)
+%         = \frac{1}{2 \tilde{\lambda}}\left[\sum ... + \tilde{\lambda} \norm{w}_2^2\right]
+%       \\
+%       \makebox[2cm][c]{$\geq$}
+%       & \frac{1}{2}\left( \sum_{\substack{k: v_k
+%         > 0 \\\xi_k\in(\xi_{k^*}, \xi_{k^*}
+%       + \delta)}} \left(w_k^{*,\tilde{\lambda}}\right)^2 +
+%       \sum_{\substack{k: v_k < 0 \\\xi_k\in(\xi_{k^*}, \xi_{k^*}
+%       + \delta)}} \left(w_k^{*,\tilde{\lambda}}\right)^2\right) \\
+%       \makebox[2cm][c]{$\overset{\text{Lem. A.6}}{\underset{\delta \text{
+%       small enough}}{\geq}} $}
+%       &
+%         \frac{1}{4}\left(\left(\frac{w_{k_+^*}^{*,\tilde{\lambda}}}
+%         {v_{k_+^*}}\right)^2\sum_{\substack{k:
+%         v_k > 0 \\\xi_k\in(\xi_{k^*}, \xi_{k^*} + \delta)}}v_k^2 +
+%       \left(\frac{w_{k_-^*}^{*,\tilde{\lambda}}}{v_{k_-^*}}\right)^2
+%       \sum_{\substack{k:  
+%         v_k < 0 \\\xi_k\in(\xi_{k^*}, \xi_{k^*} +
+%       \delta)}}v_k^2\right)\\
+%       \makebox[2cm][c]{$\stackrel{\mathbb{P}}{\geq}$}
+%       & \frac{1}{8}
+%         \left(\frac{w_{k_+^*}^{*,\tilde{\lambda}}}{v_{k^*}}\right)^2
+%         n \delta g_\xi(\xi_{k_+^*}) \mathbb{P}(v_k
+%         >0)\mathbb{E}[v_k^2|\xi_k = \xi_{k^*_+}]
+%     \end{align*}
+  
+%   \end{Lemma}
+
+\begin{Proof}[Heiss, Teichmann, and Wutte (2019, Lemma A.15)]~\\\noindent
+    \label{proof:lem15}
+  Consider $\mathcal{RN}^{*,\tilde{\lambda}}$,
+  $f^{w^{*,\tilde{\lambda}}}$, and $\kappa = \left\{1, \dots, n
+  \right\}$ instead of $\mathcal{RN}_+^{*,\tilde{\lambda}}$,
+  $f_+^{w^{*,\tilde{\lambda}}}$, and $\kappa^+$.
+  Assuming w.l.o.g. $max_{k \in
+    \kappa^+}\abs{\frac{w_k^{*,\tilde{\lambda}}}{v_k}} \geq max_{k \in
+    \kappa^-}\abs{\frac{w_k^{*,\tilde{\lambda}}}{v_k}}$
+  Lemma~ref{lem:s3} follows analogously by multiplying (58b) with two.
+\end{Proof}
+
+\begin{Proof}[Heiss, Teichmann, and Wutte (2019, Lemma
+  A.16)]~\\\noindent
+    \label{proof:lem16}
+  As we are considering $F^{\lambda,g}$ instead of
+  $F^{\lambda,g}_{+-}$ we need to substitute $2\lambda g(0)$ with
+  $\lambda g(0)$
+  and thus get
+  \[
+   \left(f^{w^{*,\tilde{\lambda}}}\right)''(x) \approx
+    \frac{w_{l_x}^{*,\tilde{\lambda}}}{v_{l_x}} n g_\xi(x)
+    \mathbb{E}\left[v_k^2|\xi_k = x\right] \stackrel{\mathbb{P}}{\pm} \varepsilon_3
+  \]
+  and use this to follow
   \[
-    \max_{k \in \left\{1,\dots,n\right\}} \frac{w^{*,
+    \lambda g(0)
-        \tilde{\lambda}}_k}{v_k} = \po_{n\to\infty}
+    \int_{\supp(g)}\hspace{-0.15cm}\frac{\left(\left(f^{w^{*,\tilde{\lambda}}}\right)''(x)\right)^2}{g(0)}dx 
+    \approx \tilde{\lambda} n
+    \int_{\supp(g)}\left(\frac{w_{l_x}^{*,\tilde{\lambda}}}{v_{l_x}}\right)^2 \hspace{-0.1cm}
+    g_xi(x) \mathbb{E}\left[v_k^2|\xi_k=x\right]dx
   \]
+  Analogous to the proof of \textcite{heiss2019} we get
+  \begin{align*}
+    \tilde{\lambda} \sum_{k \in \kappa}
+    \left(w_k^{*,\tilde{\lambda}}\right)^2
+    &= \tilde{\lambda} \sum_{k \in \kappa^+}
+      \left(w_k^{*,\tilde{\lambda}}\right)^2 + \tilde{\lambda} \sum_{k \in \kappa^-}
+      \left(w_k^{*,\tilde{\lambda}}\right)^2 \\
+    &\approx \left(\mathbb{P}[v_k <0] + \mathbb{P}[v_k >0]\right)\\
+    &\phantom{=} 
+      \int_{\supp(g_xi)}
+      \left(\frac{w_{l_x}^{*,\tilde{\lambda}}}{v_{l_x}}\right)^2
+      g_\xi(x) \mathbb{E}\left[v_k^2|\xi_k = x\right] dx
+      \stackrel{\mathbb{P}}{\pm} \varepsilon_9 \\
+    &= \int_{\supp{g_xi}}
+      \left(\frac{w_{l_x}^{*,\tilde{\lambda}}}{v_{l_x}}\right)^2
+      g_\xi(x) \mathbb{E}\left[v_k^2|\xi_k = x\right] dx
+      \stackrel{\mathbb{P}}{\pm} \varepsilon_9.      
+  \end{align*}
+  With these transformations Lemma~\ref{lem:s4} follows analogously.
+\end{Proof}
 
+\begin{Proof}[Heiss, Teichmann, and Wutte (2019, Lemma A.19)]~\\\noindent
+    \label{proof:lem19}
+  The proof works analogously if $F_{+-}^{\lambda,g}$ is substituted
+  by
+  \begin{align*}
+    F_{+-}^{\lambda,g '}(f_+, f_-) =
+    & \sum_{i =
+            1}^N \left(f(x_i^{\text{train}}) -
+                                      y_i^{\text{train}}\right)^2 \\
+    & + \lambda g(0) \left(\int_{\supp(g)}\frac{\left(f_+''(x)\right)^2}{g(x)}
+        dx + \int_{\supp(g)}\frac{\left(f''_-(x)\right)^2}{g(x)}
+        dx\right).
+  \end{align*}
+  As for $f^n = f_+^n + f_-^n$ such that $\supp(f_+^n) \cap \supp(f_-^n) =
+  \emptyset$ and $h = h_+ + h_-$ such that $\supp(h_+) \cap \supp(h_-) =
+  \emptyset$ it holds
+  \[
+    \plimn F^{\lambda, g}(f^n) = F^{\lambda, g}(h) \implies
+    \plimn F_{+-}^{\lambda,g '}(f_+,f_-) = F_{+-}^{\lambda,g '}(h_+,h_-),
+  \]
+  and all functions can be split in two functions with disjoint support,
+  Lemma~\ref{lem:s7} follows.
+\end{Proof}
+\input{Appendix_code.tex}
 
-\end{Lemma}
 \end{appendices}
 
 

TeX/bibliograpy.bib

@@ -2,15 +2,12 @@
 	series = {arXiv},
 	author = {Heiss, Jakob and Teichmann, Josef and Wutte, Hanna},
 	publisher = {Cornell University},
-	year = {2019},
+	year = {2019},	copyright = {In Copyright - Non-Commercial Use Permitted},
-	language = {en},
-	copyright = {In Copyright - Non-Commercial Use Permitted},
 	keywords = {early stopping; implicit regularization; machine learning; neural networks; spline; regression; gradient descent; artificial intelligence},
 	size = {53 p.},
-	abstract = {Today, various forms of neural networks are trained to perform approximation tasks in many fields. However, the solutions obtained are not fully understood. Empirical results suggest that typical training algorithms favor regularized solutions.These observations motivate us to analyze properties of the solutions found by gradient descent initialized close to zero, that is frequently employed to perform the training task. As a starting point, we consider one dimensional (shallow) ReLU neural networks in which weights are chosen randomly and only the terminal layer is trained. We show that the resulting solution converges to the smooth spline interpolation of the training data as the number of hidden nodes tends to infinity. Moreover, we derive a correspondence between the early stopped gradient descent and the smoothing spline regression. This might give valuable insight on the properties of the solutions obtained using gradient descent methods in general settings.},
 	DOI = {10.3929/ethz-b-000402003},
 	title = {How Implicit Regularization of Neural Networks Affects the Learned Function – Part I},
-	PAGES = {1911.02903}
+	PAGES = {1911.02903},
 }
 
 @article{Dropout,
@@ -21,7 +18,7 @@
   volume  = 15,
   number  = 56,
   pages   = {1929-1958},
-  url     = {http://jmlr.org/papers/v15/srivastava14a.html}
+  Comment url     = {http://jmlr.org/papers/v15/srivastava14a.html}
 }
 
 @article{ADADELTA,
@@ -30,12 +27,10 @@
   journal   = {CoRR},
   volume    = {abs/1212.5701},
   year      = 2012,
-  url       = {http://arxiv.org/abs/1212.5701},
+  Comment url       = {http://arxiv.org/abs/1212.5701},
   archivePrefix = {arXiv},
   eprint    = {1212.5701},
   timestamp = {Mon, 13 Aug 2018 16:45:57 +0200},
-  biburl    = {https://dblp.org/rec/journals/corr/abs-1212-5701.bib},
-  bibsource = {dblp computer science bibliography, https://dblp.org}
 }
 
 @article{backprop,
@@ -50,45 +45,31 @@ day={01},
 volume={323},
 number={6088},
 pages={533-536},
-abstract={We describe a new learning procedure, back-propagation, for networks of neurone-like units. The procedure repeatedly adjusts the weights of the connections in the network so as to minimize a measure of the difference between the actual output vector of the net and the desired output vector. As a result of the weight adjustments, internal `hidden' units which are not part of the input or output come to represent important features of the task domain, and the regularities in the task are captured by the interactions of these units. The ability to create useful new features distinguishes back-propagation from earlier, simpler methods such as the perceptron-convergence procedure1.},
 issn={1476-4687},
 doi={10.1038/323533a0},
-url={https://doi.org/10.1038/323533a0}
+Comment url={https://doi.org/10.1038/323533a0}
 }
 
 @article{MNIST,
   added-at = {2010-06-28T21:16:30.000+0200},
   author = {LeCun, Yann and Cortes, Corinna},
-  biburl = {https://www.bibsonomy.org/bibtex/2935bad99fa1f65e03c25b315aa3c1032/mhwombat},
   groups = {public},
   howpublished = {http://yann.lecun.com/exdb/mnist/},
-  interhash = {21b9d0558bd66279df9452562df6e6f3},
-  intrahash = {935bad99fa1f65e03c25b315aa3c1032},
   keywords = {MSc _checked character_recognition mnist network neural},
   lastchecked = {2016-01-14 14:24:11},
   timestamp = {2016-07-12T19:25:30.000+0200},
   title = {{MNIST} handwritten digit database},
-  url = {http://yann.lecun.com/exdb/mnist/},
+  Comment url = {http://yann.lecun.com/exdb/mnist/},
-  username = {mhwombat},
   year = 2010
 }
-
+@INPROCEEDINGS{resnet,
-@article{resnet,
+  author={Kaiming {He} and Xiangyu {Zhang} and Shaoqing {Ren} and Jian {Sun}},
-  author    = {Kaiming He and
+  booktitle={2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)}, 
-               Xiangyu Zhang and
+  title={Deep Residual Learning for Image Recognition}, 
-               Shaoqing Ren and
+  year={2016},
-               Jian Sun},
+  volume={},
-  title     = {Deep Residual Learning for Image Recognition},
+  number={},
-  journal   = {CoRR},
+  pages={770-778},}
-  volume    = {abs/1512.03385},
-  year      = 2015,
-  url       = {http://arxiv.org/abs/1512.03385},
-  archivePrefix = {arXiv},
-  eprint    = {1512.03385},
-  timestamp = {Wed, 17 Apr 2019 17:23:45 +0200},
-  biburl    = {https://dblp.org/rec/journals/corr/HeZRS15.bib},
-  bibsource = {dblp computer science bibliography, https://dblp.org}
-}
 
 @book{PRML,
    title =     {Pattern Recognition and Machine Learning},
@@ -117,6 +98,15 @@ numpages = {39}
 }
 
 @article{DBLP:journals/corr/DauphinPGCGB14,
+author = {Dauphin, Yann and Pascanu, Razvan and Gulcehre, Caglar and Cho, Kyunghyun and Ganguli, Surya and Bengio, Y.},
+year = {2014},
+month = {06},
+pages = {},
+title = {Identifying and attacking the saddle point problem in high-dimensional non-convex optimization},
+volume = {27},
+journal = {NIPS}
+}
+@article{saddle_point,
   author    = {Yann N. Dauphin and
                Razvan Pascanu and
                {\c{C}}aglar G{\"{u}}l{\c{c}}ehre and
@@ -128,11 +118,10 @@ numpages = {39}
   journal   = {CoRR},
   volume    = {abs/1406.2572},
   year      = {2014},
-  url       = {http://arxiv.org/abs/1406.2572},
+  Comment url       = {http://arxiv.org/abs/1406.2572},
   archivePrefix = {arXiv},
   eprint    = {1406.2572},
   timestamp = {Mon, 22 Jul 2019 13:15:46 +0200},
-  biburl    = {https://dblp.org/rec/journals/corr/DauphinPGCGB14.bib},
   bibsource = {dblp computer science bibliography, https://dblp.org}
 }
 
@@ -146,12 +135,10 @@ numpages = {39}
   journal   = {CoRR},
   volume    = {abs/1207.0580},
   year      = {2012},
-  url       = {http://arxiv.org/abs/1207.0580},
+  Comment url = {http://arxiv.org/abs/1207.0580},
   archivePrefix = {arXiv},
   eprint    = {1207.0580},
   timestamp = {Mon, 13 Aug 2018 16:46:10 +0200},
-  biburl    = {https://dblp.org/rec/journals/corr/abs-1207-0580.bib},
-  bibsource = {dblp computer science bibliography, https://dblp.org}
 }
 
 @inproceedings{
@@ -160,22 +147,20 @@ title={On the Variance of the Adaptive Learning Rate and Beyond},
 author={Liyuan Liu and Haoming Jiang and Pengcheng He and Weizhu Chen and Xiaodong Liu and Jianfeng Gao and Jiawei Han},
 booktitle={International Conference on Learning Representations},
 year={2020},
-url={https://openreview.net/forum?id=rkgz2aEKDr}
+Comment url={https://openreview.net/forum?id=rkgz2aEKDr}
 }
 
 @inproceedings{ADAM,
   author    = {Diederik P. Kingma and
                Jimmy Ba},
-  editor    = {Yoshua Bengio and
+  @Comment editor    = {Yoshua Bengio and
-               Yann LeCun},
+  @Comment              Yann LeCun},
   title     = {Adam: {A} Method for Stochastic Optimization},
   booktitle = {3rd International Conference on Learning Representations, {ICLR} 2015,
                San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings},
   year      = {2015},
-  url       = {http://arxiv.org/abs/1412.6980},
+  Comment url       = {http://arxiv.org/abs/1412.6980},
   timestamp = {Thu, 25 Jul 2019 14:25:37 +0200},
-  biburl    = {https://dblp.org/rec/journals/corr/KingmaB14.bib},
-  bibsource = {dblp computer science bibliography, https://dblp.org}
 }
 
 @article{transfer_learning,
@@ -187,11 +172,11 @@ url={https://openreview.net/forum?id=rkgz2aEKDr}
   pages = {020018},
   year = {2017},
   doi = {10.1063/1.4992835},
-  URL = {https://aip.scitation.org/doi/abs/10.1063/1.4992835},
   eprint = {https://aip.scitation.org/doi/pdf/10.1063/1.4992835}
 }
 
 @article{gan,
+  author = "Maayan Frid-Adar and Idit Diamant and Eyal Klang and Michal Amitai and Jacob Goldberger and Hayit Greenspan",
   title = "GAN-based synthetic medical image augmentation for increased CNN performance in liver lesion classification",
   journal = "Neurocomputing",
   volume = 321,
@@ -199,6 +184,112 @@ url={https://openreview.net/forum?id=rkgz2aEKDr}
   year = 2018,
   issn = "0925-2312",
   doi = "https://doi.org/10.1016/j.neucom.2018.09.013",
-  url = "http://www.sciencedirect.com/science/article/pii/S0925231218310749",
+  Comment 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},
+Comment 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,
+  Comment url       = {http://arxiv.org/abs/1708.04896},
+  archivePrefix = {arXiv},
+  eprint    = {1708.04896},
+  timestamp = {Mon, 13 Aug 2018 16:47:52 +0200},
+}
+
+@misc{draw_convnet,
+  title = {Python script for illustrating Convolutional Neural Network (ConvNet)},
+  howpublished = {\url{https://github.com/gwding/draw_convnet}},
+  note = {Accessed: 30.08.2020},
+  author = {Gavin Weiguang Ding},
+  year = 2018
+}
+
+@book{Haykin,
+  added-at = {2009-06-26T15:25:19.000+0200},
+  author = {Haykin, Simon},
+  note = {2nd edition},
+  publisher = {Prentice Hall},
+  title = {Neural Networks: {A} Comprehensive Foundation},
+  year = 1999
+}
+
+@book{Goodfellow,
+    title={Deep Learning},
+    author={Ian Goodfellow and Yoshua Bengio and Aaron Courville},
+    publisher={MIT Press},
+    note={\url{http://www.deeplearningbook.org}},
+    year=2016
+}
+
+@article{ruder,
+  author    = {Sebastian Ruder},
+  title     = {An overview of gradient descent optimization algorithms},
+  journal   = {CoRR},
+  volume    = {abs/1609.04747},
+  year      = {2016},
+  url       = {http://arxiv.org/abs/1609.04747},
+  archivePrefix = {arXiv},
+  eprint    = {1609.04747},
+  timestamp = {Mon, 13 Aug 2018 16:48:10 +0200},
+  biburl    = {https://dblp.org/rec/journals/corr/Ruder16.bib},
+  bibsource = {dblp computer science bibliography, https://dblp.org}
+}
+
+@incollection{goodfellow_gan,
+title = {Generative Adversarial Nets},
+author = {Goodfellow, Ian and Pouget-Abadie, Jean and Mirza, Mehdi and Xu, Bing and Warde-Farley, David and Ozair, Sherjil and Courville, Aaron and Bengio, Yoshua},
+booktitle = {Advances in Neural Information Processing Systems 27},
+pages = {2672--2680},
+year = {2014},
+publisher = {Curran Associates, Inc.},
+url = {http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf}
+}
+
+@book{hastie01statisticallearning,
+  added-at = {2008-05-16T16:17:42.000+0200},
+  address = {New York, NY, USA},
+  author = {Hastie, Trevor and Tibshirani, Robert and Friedman, Jerome},
+  biburl = {https://www.bibsonomy.org/bibtex/2f58afc5c9793fcc8ad8389824e57984c/sb3000},
+  interhash = {d585aea274f2b9b228fc1629bc273644},
+  intrahash = {f58afc5c9793fcc8ad8389824e57984c},
+  keywords = {ml statistics},
+  publisher = {Springer New York Inc.},
+  series = {Springer Series in Statistics},
+  timestamp = {2008-05-16T16:17:43.000+0200},
+  title = {The Elements of Statistical Learning},
+  year = 2001
 }

TeX/introduction.tex

@@ -1,6 +1,74 @@
 \section{Introduction}
 
+Neural networks have become a widely used model for a plethora of
+applications.
+They are an attractive choice as they are  able to
+model complex data with relatively little additional input to the
+training data needed.
+Additionally, as the price of parallelized computing
+power in the form of graphics processing unit has decreased drastically over the last
+years, it has become far more accessible to train and use large
+neural networks.
+Furthermore, highly optimized and parallelized frameworks for tensor
+operations have been developed.
+With these frameworks, such as TensorFlow and PyTorch, building neural
+networks has become a much more straightforward process.
+% Furthermore, with the development of highly optimized and
+% parallelized implementations of mathematical operations needed for
+% neural networks, such as TensorFlow or PyTorch, building neural network
+% models has become a much more straightforward process.
+% For example the flagship consumer GPU GeForce RTX 3080 of NVIDIA's current
+% generation has 5.888 CUDS cores at a ... price of 799 Euro compared
+% to the last generations flagship GeForce RTX 2080 Ti with 4352 CUDA
+% cores at a ... price of 1259 Euro. These CUDA cores are computing
+% cores specialized for tensor operations, which are necessary in
+% fitting and using neural networks.
 
+In this thesis we want to get an understanding of the behavior of neural %
+networks and 
+how we can use them for problems with a complex relationship between
+in- and output.
+In Section 2 we introduce the mathematical construct of neural
+networks and how to fit them to training data.
+
+To gain some insight about the learned function,
+we examine a simple class of neural networks that contain only one
+hidden layer.
+In Section~\ref{sec:shallownn} we proof a relation between such networks and
+functions that minimize the distance to training data 
+with respect to its second derivative.
+
+An interesting application of neural networks is the task of
+classifying images.
+However, for such complex problems the number of parameters in fully
+connected neural networks can exceed what is
+feasible for training.
+In Section~\ref{sec:cnn} we explore the addition of convolution to neural
+networks to reduce the number of parameters.
+
+As these large networks are commonly trained using gradient decent
+algorithms we compare the performance of different algorithms based on
+gradient descent in Section~4.4.
+% and
+% show that it is beneficial to only use small subsets of the training
+% data in each iteration rather than using the whole data set to update
+% the parameters.
+Most statistical models especially these with large amounts of
+trainable parameters can struggle with overfitting the data.
+In Section 4.5 we examine the impact of two measures designed to combat
+overfitting.
+
+In some applications such as working with medical images the data
+available for training can be scarce, which results in the networks
+being prone to overfitting.
+As these are interesting applications of neural networks we examine
+the benefit of the measures to combat overfitting for 
+scenarios with limited amounts of training data.
+
+% As in some applications such as medical imaging one might be limited
+% to very small training data we study the impact of two measures in
+% improving the accuracy in such a case by trying to ... the model from
+% overfitting the data.
 
 
 

TeX/introduction_nn.tex

@@ -1,19 +1,26 @@
 
 \section{Introduction to Neural Networks}
 
-Neural Networks (NN) are a mathematical construct inspired by the
+This chapter is based on \textcite[Chapter~6]{Goodfellow} and \textcite{Haykin}.
-... of brains in mammals. It consists of an array of neurons that
+
-receive inputs and compute a accumulated output. These neurons are
+Neural Networks are a mathematical construct inspired by the
-arranged in layers, with one input and output layer and a arbirtary
+structure of brains in mammals. They consist of an array of neurons that
-amount of hidden layer between them.
+receive inputs and compute an accumulated output. These neurons are
-The amount of neurons in the in- and output layers correspond to the
+arranged in layers, with one input and output layer
+and an arbitrary
+amount of hidden layers between them.
+The number of neurons in the in- and output layers correspond to the
 desired dimensions of in- and outputs of the model.
-In conventional neural networks the information is passed ... from the
+
-input layer towards the output layer hence they are often called feed
+In conventional neural networks, the information is fed forward from the
-forward networks. Each neuron in a layer has the outputs of all
+input layer towards the output layer, hence they are often called
-neurons in the preceding layer as input (fully connected). A
+feed forward networks. Each neuron in a layer has the outputs of all 
-illustration of a example neuronal network is given in
+neurons in the preceding layer as input and computes an accumulated
-Figure~\ref{fig:nn} and one of a neuron in Figure~\ref{fig:neuron}
+value from these (fully connected).
+% An illustration of an example neural network is given in
+% Figure~\ref{fig:nn} and one of a neuron in Figure~\ref{fig:neuron}.
+Illustrations of a neural network and the structure of a neuron are given
+in Figure~\ref{fig:nn} and Figure~\ref{fig:neuron}.
 
 \tikzset{%
   every neuron/.style={
@@ -39,16 +46,16 @@ Figure~\ref{fig:nn} and one of a neuron in Figure~\ref{fig:neuron}
             {\arrow[scale=1.5,>=stealth]{>}}},postaction={decorate}}}
       
       \foreach \m/\l [count=\y] in {1,2,3,missing,4}
-      \node [every neuron/.try, neuron \m/.try] (input-\m) at (0,2.5-\y) {};
+      \node [every neuron/.try, neuron \m/.try] (input-\m) at (0,2.55-\y*0.85) {};
       
       \foreach \m [count=\y] in {1,missing,2}
-      \node [every neuron/.try, neuron \m/.try ] (hidden1-\m) at (2,2-\y*1.25) {};
+      \node [every neuron/.try, neuron \m/.try ] (hidden1-\m) at (2.5,2.5-\y*1.25) {};
       
       \foreach \m [count=\y] in {1,missing,2}
-      \node [every neuron/.try, neuron \m/.try ] (hidden2-\m) at (5,2-\y*1.25) {};
+      \node [every neuron/.try, neuron \m/.try ] (hidden2-\m) at (5,2.5-\y*1.25) {};
         
       \foreach \m [count=\y] in {1,missing,2}
-      \node [every neuron/.try, neuron \m/.try ] (output-\m) at (7,1.5-\y) {};
+      \node [every neuron/.try, neuron \m/.try ] (output-\m) at (7,1.5-\y*0.75) {};
 
       \foreach \l [count=\i] in {1,2,3,d_i}
       \draw [myptr] (input-\i)+(-1,0) -- (input-\i)
@@ -84,7 +91,7 @@ Figure~\ref{fig:nn} and one of a neuron in Figure~\ref{fig:neuron}
       \node[fill=white,scale=1.5,inner xsep=10pt,inner ysep=10mm] at ($(hidden1-1)!.5!(hidden2-2)$) {$\dots$};
         
     \end{tikzpicture}}%}
-  \caption[Illustration of a neural network]{Illustration of a neural network with $d_i$ inputs, $l$
+  \caption[Illustration of a Neural Network]{Illustration of a neural network with $d_i$ inputs, $l$
     hidden layers with $n_{\cdot}$ nodes in each layer, as well as
     $d_o$ outputs.
   }
@@ -93,102 +100,62 @@ Figure~\ref{fig:nn} and one of a neuron in Figure~\ref{fig:neuron}
 
 \subsection{Nonlinearity of Neural Networks}
 
-The arguably most important feature of neural networks that sets them
+The arguably most important feature of neural networks which sets them
 apart from linear models is the activation function implemented in the
-neurons. As seen in Figure~\ref{fig:neuron} on the weighted sum of the
+neurons. As illustrated in Figure~\ref{fig:neuron} on the weighted sum of the
-inputs a activation function $\sigma$ is applied in order to obtain
+inputs an activation function $\sigma$ is applied resulting in the
-the output resulting in the output of the $k$-th. neuron in a layer
+output of the $k$-th neuron in a layer $l$ with $m$ nodes in layer $l-1$
 being given by
-\[
+\begin{align*}
-  o_k = \sigma\left(b_k + \sum_{j=1}^m w_{k,j} i_j\right)
+  o_{l,k} = \sigma\left(b_{l,k} + \sum_{j=1}^{m} w_{l,k,j}
-\]
+            o_{l-1,j}\right),
-for weights $w_{k,j}$ and biases $b_k$.
+\end{align*}
+for weights $w_{l,k,j}$ and biases $b_{l,k}$. For a network with $L$
+hidden layers and inputs $o_{0}$ the final outputs of the network
+are thus given by $o_{L+1}$.
 The activation function is usually chosen nonlinear (a linear one
-would result in the entire model collapsing into a linear one\todo{beweis?}) which
+would result in the entire network collapsing into a linear model) which
 allows it to better model data where the relation of in- and output is
 of nonlinear nature.
-There are two types of activation functions, saturating and not
+There are two types of activation functions, saturating and
-saturating ones. Popular examples for the former are sigmoid
+non-saturating ones. Popular examples for the former are sigmoid 
-functions where most commonly the standard logisitc function or tangen
+functions where most commonly the standard logistic function or tangens
 hyperbolicus are used
-as they have easy to compute derivatives which is desirable for gradient
+as they have easy to compute derivatives which is desirable for
-based optimization algorithms. The standard logistic function (often
+gradient-based optimization algorithms. The standard logistic function
-referred to simply as sigmoid function) is given by
+(often simply referred to as sigmoid function) is given by
 \[
   f(x) = \frac{1}{1+e^{-x}} 
 \]
-and has a realm of $[0,1]$. Its usage as an activation function is
+and has a realm of $[0,1]$. The tangens hyperbolicus is given by
-motivated by modeling neurons which
-are close to deactive until a certain threshold where they grow in
-intensity until they are fully
-active, which is similar to the behavior of neurons in
-brains\todo{besser schreiben}. The tangens hyperbolicus is given by
 \[
   \tanh(x) = \frac{2}{e^{2x}+1}
 \]
-
+and has a realm of $[-1,1]$. Both functions result in neurons that are
-The downside of these saturating activation functions is that given
+close to inactive until a certain threshold is reached where they grow
-their saturating nature their derivatives are close to zero for large or small
+until saturation.
-input values which can slow or hinder the progress of gradient based methods.
+The downside of these saturating activation functions is, that their
-
+derivatives are close to zero on most of their realm, only assuming
-The nonsaturating activation functions commonly used are the recified
+larger values in proximity to zero.
-linear using (ReLU) or the leaky RelU. The ReLU is given by
+This can hinder the progress of gradient-based methods.
-\[
+
+The non-saturating activation functions commonly used are the rectified
+linear unit (ReLU) or the leaky ReLU. The ReLU is given by
+\begin{equation}
   r(x) = \max\left\{0, x\right\}.
-\]
+  \label{eq:relu}
+\end{equation}
 This has the benefit of having a constant derivative for values larger
-than zero. However the derivative being zero has the same downside for
+than zero. However, the derivative being zero for negative values has
-fitting the model with gradient based methods. The leaky ReLU is
+the same downside for
+fitting the model with gradient-based methods. The leaky ReLU is
 an attempt to counteract this problem by assigning a small constant
-derivative to all values smaller than zero and for scalar $\alpha$ is given by
+derivative to all values smaller than zero and for a scalar $\alpha$ is given by
 \[
   l(x) = \max\left\{0, x\right\} + \alpha \min \left\{0, x\right\}.
 \]
-In order to illustrate these functions plots of them are given in Figure~\ref{fig:activation}.
+In Figure~\ref{fig:activation} visualizations of these functions are given.
-
+%In order to illustrate these functions plots of them are given in Figure~\ref{fig:activation}.
-\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}
-    \caption{\titlecap{standard logistic function}}
-  \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}
-    \caption{\titlecap{tangens hyperbolicus}}
-  \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}
-    \caption{ReLU}
-  \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}
-    \caption{Leaky ReLU, $\alpha = 0.1$}
-  \end{subfigure}
-  \caption{Plots of the activation functions}
-  \label{fig:activation}
-\end{figure}
 
 
 \begin{figure}
@@ -274,17 +241,62 @@ In order to illustrate these functions plots of them are given in Figure~\ref{fi
     % \draw [->] (hidden-\i) -- (output-\j);
 
   \end{tikzpicture}
-  \caption{Structure of a single neuron}
+  \caption[Structure of a Single Neuron]{Structure of a single neuron.}
   \label{fig:neuron}
 \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}
+    \caption{Standard Logistic Function}
+  \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}
+    \caption{Tangens Hyperbolicus}
+  \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}
+    \caption{ReLU}
+  \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}
+    \caption{Leaky ReLU, $\alpha = 0.1$}
+  \end{subfigure}
+  \caption[Plots of the Activation Functions]{Plots of the activation functions.}
+  \label{fig:activation}
+\end{figure}
+
 \clearpage
 \subsection{Training Neural Networks}
 
-As neural networks are a PARAMETRIC model we need to fit it to input
+As neural networks are parametric models we need to fit the
-data in order to get meaningfull  OUTPUT from the network in order to
+parameters to the input
-do this we first need to discuss how we interpret the output of the
+data to get meaningful predictions from the network. In order
-neural network.
+to accomplish this we need to discuss how we interpret the output of the
+neural network and assess the quality of predictions.
 
 % After a neural network model is designed, like most statistical models
 % it has to be fit to the data. In the machine learning context this is
@@ -302,20 +314,20 @@ neural network.
 % data-point in fitting the model, where usually some distance between
 % the model output and the labels is minimized.
 
-\subsubsection{\titlecap{nonliniarity in last layer}}
+\subsubsection{Nonlinearity in the Last Layer}
 
-Given the nature of the neural net the output of the last layer are
+Given the nature of the neural net, the outputs of the last layer are
-real numbers. For regression tasks this is desirable, for
+real numbers. For regression tasks, this is desirable, for
 classification problems however some transformations might be
 necessary.
 As the goal in the latter is to predict a certain class or classes for
-an object the output needs to be of a form that allows this
+an object, the output needs to be of a form that allows this
 interpretation.
 Commonly the nodes in the output layer each correspond to a class and
 the class chosen as prediction is the one with the highest value at
 the corresponding output node.
-This corresponds to a transformation of the output
+This can be modeled as a transformation of the output
-vector $o$ into a one-hot vector
+vector $o \in \mathbb{R}^n$ into a one-hot vector
 \[
   \text{pred}_i =
   \begin{cases}
@@ -323,19 +335,20 @@ vector $o$ into a one-hot vector
     0,& \text{else}.
   \end{cases}
 \]
-This however makes training the model with gradient based methods impossible, as the derivative of
+This however makes training the model with gradient-based methods impossible, as the derivative of
 the transformation is either zero or undefined.
-A continuous transformation that is close to the argmax one is given by
+An continuous transformation that is close to argmax is given by
 softmax
-\[
+\begin{equation}
   \text{softmax}(o)_i = \frac{e^{o_i}}{\sum_j e^{o_j}}.
-\]
+  \label{eq:softmax}
+\end{equation}
 The softmax function transforms the realm of the output to the interval $[0,1]$
 and the individual values sum to one, thus the output can be interpreted as
-a probability for each class given the input.
+a probability for each class conditioned on the input.
-Additionally to being differentiable this allows for evaluataing the
+Additionally, to being differentiable this allows to evaluate the
-cetainiy of a prediction, rather than just whether it is accurate.
+certainty of a prediction, rather than just whether it is accurate.
-A similar effect is obtained when for a binary or two class problem the
+A similar effect is obtained when for a binary or two-class problem the
 sigmoid function
 \[
   f(x) = \frac{1}{1 + e^{-x}}
@@ -343,7 +356,6 @@ sigmoid function
 is used and the output $f(x)$ is interpreted as the probability for
 the first class and $1-f(x)$ for the second class.
 
-\todo{vielleicht additiv invarianz}
 % Another property that makes softmax attractive is the invariance to addition
 % \[
 %   \text{sofmax}(o) = \text{softmax}(o + c
@@ -379,57 +391,62 @@ the first class and $1-f(x)$ for the second class.
 % way to circumvent this problem is to normalize the output vector is
 % such a way that the entries add up to one, this allows for the
 % interpretation of probabilities assigned to each class.
-
+\clearpage
 \subsubsection{Error Measurement}
 
-In order to make assessment about the quality of a network $\mathcal{NN}$ and train
+In order to train the network we need to be able to assess the quality
-it we need to discuss how we measure error. The choice of the error
+of predictions using some error measure. 
-function is highly dependent on the type of the problem. For
+The choice of the error
-regression problems a commonly used error measure is the mean squared
+function is highly dependent on the type of problem. For
+regression problems, a commonly used error measure is the mean squared
 error (MSE)
-which for a function $f$ and data $(x_i,y_i), i=1,\dots,n$ is given by
+which for a function $f$ and data $(x_i,y_i), i \in \left\{1,\dots,n\right\}$ is given by
 \[
   MSE(f) = \frac{1}{n} \sum_i^n \left(f(x_i) - y_i\right)^2.
 \]
-However depending on the problem error measures with differnt
+However, depending on the problem error measures with different
-properties might be needed, for example in some contexts it is
+properties might be needed. For example in some contexts it is
-required to consider a proportional rather than absolute error as is
+required to consider a proportional rather than absolute error.
-common in time series models. \todo{komisch}
 
 As discussed above the output of a neural network for a classification
 problem can be interpreted as a probability distribution over the classes
-conditioned on the input. In this case it is desirable to
+conditioned on the input. In this case, it is desirable to
 use error functions designed to compare probability distributions. A
-widespread error function for this use case is the cross entropy (\textcite{PRML}),
+widespread error function for this use case is the categorical cross entropy (\textcite{PRML}),
 which for two discrete distributions $p, q$ with the same realm $C$ is given by
 \[
   H(p, q) = \sum_{c \in C} p(c) \ln\left(\frac{1}{q(c)}\right),
 \]
-which compares a $q$ to a true underlying distribution $p$.  
+comparing $q$ to a target density $p$.  
-For a data set $(x_i,y_i), i = 1,\dots,n$ where each $y_{i,c}$
+For a data set $(x_i,y_i), i \in \left\{1,\dots,n\right\}$ where each $y_{i,c}$
-corresponds to the probability of class $c$ given $x_i$ and predictor
+corresponds to the probability of class $c$ given $x_i$ and a predictor
 $f$ we get the loss function
-\[
+\begin{equation}
-  Bla = \sum_{i=1}^n H(y_i, f(x_i)).
+  CE(f) = \sum_{i=1}^n H(y_i, f(x_i)).
-\]
+  \label{eq:cross_entropy}
+\end{equation}
 
--Maximum Likelihood
+% \todo{Den satz einbauen}
--Ableitung mit softmax pseudo linear -> fast improvemtns possible
+% -Maximum Likelihood
+% -Ableitung mit softmax pseudo linear -> fast improvemtns possible
 
 \subsubsection{Gradient Descent Algorithm}
 
 Trying to find the optimal parameter for fitting the model to the data
 can be a hard problem. Given the complex nature of a neural network
-with many layers and neurons it is hard to predict the impact of
+with many layers and neurons, it is hard to predict the impact of
 single parameters on the accuracy of the output.
-Thus applying numeric optimization algorithms is the only
+Thus using numeric optimization algorithms is the only
-feasible way to fit the model. A attractive algorithm for training
+feasible way to fit the model.
-neural networks is gradient descent where each parameter $\theta_i$ is
+
-iterative changed according to the gradient regarding the error
+An attractive algorithm for training
-measure and a step size $\gamma$. For this all parameters are
+neural networks is gradient descent. Here all parameters are
-initialized (often random or close to zero) and then iteratively
+initialized with certain values (often random or close to zero) and
-updated until a certain stopping criterion is hit, mostly either being a fixed
+then iteratively updated. The updates are made in the direction of the
-number of iterations or a desired upper limit for the error measure.
+gradient regarding the error with a step size $\gamma$ until a
+specified stopping criterion is hit.
+% This  mostly either being a fixed
+% number of iterations or a desired upper limit for the error measure.
 % For a function $f_\theta$ with parameters $\theta \in \mathbb{R}^n$
 % and a error function $L(f_\theta)$ the gradient descent algorithm is
 % given in \ref{alg:gd}.
@@ -452,16 +469,21 @@ number of iterations or a desired upper limit for the error measure.
 
 The algorithm for gradient descent is given in
 Algorithm~\ref{alg:gd}. In the context of fitting a neural network
-$f_\theta$ corresponds to the error measurement of the network
+$f_\theta$ corresponds to an error measurement of a neural network
-$L\left(\mathcal{NN}_{\theta}\right)$ where $\theta$ is a vector
+$\mathcal{NN}_{\theta}$ where $\theta$ is a vector
 containing all the weights and biases of the network.
-As ca be seen this requires computing the derivative of the network
+As can be seen, this requires computing the derivative of the network
 with regard to each variable. With the number of variables getting
 large in networks with multiple layers of high neuron count naively
-computing these can get quite memory and computational expensive. But
+computing the derivatives can get quite memory and computational
-by using the chain rule and exploiting the layered structure we can
+expensive.
-compute the gradient much more efficiently by using backpropagation
+By using the chain rule and exploiting the layered structure we can
-introduced by \textcite{backprop}. 
+compute the parameter update much more efficiently. This practice is
+called backpropagation and was introduced for use in neural networks by
+\textcite{backprop}. The algorithm
+for one data point is given in Algorithm~\ref{alg:backprop}, but for all error
+functions that are sums of errors for single data points (MSE, cross
+entropy) backpropagation works analogously for larger training data. 
 
 % \subsubsection{Backpropagation}
 
@@ -475,10 +497,34 @@ introduced by \textcite{backprop}.
 % for each layer from output layer towards the first layer while only
 % needing to ....
 
-\[
+\begin{algorithm}[H]
-  \frac{\partial L(...)}{}
+  \SetAlgoLined
-\]
+  \KwInput{Inputs $o_0$, neural network
-\todo{Backprop richtig aufschreiben}
+    with $L$ hidden layers, weights $w$, and biases $b$ for $n_l$
+    nodes as well as an activation function $\sigma_l$ in layer $l$
+    and loss function $\tilde{L}$.}
+  Forward Propagation:
+  \For{$l \in \left\{1, \dots, L+1\right\}$}{
+    Compute values for layer $l$:
+    $z_{l,k} \leftarrow b_{l,k} + w_{l,k}^{\mathrm{T}} o_{l-1}, k \in \left\{1,\dots,n_l\right\}$\;
+    $o_{l,k} \leftarrow \sigma_l(z_{l,k}), k \in \left\{1,\dots,n_l\right\}$ \;
+  }
+  Calculate derivative for output layer: $\delta_{L+1, k} \leftarrow
+  \frac{\partial\tilde{L}(o_{L+1})}{\partial o_{L+1,k}} \sigma_{L+1}'(z_{L+1,k})$\;
+  Back propagate the error:
+  \For{$l \in \left\{L,\dots,1\right\}$}{
+    $\delta_{l,k} \leftarrow w_{l+1,k}^{\mathrm{T}} \delta_{l+1}
+    \sigma_{l}'(z_{l,k}), k=1,\dots,n_k$
+  }
+  Calculate gradients:
+  $\frac{\partial\tilde{L}}{\partial w_{l,k,j}} =
+  \delta_{l,k}o_{l-1,j}$,
+  $\frac{\partial\tilde{L}}{\partial b_{l,k}} =
+  \delta_{l,k}$\;
+  
+  \caption{Backpropagation for one data point}
+  \label{alg:backprop}
+\end{algorithm}
 
 %%% Local Variables:
 %%% mode: latex

TeX/main.lot

@@ -0,0 +1,6 @@
+\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax 
+\babel@toc {english}{}
+\defcounter {refsection}{0}\relax 
+\contentsline {table}{\numberline {4.1}{\ignorespaces Values of Test Accuracies for Models Trained on Subsets of MNIST Handwritten Digits}}{41}{table.4.1}% 
+\defcounter {refsection}{0}\relax 
+\contentsline {table}{\numberline {4.2}{\ignorespaces Values of Test Accuracies for Models Trained on Subsets of Fashion MNIST}}{41}{table.4.2}% 

TeX/main.out

@@ -0,0 +1,25 @@
+\BOOKMARK [1][-]{section.1}{Introduction}{}% 1
+\BOOKMARK [1][-]{section.2}{Introduction to Neural Networks}{}% 2
+\BOOKMARK [2][-]{subsection.2.1}{Nonlinearity of Neural Networks}{section.2}% 3
+\BOOKMARK [2][-]{subsection.2.2}{Training Neural Networks}{section.2}% 4
+\BOOKMARK [3][-]{subsubsection.2.2.1}{Nonlinearity in the Last Layer}{subsection.2.2}% 5
+\BOOKMARK [3][-]{subsubsection.2.2.2}{Error Measurement}{subsection.2.2}% 6
+\BOOKMARK [3][-]{subsubsection.2.2.3}{Gradient Descent Algorithm}{subsection.2.2}% 7
+\BOOKMARK [1][-]{section.3}{Shallow Neural Networks}{}% 8
+\BOOKMARK [2][-]{subsection.3.1}{Convergence Behavior of One-Dimensional Randomized Shallow Neural Networks}{section.3}% 9
+\BOOKMARK [2][-]{subsection.3.2}{Simulations}{section.3}% 10
+\BOOKMARK [1][-]{section.4}{Application of Neural Networks to Higher Complexity Problems}{}% 11
+\BOOKMARK [2][-]{subsection.4.1}{Convolution}{section.4}% 12
+\BOOKMARK [2][-]{subsection.4.2}{Convolutional Neural Networks}{section.4}% 13
+\BOOKMARK [2][-]{subsection.4.3}{Stochastic Training Algorithms}{section.4}% 14
+\BOOKMARK [2][-]{subsection.4.4}{Modified Stochastic Gradient Descent}{section.4}% 15
+\BOOKMARK [2][-]{subsection.4.5}{Combating Overfitting}{section.4}% 16
+\BOOKMARK [3][-]{subsubsection.4.5.1}{Dropout}{subsection.4.5}% 17
+\BOOKMARK [3][-]{subsubsection.4.5.2}{Manipulation of Input Data}{subsection.4.5}% 18
+\BOOKMARK [3][-]{subsubsection.4.5.3}{Comparisons}{subsection.4.5}% 19
+\BOOKMARK [3][-]{subsubsection.4.5.4}{Effectiveness for Small Training Sets}{subsection.4.5}% 20
+\BOOKMARK [1][-]{section.5}{Summary and Outlook}{}% 21
+\BOOKMARK [1][-]{section*.27}{Appendices}{}% 22
+\BOOKMARK [1][-]{Appendix.1.A}{Notes on Proofs of Lemmata in Section 3.1}{}% 23
+\BOOKMARK [1][-]{Appendix.1.B}{Implementations}{}% 24
+\BOOKMARK [1][-]{Appendix.1.C}{Additional Comparisons}{}% 25

TeX/main.tex

@@ -1,4 +1,4 @@
-\documentclass[a4paper, 12pt, draft=true]{article}
+\documentclass[a4paper, 12pt]{article}
 
 %\usepackage[margin=1in]{geometry}
 %\geometry{a4paper, left=30mm, right=40mm,top=25mm, bottom=20mm}
@@ -6,6 +6,7 @@
 \usepackage[english]{babel}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
+\usepackage{hyperref}
 \usepackage{textcomp}
 %\usepackage{libertine}
 \usepackage{amsmath}
@@ -34,10 +35,18 @@
 \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{titlecaps}
+\usepackage{afterpage}
+\usepackage{xcolor}
+\usepackage{chngcntr}
+%\usepackage{hyperref}
+% \hypersetup{
+%   linktoc=all,     %set to all if you want both sections and subsections linked
+% }
+\allowdisplaybreaks  
 
 \captionsetup[sub]{justification=centering}
 
@@ -52,9 +61,125 @@
 \pgfplotsset{compat = 1.16}
 \usepackage[export]{adjustbox}
 
-
+\definecolor{maroon}{cmyk}{0, 0.87, 0.68, 0.32}
-
+\definecolor{halfgray}{gray}{0.55}
-\usepackage[style=authoryear, backend=bibtex]{biblatex}
+\definecolor{ipython_frame}{RGB}{207, 207, 207}
+\definecolor{ipython_bg}{RGB}{247, 247, 247}
+\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}{110, 64, 130}
+
+\usepackage{listings}
+\usepackage{float}
+
+\newfloat{lstfloat}{htbp}{lop}
+\floatname{lstfloat}{Listing}
+\def\lstfloatautorefname{Listing}
+
+\lstset{
+    breaklines=true,
+    %
+    extendedchars=true,
+    literate=
+    {á}{{\'a}}1 {é}{{\'e}}1 {í}{{\'i}}1 {ó}{{\'o}}1 {ú}{{\'u}}1
+    {Á}{{\'A}}1 {É}{{\'E}}1 {Í}{{\'I}}1 {Ó}{{\'O}}1 {Ú}{{\'U}}1
+    {à}{{\`a}}1 {è}{{\`e}}1 {ì}{{\`i}}1 {ò}{{\`o}}1 {ù}{{\`u}}1
+    {À}{{\`A}}1 {È}{{\'E}}1 {Ì}{{\`I}}1 {Ò}{{\`O}}1 {Ù}{{\`U}}1
+    {ä}{{\"a}}1 {ë}{{\"e}}1 {ï}{{\"i}}1 {ö}{{\"o}}1 {ü}{{\"u}}1
+    {Ä}{{\"A}}1 {Ë}{{\"E}}1 {Ï}{{\"I}}1 {Ö}{{\"O}}1 {Ü}{{\"U}}1
+    {â}{{\^a}}1 {ê}{{\^e}}1 {î}{{\^i}}1 {ô}{{\^o}}1 {û}{{\^u}}1
+    {Â}{{\^A}}1 {Ê}{{\^E}}1 {Î}{{\^I}}1 {Ô}{{\^O}}1 {Û}{{\^U}}1
+    {œ}{{\oe}}1 {Œ}{{\OE}}1 {æ}{{\ae}}1 {Æ}{{\AE}}1 {ß}{{\ss}}1
+    {ç}{{\c c}}1 {Ç}{{\c C}}1 {ø}{{\o}}1 {å}{{\r a}}1 {Å}{{\r A}}1
+    {€}{{\EUR}}1 {£}{{\pounds}}1
+}
+
+%%
+%% Python definition (c) 1998 Michael Weber
+%% Additional definitions (2013) Alexis Dimitriadis
+%% 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},%
+    %
+    % 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,val},%
+    %
+    sensitive=true,%
+    morecomment=[l]\#,%
+    morestring=[b]',%
+    morestring=[b]",%
+    %
+    morestring=[s]{'''}{'''},% used for documentation text (mulitiline strings)
+    morestring=[s]{"""}{"""},% added by Philipp Matthias Hahn
+    %
+    morestring=[s]{r'}{'},% `raw' strings
+    morestring=[s]{r"}{"},%
+    morestring=[s]{r'''}{'''},%
+    morestring=[s]{r"""}{"""},%
+    morestring=[s]{u'}{'},% unicode strings
+    morestring=[s]{u"}{"},%
+    morestring=[s]{u'''}{'''},%
+    morestring=[s]{u"""}{"""},%
+    %
+    % {replace}{replacement}{lenght of replace}
+    % *{-}{-}{1} will not replace in comments and so on
+    literate=
+    {á}{{\'a}}1 {é}{{\'e}}1 {í}{{\'i}}1 {ó}{{\'o}}1 {ú}{{\'u}}1
+    {Á}{{\'A}}1 {É}{{\'E}}1 {Í}{{\'I}}1 {Ó}{{\'O}}1 {Ú}{{\'U}}1
+    {à}{{\`a}}1 {è}{{\`e}}1 {ì}{{\`i}}1 {ò}{{\`o}}1 {ù}{{\`u}}1
+    {À}{{\`A}}1 {È}{{\'E}}1 {Ì}{{\`I}}1 {Ò}{{\`O}}1 {Ù}{{\`U}}1
+    {ä}{{\"a}}1 {ë}{{\"e}}1 {ï}{{\"i}}1 {ö}{{\"o}}1 {ü}{{\"u}}1
+    {Ä}{{\"A}}1 {Ë}{{\"E}}1 {Ï}{{\"I}}1 {Ö}{{\"O}}1 {Ü}{{\"U}}1
+    {â}{{\^a}}1 {ê}{{\^e}}1 {î}{{\^i}}1 {ô}{{\^o}}1 {û}{{\^u}}1
+    {Â}{{\^A}}1 {Ê}{{\^E}}1 {Î}{{\^I}}1 {Ô}{{\^O}}1 {Û}{{\^U}}1
+    {œ}{{\oe}}1 {Œ}{{\OE}}1 {æ}{{\ae}}1 {Æ}{{\AE}}1 {ß}{{\ss}}1
+    {ç}{{\c c}}1 {Ç}{{\c C}}1 {ø}{{\o}}1 {å}{{\r a}}1 {Å}{{\r A}}1
+    {€}{{\EUR}}1 {£}{{\pounds}}1
+    %
+    {^}{{{\color{ipython_purple}\^{}}}}1
+    {=}{{{\color{ipython_purple}=}}}1
+    %
+    {+}{{{\color{ipython_purple}+}}}1
+    {*}{{{\color{ipython_purple}$^\ast$}}}1
+    {/}{{{\color{ipython_purple}/}}}1
+    %
+    {+=}{{{+=}}}1
+    {-=}{{{-=}}}1
+    {*=}{{{$^\ast$=}}}1
+    {/=}{{{/=}}}1,
+    literate=
+    *{-}{{{\color{ipython_purple}-}}}1
+     {?}{{{\color{ipython_purple}?}}}1,
+    %
+    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,
+    numberstyle=\tiny\color{halfgray},
+    %
+    %
+    backgroundcolor=\color{ipython_bg},
+    %   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},
+}
+
+\usepackage[authordate, backend=bibtex, firstinits = true]{biblatex-chicago}
 \urlstyle{same}
 \bibliography{bibliograpy.bib}
 \numberwithin{figure}{section}
@@ -73,11 +198,13 @@
 \newtheorem{Algorithm}[Theorem]{Algorithm}
 \newtheorem{Example}[Theorem]{Example}
 \newtheorem{Assumption}[Theorem]{Assumption}
+\newtheorem{Proof}[Theorem]{Proof}
 
 
 \DeclareMathOperator*{\plim}{\mathbb{P}\text{-}\lim}
 \DeclareMathOperator{\supp}{supp}
 \DeclareMathOperator*{\argmin}{arg\,min}
+\DeclareMathOperator*{\argmax}{arg\,max}
 \DeclareMathOperator*{\po}{\mathbb{P}\text{-}\mathcal{O}}
 \DeclareMathOperator*{\equals}{=}
 \begin{document}
@@ -103,19 +230,36 @@
 %\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}}
+  \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{Neural Networks and their Application on Higher Complexity Problems}\\
+  \vspace{1cm}
+  \huge \textbf{Tim Tobias Arndt}\\
+  \vspace{1cm}
+  \Large \textbf{October 2020}
 \end{center}
-\hrulefill
+
 \pagenumbering{gobble}
-\newpage
+\clearpage
 %\setcounter{tocdepth}{4}
 \tableofcontents
 \clearpage
 \listoffigures
-\listoftodos
+\listoftables
 \newpage
 \pagenumbering{arabic}
 % Introduction
@@ -137,7 +281,7 @@
 \input{further_applications_of_nn}
 
 \newpage
-
+\DeclareNameAlias{sortname}{last-first}
 \printbibliography
 
 % Appendix A

TeX/theo_3_8.tex

@@ -5,17 +5,18 @@
 %%% TeX-master: "main"
 %%% End:
 \section{Shallow Neural Networks}
-
+\label{sec:shallownn}
 % In order to get a some understanding of the behavior of neural
 % networks we study a simplified class of networks called shallow neural
 % networks in this chapter.
 % We consider shallow neural networks consist of a single
 % hidden layer and
-In order to get some understanding of the behavior of neural networks
+To get some understanding of the behavior of neural networks
 we examine a simple class of networks in this chapter. We consider
 networks that contain only one hidden layer and have a single output
-node. We call these networks shallow neural networks.
+node and call these networks shallow neural networks.
-\begin{Definition}[Shallow neural network]
+\begin{Definition}[Shallow neural network, Heiss, Teichmann, and
+  Wutte (2019, Definition 1.4)]
   For a input dimension $d$ and a Lipschitz continuous activation function $\sigma:
   \mathbb{R} \to \mathbb{R}$ we define a shallow neural network with
   $n$ hidden nodes as
@@ -84,8 +85,8 @@ with
 %   \label{fig:shallowNN}
 % \end{figure}
 
-As neural networks with a large amount of nodes have a large amount of
+As neural networks with a large number of nodes have a large amount of
-parameters that can be tuned it can often fit the data quite well. If
+tunable parameters it can often fit data quite well. If
 a ReLU activation function
 \[
   \sigma(x) \coloneqq \max{(0, x)}
@@ -105,9 +106,9 @@ on MSE will perfectly fit the data.
   minimizing squared error loss.
   \proof
   W.l.o.g. all values $x_{ij}^{\text{train}} \in [0,1],~\forall i \in
-  \left\{1,\dots\right\}, j \in \left\{1,\dots,d\right\}$. Now we
+  \left\{1,\dots, t\right\}, j \in \left\{1,\dots,d\right\}$. Now we
-  chose $v^*$ in order to calculate a unique value for all
+  chose $v^*$ such that the vector-product with $x_i^{\text{train}}$
-  $x_i^{\text{train}}$:
+  results is distinct values for all $i \in \left\{1,\dots,t\right\}$:
   \[
     v^*_{k,j} = v^*_{j} = 10^{j-1}, ~ \forall k \in \left\{1,\dots,n\right\}.
   \]
@@ -141,30 +142,32 @@ on MSE will perfectly fit the data.
   and $\vartheta^* = (w^*, b^*, v^*, c = 0)$ we get
   \[
     \mathcal{NN}_{\vartheta^*} (x_i^{\text{train}}) = \sum_{k =
-      1}^{i-1} w_k\left(\left(v^*\right)^{\mathrm{T}}
+      1}^{i-1} w_k\left(b_k^* + \left(v^*\right)^{\mathrm{T}}
-      x_i^{\text{train}}\right) + w_i\left(\left(v^*\right)^{\mathrm{T}}
+      x_i^{\text{train}}\right) + w_i\left(b_i^* +\left(v^*\right)^{\mathrm{T}}
       x_i^{\text{train}}\right) = y_i^{\text{train}}.
   \]
   As the squared error of $\mathcal{NN}_{\vartheta^*}$ is zero all
   squared error loss minimizing shallow networks with at least $t$ hidden
-  nodes will perfectly fit the data.
+  nodes will perfectly fit the data. \qed
-  \qed
   \label{theo:overfit}
 \end{Theorem}
 
-However this behavior is often not desired as over fit models generally
+However, this behavior is often not desired as overfit models tend to
-have bad generalization properties especially if noise is present in
+have bad generalization properties, especially if noise is present in
 the data. This effect is illustrated in
-Figure~\ref{fig:overfit}. Here a shallow neural network that perfectly fits the
+Figure~\ref{fig:overfit}.
-training data regarding MSE is \todo{Formulierung}
+
-constructed according to the proof of Theorem~\ref{theo:overfit} and
+Here a shallow neural network is
-compared to a regression spline 
+constructed according to the proof of Theorem~\ref{theo:overfit} to
+perfectly fit some data and
+compared to a cubic smoothing spline 
 (Definition~\ref{def:wrs}). While the neural network
 fits the data better than the spline, the spline represents the
 underlying mechanism that was used to generate the data more accurately. The better
 generalization of the spline compared to the network is further
-demonstrated by the better validation error computed on newly generated
+demonstrated by the better performance on newly generated
 test data.
+
 In order to improve the accuracy of the model we want to reduce
 overfitting. A possible way to achieve this is by explicitly
 regularizing the network through the cost function as done with
@@ -172,48 +175,48 @@ ridge penalized networks
 (Definition~\ref{def:rpnn}) where large weights $w$ are punished. In
 Theorem~\ref{theo:main1} we will
 prove that this will result in the shallow neural network converging to
-regressions splines as the amount of nodes in the hidden layer is
+a form of splines as the number of nodes in the hidden layer is
 increased.
 
-
+\vfill
-
+
-
+\begin{figure}[h]
-\begin{figure}
+  \pgfplotsset{
-    \pgfplotsset{
+    compat=1.11,
-      compat=1.11,
+    legend image code/.code={
-legend image code/.code={
+      \draw[mark repeat=2,mark phase=2]
-  \draw[mark repeat=2,mark phase=2]
+      plot coordinates {
-plot coordinates {
+        (0cm,0cm)
-  (0cm,0cm)
+        (0.15cm,0cm)        %% default is (0.3cm,0cm)
-(0.15cm,0cm)        %% default is (0.3cm,0cm)
+        (0.3cm,0cm)         %% default is (0.6cm,0cm)
-(0.3cm,0cm)         %% default is (0.6cm,0cm)
+      };%
-};%
+    }
-}
+  }
-}
+  \begin{tikzpicture}
-    \begin{tikzpicture}
+    \begin{axis}[tick style = {draw = none}, width = \textwidth,
-      \begin{axis}[tick style = {draw = none}, width = \textwidth,
+      height = 0.6\textwidth]
-        height = 0.6\textwidth]
+      \addplot table
-        \addplot table
+      [x=x, y=y, col sep=comma, only marks,mark options={scale =
-        [x=x, y=y, col sep=comma, only marks,mark options={scale =
+        0.7}] {Figures/Data/overfit.csv};  
-          0.7}] {Plots/Data/overfit.csv};  
+      \addplot [red, line width=0.8pt] table [x=x_n, y=s_n, col
-        \addplot [red, line width=0.8pt] table [x=x_n, y=s_n, col
+      sep=comma] {Figures/Data/overfit.csv}; 
-        sep=comma, forget plot] {Plots/Data/overfit.csv}; 
+      \addplot [black, line width=0.8pt] table [x=x_n, y=y_n, col
-        \addplot [black, line width=0.8pt] table [x=x_n, y=y_n, col
+      sep=comma] {Figures/Data/overfit.csv}; 
-        sep=comma] {Plots/Data/overfit.csv}; 
+      \addplot [black, line width=0.8pt, dashed] table [x=x, y=y, col
-        \addplot [black, line width=0.8pt, dashed] table [x=x, y=y, col
+      sep=comma] {Figures/Data/overfit_spline.csv};
-        sep=comma] {Plots/Data/overfit_spline.csv};
+      
-
+      \addlegendentry{\footnotesize{Data}};
-        \addlegendentry{\footnotesize{data}};
+      \addlegendentry{\footnotesize{Truth}};
-        \addlegendentry{\footnotesize{$\mathcal{NN}_{\vartheta^*}$}};
+      \addlegendentry{\footnotesize{$\mathcal{NN}_{\vartheta^*}$}};
-        \addlegendentry{\footnotesize{spline}};
+      \addlegendentry{\footnotesize{Spline}};
-      \end{axis}
+    \end{axis}
-    \end{tikzpicture}
+  \end{tikzpicture}
-  \caption[Overfitting of shallow neural networks]{For data of the form $y=\sin(\frac{x+\pi}{2 \pi}) +
+  \caption[Overfitting of Shallow Neural Networks]{For data of the form $y=\sin(\frac{x+\pi}{2 \pi}) +
     \varepsilon,~ \varepsilon \sim \mathcal{N}(0,0.4)$
-    (\textcolor{blue}{blue dots}) the neural network constructed 
+    (\textcolor{blue}{blue}) the neural network constructed 
     according to the proof of Theorem~\ref{theo:overfit} (black) and the
     underlying signal (\textcolor{red}{red}). While the network has no
-    bias a regression spline (black dashed) fits the data much
+    bias a cubic smoothing spline (black, dashed) fits the data much
     better. For a test set of size 20 with uniformly distributed $x$
     values and responses of the same fashion as the training data the MSE of the neural network is
     0.30, while the MSE of the spline is only 0.14 thus generalizing
@@ -222,31 +225,43 @@ plot coordinates {
   \label{fig:overfit}
 \end{figure}
 
+\vfill
+
 \clearpage
-\subsection{Convergence Behaviour of 1-dim. Randomized Shallow Neural
+\subsection{Convergence Behavior of One-Dimensional Randomized Shallow
-  Networks} 
+  Neural Networks}
-
+\label{sec:conv}
-
+
-This section is based on \textcite{heiss2019}. We will analyze the
+This section is based on \textcite{heiss2019}.
-connection between randomized shallow
+In this section, we examine the convergence behavior of certain shallow
-Neural Networks with one dimensional input with a ReLU as activation
+neural networks.
-function for all neurons and regression splines.
+We consider shallow neural networks with a one dimensional input where the parameters in the
-% \[
+hidden layer are randomized resulting in only the weights is the
-%   \sigma(x) = \max\left\{0,x\right\}.
+output layer being trainable.
-% \]
+Additionally, we assume all neurons use a ReLU as an activation function
-We will see that the punishment of the size of the weights in training
+and call such networks randomized shallow neural networks. 
-the randomized shallow
+
-Neural Network will result in a learned function that minimizes the second
+% We will analyze the
-derivative as the amount of hidden nodes is grown to infinity. In order
+% connection between randomized shallow
-to properly formulate this relation we will first need to introduce
+% Neural Networks with one dimensional input with a ReLU as activation
-some definitions, all neural networks introduced in the following will
+% function for all neurons and cubic smoothing splines.
-use a ReLU as activation at all neurons.
+% % \[
-
+% %   \sigma(x) = \max\left\{0,x\right\}.
-A randomized shallow network is characterized by only the weight
+% % \]
-parameter of the output layer being trainable, whereas the other
+% We will see that the punishment of the size of the weights in training
-parameters are random numbers.
+% the randomized shallow
-
+% Neural Network will result in a learned function that minimizes the second
-\begin{Definition}[Randomized shallow neural network]
+% derivative as the amount of hidden nodes is grown to infinity. In order
+% to properly formulate this relation we will first need to introduce
+% some definitions, all neural networks introduced in the following will
+% use a ReLU as activation at all neurons.
+
+% A randomized shallow network is characterized by only the weight
+% parameter of the output layer being trainable, whereas the other
+% parameters are random numbers.
+
+\begin{Definition}[Randomized shallow neural network, Heiss, Teichmann, and
+  Wutte (2019, Definition 2.1)]
   For an input dimension $d$, let $n \in \mathbb{N}$ be the number of
   hidden nodes and $v(\omega) \in \mathbb{R}^{i \times n}, b(\omega)
   \in \mathbb{R}^n$ randomly drawn weights. Then for a weight vector
@@ -257,15 +272,27 @@ parameters are random numbers.
 \]
 \label{def:rsnn}
 \end{Definition}
-We call a one dimensional randomized shallow neural network were the
+% We call a one dimensional randomized shallow neural network were the
-$L^2$ norm of the trainable weights $w$ are penalized in the loss
+%  are penalized in the loss
-function ridge penalized neural networks.
+% function ridge penalized neural networks.
+
+We will prove that if we penalize the amount of the trainable weights
+when fitting a randomized shallow neural network it will
+converge to a function that minimizes the distance to the training
+data with respect to its second derivative as the amount of nodes is increased.
+We call such a network that is fitted according to MSE and a penalty term for
+the $L^2$ norm of the trainable weights $w$ a ridge penalized neural network.
+% $\lam$
+% We call a randomized shallow neural network trained on MSE and
+% punished for the amount of the weights $w$ according to a
+% ... $\lambda$ ridge penalized neural networks.
 
 % We call a randomized shallow neural network where the size of the trainable
 % weights is punished in the error function a ridge penalized
 % neural network. For a tuning parameter $\tilde{\lambda}$ .. the extent
 % of penalization we get:
-\begin{Definition}[Ridge penalized Neural Network]
+\begin{Definition}[Ridge penalized Neural Network, Heiss, Teichmann, and
+  Wutte (2019, Definition 3.2)]
   \label{def:rpnn}
   Let $\mathcal{RN}_{w, \omega}$ be a randomized shallow neural
   network, as introduced in Definition~\ref{def:rsnn} and tuning
@@ -276,7 +303,7 @@ function ridge penalized neural networks.
     \mathcal{RN}^{*, \tilde{\lambda}}_{\omega}(x) \coloneqq
     \mathcal{RN}_{w^{*, \tilde{\lambda}}(\omega), \omega}
   \]
-  with
+  with \
   \[
     w^{*,\tilde{\lambda}}(\omega) :\in \argmin_{w \in
       \mathbb{R}^n} \underbrace{ \left\{\overbrace{\sum_{i = 1}^N \left(\mathcal{RN}_{w,
@@ -292,7 +319,7 @@ having minimal weights, resulting in the \textit{minimum norm
   network} $\mathcal{RN}_{w^{\text{min}}, \omega}$.
 \[
   \mathcal{RN}_{w^{\text{min}}, \omega} \text{ randomized shallow
-    Neural network with weights } w^{\text{min}}:
+    neural network with weights } w^{\text{min}}\colon
 \]
 \[
   w^{\text{min}} \in \argmin_{w \in \mathbb{R}^n} \norm{w}, \text{
@@ -304,18 +331,18 @@ For $\tilde{\lambda} \to \infty$ the learned
 function will resemble the data less and with the weights
 approaching $0$ will converge to the constant $0$ function.
 
-In order to make the notation more convinient in the following the
+To make the notation more convenient, in the following the
-$\omega$ used to express the realised random parameters will no longer
+$\omega$ used to express the realized random parameters will no longer
 be explicitly mentioned.
 
 We call a function that minimizes the cubic distance between training points
-and the function with respect\todo{richtiges wort} to the second
+and the function with regard to the second
-derivative of the function a regression spline.
+derivative of the function a cubic smoothing spline.
 
-\begin{Definition}[Regression Spline]
+\begin{Definition}[Cubic Smoothing Spline]
   Let $x_i^{\text{train}}, y_i^{\text{train}} \in \mathbb{R}, i \in
-  \left\{1,\dots,N\right\}$ be trainig data. for a given $\lambda \in
+  \left\{1,\dots,N\right\}$ be training data. for a given $\lambda \in
-  \mathbb{R}$ the regression spline is given by
+  \mathbb{R}$ the cubic smoothing spline is given by
   \[
     f^{*,\lambda} :\in \argmin_{f \in
       \mathcal{C}^2}\left\{\sum_{i=1}^N
@@ -324,12 +351,12 @@ derivative of the function a regression spline.
   \]
 \end{Definition}
 
-We will show that for specific hyper parameters the ridge penalized
+We will show that for specific hyperparameters the ridge penalized
 shallow neural networks converge to a slightly modified variant of the
-regression spline. We will need to incorporate the densities of the
+cubic smoothing spline. We need to incorporate the densities of the
 random parameters in the loss function of the spline to ensure
 convergence. Thus we define
-the adapted weighted regression spline where the loss for the second
+the adapted weighted cubic smoothing spline where the loss for the second
 derivative is weighted by a function $g$ and the support of the second
 derivative of $f$ has to be a subset the support of $g$. The formal
 definition is given in Definition~\ref{def:wrs}.
@@ -340,19 +367,20 @@ definition is given in Definition~\ref{def:wrs}.
 % spline that allows for weighting the penalty term for the second
 % derivative with a weight function $g$. This is needed to ...the
 % distributions of the random parameters ... We call this the adapted
-% weighted regression spline.
+% weighted cubic smoothing spline.
 
-% Now we take a look at weighted regression splines. Later we will prove
+% Now we take a look at weighted cubic smoothing splines. Later we will prove
 % that the ridge penalized neural network as defined in
-% Definition~\ref{def:rpnn} converges a weighted regression spline, as
+% Definition~\ref{def:rpnn} converges a weighted cubic smoothing spline, as
 % the amount of hidden nodes is grown to inifity.
 
-\begin{Definition}[Adapted Weighted regression spline]
+\begin{Definition}[Adapted weighted cubic smoothing spline, Heiss, Teichmann, and
+  Wutte (2019, Definition 3.5)]
   \label{def:wrs}
   Let $x_i^{\text{train}}, y_i^{\text{train}} \in \mathbb{R}, i \in
-  \left\{1,\dots,N\right\}$ be trainig data. For a given $\lambda \in \mathbb{R}_{>0}$
+  \left\{1,\dots,N\right\}$ be training data. For a given $\lambda \in \mathbb{R}_{>0}$
   and a function $g: \mathbb{R} \to \mathbb{R}_{>0}$ the weighted
-  regression spline $f^{*, \lambda}_g$ is given by
+  cubic smoothing spline $f^{*, \lambda}_g$ is given by
 
   \[
     f^{*, \lambda}_g :\in \argmin_{\substack{f \in \mathcal{C}^2(\mathbb{R})
@@ -361,43 +389,41 @@ definition is given in Definition~\ref{def:wrs}.
         \lambda g(0) \int_{\supp(g)}\frac{\left(f''(x)\right)^2}{g(x)}
         dx\right\}}_{\eqqcolon F^{\lambda, g}(f)}.
   \]
-  \todo{Anforderung an Ableitung von f, doch nicht?}
+%  \todo{Anforderung an Ableitung von f, doch nicht?}
 \end{Definition}
-
 Similarly to ridge weight penalized neural networks the parameter
 $\lambda$ controls a trade-off between accuracy on the training data
-and smoothness or low second dreivative. For $g \equiv 1$ and $\lambda \to 0$ the
+and smoothness or low second derivative. For $g \equiv 1$ and $\lambda \to 0$ the
 resulting function $f^{*, 0+}$ will interpolate the training data while minimizing
 the second derivative. Such a function is known as cubic spline
 interpolation.
-\todo{cite cubic spline}
+\vspace{-0.2cm}
-
 \[
   f^{*, 0+} \text{ smooth spline interpolation: }
 \]
 \[
   f^{*, 0+} \coloneqq \lim_{\lambda \to 0+} f^{*, \lambda}_1 \in
-  \argmin_{\substack{f \in \mathcal{C}^2\mathbb{R}, \\ f(x_i^{\text{train}}) =
+  \argmin_{\substack{f \in \mathcal{C}^2(\mathbb{R}), \\ f(x_i^{\text{train}}) =
       y_i^{\text{train}}}} = \left( \int _{\mathbb{R}} (f''(x))^2dx\right).
 \]
-
 For $\lambda \to \infty$ on the other hand $f_g^{*\lambda}$ converges
 to linear regression of the data.
 
 We use two intermediary functions in order to show the convergence of
-the ridge penalized shallow neural network to adapted regression splines.
+the ridge penalized shallow neural network to adapted cubic smoothing splines.
 % In order to show that ridge penalized shallow neural networks converge
-% to adapted regression splines for a growing amount of hidden nodes we
+% to adapted cubic smoothing splines for a growing amount of hidden nodes we
 % define two intermediary functions.
-One being a smooth approximation of
+One being a smooth approximation of a
-the neural network, and a randomized shallow neural network designed
+neural network and the other being a randomized shallow neural network designed
 to approximate a spline.
-In order to properly BUILD these functions we need to take the points
+In order to properly construct these functions, we need to take the points
-of the network into consideration where the TRAJECTORY changes or
+of the network into consideration where the trajectory of the learned
-their points of discontinuity
+function changes
+(or their points of discontinuity).
 As we use the ReLU activation the function learned by the
 network will possess points of discontinuity where a neuron in the hidden
-layer gets activated (goes from 0 -> x>0). We formalize these points
+layer gets activated and their output is no longer zero. We formalize these points
 as kinks in Definition~\ref{def:kink}.
 \begin{Definition}
   \label{def:kink}
@@ -415,9 +441,9 @@ as kinks in Definition~\ref{def:kink}.
     \item Let $\xi_k \coloneqq -\frac{b_k}{v_k}$ be the k-th kink of $\mathcal{RN}_w$.
     \item Let $g_{\xi}(\xi_k)$ be the density of the kinks $\xi_k =
     - \frac{b_k}{v_k}$ in accordance to the distributions of $b_k$ and
-    $v_k$.
+    $v_k$. With $\supp(g_\xi) = \left[C_{g_\xi}^l, C_{g_\xi}^u\right]$.
     \item Let $h_{k,n} \coloneqq \frac{1}{n g_{\xi}(\xi_k)}$ be the
-    average estmated distance from kink $\xi_k$ to the next nearest
+    average estimated distance from kink $\xi_k$ to the next nearest
     one.
   \end{enumerate}
 \end{Definition}
@@ -433,40 +459,36 @@ network by applying the kernel similar to convolution.
   corresponding kink density $g_{\xi}$ as given by
   Definition~\ref{def:kink}.
   In order to smooth the RSNN consider following kernel for every $x$:
-
+  \begin{align*}
-  \[
+    \kappa_x(s) &\coloneqq \mathds{1}_{\left\{\abs{s} \leq \frac{1}{2 \sqrt{n}
-    \kappa_x(s) \coloneqq \mathds{1}_{\left\{\abs{s} \leq \frac{1}{2 \sqrt{n}
+          g_{\xi}(x)}\right\}}(s)\sqrt{n} g_{\xi}(x), \, \forall s \in \mathbb{R}\\
-          g_{\xi}(x)}\right\}}(s)\sqrt{n} g_{\xi}(x), \, \forall s \in \mathbb{R}
+  \intertext{Using this kernel we define a smooth approximation of
-  \]
+  $\mathcal{RN}_w$ by}
-
+    f^w(x) &\coloneqq \int_{\mathds{R}} \mathcal{RN}_w(x-s)
-  Using this kernel we define a smooth approximation of
+             \kappa_x(s) ds.
-  $\mathcal{RN}_w$ by
+  \end{align*}
-
-  \[
-    f^w(x) \coloneqq \int_{\mathds{R}} \mathcal{RN}_w(x-s) \kappa_x(s) ds.
-  \]
 \end{Definition}
-
 Note that the kernel introduced in Definition~\ref{def:srsnn}
-satisfies $\int_{\mathbb{R}}\kappa_x dx = 1$. While $f^w$ looks highly
+satisfies $\int_{\mathbb{R}}\kappa_x dx = 1$. While $f^w$ looks
 similar to a convolution, it differs slightly as the kernel $\kappa_x(s)$
 is dependent on $x$. Therefore only $f^w = (\mathcal{RN}_w *
 \kappa_x)(x)$ is well defined, while $\mathcal{RN}_w * \kappa$ is not.
-We use $f^{w^{*,\tilde{\lambda}}}$ do describe the spline
+We use $f^{w^{*,\tilde{\lambda}}}$ to describe the spline
-approximating the ... ridge penalized network
+approximating the ridge penalized network
-$\mathrm{RN}^{*,\tilde{\lambda}}$.
+$\mathcal{RN}^{*,\tilde{\lambda}}$.
 
-Next we construct a randomized shallow neural network which
+Next, we construct a randomized shallow neural network that
-approximates a spline independent from the realization of the random
+is designed to be close to a spline, independent from the realization of the random
-parameters. In order to achieve this we ...
+parameters, by approximating the splines curvature between the
+kinks.
 
-\begin{Definition}[Spline approximating Randomised Shallow Neural
+\begin{Definition}[Spline approximating Randomized Shallow Neural
   Network]
   \label{def:sann}
-  Let $\mathcal{RN}$ be a randomised shallow Neural Network according
+  Let $\mathcal{RN}$ be a randomized shallow Neural Network according
   to Definition~\ref{def:rsnn} and $f^{*, \lambda}_g$ be the weighted
-  regression spline as introduced in Definition~\ref{def:wrs}. Then
+  cubic smoothing spline as introduced in Definition~\ref{def:wrs}. Then
-  the randomised shallow neural network approximating $f^{*,
+  the randomized shallow neural network approximating $f^{*,
     \lambda}_g$ is given by 
   \[
     \mathcal{RN}_{\tilde{w}}(x) = \sum_{k = 1}^n \tilde{w}_k \sigma(b_k + v_k x),
@@ -474,7 +496,7 @@ parameters. In order to achieve this we ...
   with the weights $\tilde{w}_k$ defined as
   \[
     \tilde{w}_k \coloneqq \frac{h_{k,n} v_k}{\mathbb{E}[v^2 \vert \xi
-      = \xi_k]}  (f_g^{*, \lambda})''(\xi_k).
+      = \xi_k]}  \left(f_g^{*, \lambda}\right)''(\xi_k).
   \]
 \end{Definition}
 
@@ -488,16 +510,16 @@ derivative of $\mathcal{RN}_{\tilde{w}}(x)$ which is given by
   x}} \tilde{w}_k v_k \nonumber \\
               &= \frac{1}{n} \sum_{\substack{k \in \mathbb{N} \\
   \xi_k < x}} \frac{v_k^2}{g_{\xi}(\xi_k) \mathbb{E}[v^2 \vert \xi
-  = \xi_k]} (f_g^{*, \lambda})''(\xi_k). \label{eq:derivnn}
+  = \xi_k]} \left(f_g^{*, \lambda}\right)''(\xi_k). \label{eq:derivnn}
 \end{align}
-As the expression (\ref{eq:derivnn}) behaves similary to a
+As the expression (\ref{eq:derivnn}) behaves similarly to a
 Riemann-sum for $n \to \infty$ it will converge in probability to the
-first derivative of $f^{*,\lambda}_g$. A formal proof of this behaviour
+first derivative of $f^{*,\lambda}_g$. A formal proof of this behavior
 is given in Lemma~\ref{lem:s0}.
 
 In order to ensure the functions used in the proof of the convergence
-are well defined we need to assume some properties of the random
+are well defined we need to make some assumptions about properties of the random
-parameters and their densities
+parameters and their densities.
 
 % In order to formulate the theorem describing the convergence of $RN_w$
 % we need to make a couple of assumptions.
@@ -506,7 +528,7 @@ parameters and their densities
 \begin{Assumption}~
   \label{ass:theo38}
   \begin{enumerate}[label=(\alph*)]
-    \item The probability density fucntion of the kinks $\xi_k$,
+    \item The probability density function of the kinks $\xi_k$,
     namely $g_{\xi}$ as defined in Definition~\ref{def:kink} exists
     and is well defined.
     \item The density function $g_\xi$
@@ -514,14 +536,14 @@ parameters and their densities
     \item The density function $g_{\xi}$ is uniformly continuous on $\supp(g_{\xi})$.
     \item $g_{\xi}(0) \neq 0$.
     \item $\frac{1}{g_{\xi}}\Big|_{\supp(g_{\xi})}$ is uniformly
-    continous on $\supp(g_{\xi})$.
+    continuous on $\supp(g_{\xi})$.
     \item The conditional distribution $\mathcal{L}(v_k|\xi_k = x)$
-    is uniformly continous on $\supp(g_{\xi})$.
+    is uniformly continuous on $\supp(g_{\xi})$.
     \item $\mathbb{E}\left[v_k^2\right] < \infty$.
   \end{enumerate}
 \end{Assumption}
 
-As we will prove the convergence of in the Sobolev space, we hereby
+As we will prove the convergence of in the Sobolev Space, we hereby
 introduce it and the corresponding induced norm.
 
 \begin{Definition}[Sobolev Space]
@@ -529,7 +551,7 @@ introduce it and the corresponding induced norm.
   define the Sobolev space $W^{k,p}(K)$ as the space containing all
   real valued functions $u \in L^p(K)$ such that for every multi-index
   $\alpha \in \mathbb{N}^n$ with $\abs{\alpha} \leq
-  k$ the mixed parial derivatives
+  k$ the mixed partial derivatives
   \[
     u^{(\alpha)} = \frac{\partial^{\abs{\alpha}} u}{\partial
       x_1^{\alpha_1} \dots \partial x_n^{\alpha_n}}
@@ -538,9 +560,8 @@ introduce it and the corresponding induced norm.
   \[
     \norm{u^{(\alpha)}}_{L^p} < \infty.
   \]
-  \todo{feritg machen}
   \label{def:sobonorm}
-  The natural norm of the sobolev space is given by
+  The natural norm of the Sobolev Space is given by
   \[
     \norm{f}_{W^{k,p}(K)} =
     \begin{cases}
@@ -554,18 +575,21 @@ introduce it and the corresponding induced norm.
   \]
 \end{Definition}
 
-With the important definitions and assumptions in place we can now
+With the important definitions and assumptions in place, we can now
-formulate the main theorem ... the convergence of ridge penalized
+formulate the main theorem.
-random neural networks to adapted regression splines when the
+% ... the convergence of ridge penalized
-parameters are chosen accordingly.
+% random neural networks to adapted cubic smoothing splines when the
+% parameters are chosen accordingly.
 
-\begin{Theorem}[Ridge weight penaltiy corresponds to weighted regression spline]
+\begin{Theorem}[Ridge Weight Penalty Corresponds to Weighted Cubic
+  Smoothing Spline]
   \label{theo:main1}
-  For $N \in \mathbb{N}$ arbitrary training data
+  For $N \in \mathbb{N}$, arbitrary training data 
-  \(\left(x_i^{\text{train}}, y_i^{\text{train}}
+  $\left(x_i^{\text{train}}, y_i^{\text{train}}
-  \right)\) and $\mathcal{RN}^{*, \tilde{\lambda}}, f_g^{*, \lambda}$
+  \right)~\in~\mathbb{R}^2$, with $i \in \left\{1,\dots,N\right\}$,
+  and $\mathcal{RN}^{*, \tilde{\lambda}}, f_g^{*, \lambda}$ 
   according to Definition~\ref{def:rpnn} and Definition~\ref{def:wrs}
-  respectively with Assumption~\ref{ass:theo38} it holds
+  respectively with Assumption~\ref{ass:theo38} it holds that
 
   \begin{equation}
     \label{eq:main1}
@@ -581,7 +605,7 @@ parameters are chosen accordingly.
   \end{align*}
 \end{Theorem}
 As mentioned above we will prof Theorem~\ref{theo:main1} utilizing
-the ... functions. We show that 
+intermediary functions. We show that 
 \begin{equation}
   \label{eq:main2}
   \plimn \norm{\mathcal{RN}^{*, \tilde{\lambda}} - f^{w^*}}_{W^{1,
@@ -593,16 +617,16 @@ and
   \plimn \norm{f^{w^*} - f_g^{*, \lambda}}_{W^{1,\infty}(K)} = 0
 \end{equation}
 and then get (\ref{eq:main1}) using the triangle inequality. In
-order to prove (\ref{eq:main2}) and (\ref{eq:main3}) we will need to
+order to prove (\ref{eq:main2}) and (\ref{eq:main3}) we need to
-introduce a number of auxiliary lemmmata, proves of these will be
+introduce a number of auxiliary lemmata, proves of which are
-provided in the appendix.
+given in \textcite{heiss2019} and Appendix~\ref{appendix:proofs}.
 
 
 
-\begin{Lemma}[Poincar\'e typed inequality]
+\begin{Lemma}[Poincar\'e Typed Inequality]
   \label{lem:pieq}
   Let \(f:\mathbb{R} \to \mathbb{R}\) differentiable with \(f' :
-  \mathbb{R} \to \mathbb{R}\) Lesbeque integrable. Then for \(K=[a,b]
+  \mathbb{R} \to \mathbb{R}\) Lebesgue integrable. Then for \(K=[a,b]
   \subset \mathbb{R}\) with \(f(a)=0\) it holds that
   \begin{equation*}
     \label{eq:pti1}
@@ -610,14 +634,15 @@ provided in the appendix.
     \norm{f}_{w^{1,\infty}(K)} \leq C_K^{\infty}
     \norm{f'}_{L^{\infty}(K)}.
   \end{equation*}
-  If additionaly \(f'\) is differentiable with \(f'': \mathbb{R} \to
+  If additionally \(f'\) is differentiable with \(f'': \mathbb{R} \to
-  \mathbb{R}\) Lesbeque integrable then additionally
+  \mathbb{R}\) Lebesgue integrable then
   \begin{equation*}
     \label{eq:pti2}
     \exists C_K^2 \in \mathbb{R}_{>0} : \norm{f}_{W^{1,\infty}(K)} \leq
     C_K^2 \norm{f''}_{L^2(K)}.
   \end{equation*}
-  \proof The proof is given in the appendix...
+ % \proof The proof is given in the appendix...
+
   % With the fundamental theorem of calculus, if
   % \(\norm{f}_{L^{\infty}(K)}<\infty\) we get
   % \begin{equation}
@@ -654,11 +679,12 @@ provided in the appendix.
 \begin{Lemma}
   \label{lem:cnvh}
   Let $\mathcal{RN}$ be a shallow Neural network. For \(\varphi :
-  \mathbb{R}^2 \to \mathbb{R}\) uniformly continous such that
+  \mathbb{R}^2 \to \mathbb{R}\) uniformly continuous such that
   \[
     \forall x \in  \supp(g_{\xi}) : \mathbb{E}\left[\varphi(\xi, v)
       \frac{1}{n g_{\xi}(\xi)} \vert \xi = x \right] < \infty,
   \]
+  \clearpage
   it holds, that
   \[
     \plimn \sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k)
@@ -667,7 +693,7 @@ provided in the appendix.
     \mathbb{E}\left[\varphi(\xi, v) \vert \xi = x \right] dx
   \]
   uniformly in \(T \in K\).
-  \proof The proof is given in appendix...
+ % \proof The proof is given in appendix...
   % For \(T \leq C_{g_{\xi}}^l\) both sides equal 0, so it is sufficient to
   % consider \(T > C_{g_{\xi}}^l\). With \(\varphi\) and
   % \(\nicefrac{1}{g_{\xi}}\) uniformly continous in \(\xi\),
@@ -712,7 +738,7 @@ provided in the appendix.
   %       \kappa : \xi_m \in [\delta l, \delta(l +
   %       1)]\right\}}}{ng_{\xi}(l\delta)}\right) \pm \varepsilon .\\
   %   \intertext{We use the mean to approximate the number of kinks in
-  %   each $\delta$-strip, as it follows a bonomial distribution this
+  %   each $\delta$-strip, as it follows a binomial distribution this
   %   amounts to
   %   \[
   %     \mathbb{E}\left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l,
@@ -722,15 +748,16 @@ provided in the appendix.
   %   \]
   %   Bla Bla Bla $v_k$}
   %   \circled{1} & \approx 
-  % \end{align*}
+                    %     \end{align*}
+  \proof Notes on the proof are given in Proof~\ref{proof:lem9}.
 \end{Lemma}
 
-\begin{Lemma}[Step 0]
+\begin{Lemma}
-  For any $\lambda > 0$, training data $(x_i^{\text{train}}
+  For any $\lambda > 0$, $N \in \mathbb{N}$, training data $(x_i^{\text{train}}
   y_i^{\text{train}}) \in \mathbb{R}^2$, with $ i \in
-  \left\{1,\dots,N\right\}$ and subset $K \subset \mathbb{R}$ the spline approximating randomized
+  \left\{1,\dots,N\right\}$, and subset $K \subset \mathbb{R}$ the spline approximating randomized
   shallow neural network $\mathcal{RN}_{\tilde{w}}$ converges to the
-  regression spline $f^{*, \lambda}_g$ in
+  cubic smoothing spline $f^{*, \lambda}_g$ in
   $\norm{.}_{W^{1,\infty}(K)}$ as the node count $n$ increases,
   \begin{equation}
     \label{eq:s0}
@@ -744,77 +771,91 @@ provided in the appendix.
         \lambda}_g)'}_{L^{\infty}} = 0.
   \]
   This can be achieved by using Lemma~\ref{lem:cnvh} with $\varphi(\xi_k,
-  v_k) = \frac{v_k^2}{\mathbb{E}[v^2|\xi = z]} (f^{*, \lambda}_w)''(\xi_k) $
+  v_k) = \frac{v_k^2}{\mathbb{E}[v^2|\xi = z]} (f^{*, \lambda}_g)''(\xi_k) $
   thus obtaining
   \begin{align*}
-    \plimn \frac{\partial \mathcal{RN}_{\tilde{w}}}{\partial x}
+    \plimn \frac{\partial \mathcal{RN}_{\tilde{w}}}{\partial x} (x)
-    \stackrel{(\ref{eq:derivnn})}{=}
+    \equals^{(\ref{eq:derivnn})}_{\phantom{\text{Lemma 3.1.4}}}
- & \plimn \sum_{\substack{k \in \mathbb{N}                         \\ 
+    %\stackrel{(\ref{eq:derivnn})}{=}
-    \xi_k < x}} \frac{v_k^2}{\mathbb{E}[v^2 \vert \xi
+    &
-    = \xi_k]} (f_g^{*, \lambda})''(\xi_k) h_{k,n}
+      \plimn \sum_{\substack{k \in \mathbb{N}   \\                      
-    \stackrel{\text{Lemma}~\ref{lem:cnvh}}{=}                      \\
+      \xi_k < x}} \frac{v_k^2}{\mathbb{E}[v^2 \vert \xi
-    \stackrel{\phantom{(\ref{eq:derivnn})}}{=}
+      = \xi_k]} (f_g^{*, \lambda})''(\xi_k) h_{k,n} \\ 
- & 
+      \stackrel{\text{Lemma}~\ref{lem:cnvh}}{=}                    
-      \int_{\min\left\{C_{g_{\xi}}^l,T\right\}}^{min\left\{C_{g_{\xi}}^u,T\right\}}
+      %\stackrel{\phantom{(\ref{eq:derivnn})}}{=}
+    & 
+      \int_{\max\left\{C_{g_{\xi}}^l,x\right\}}^{\min\left\{C_{g_{\xi}}^u,x\right\}}
       \mathbb{E}\left[\frac{v^2}{\mathbb{E}[v^2|\xi = z]} (f^{*,
-      \lambda}_w)''(\xi) \vert
+      \lambda}_g)''(\xi) \vert
-      \xi = x \right] dx \equals^{\text{Tower-}}_{\text{property}} \\
+      \xi = z \right] dz\\
-    \stackrel{\phantom{(\ref{eq:derivnn})}}{=}
+      \mathmakebox[\widthof{$\stackrel{\text{Lemma 3.14}}{=}$}][c]{\equals^{\text{Tower-}}_{\text{property}}}
- & 
+       %\stackrel{\phantom{(\ref{eq:derivnn})}}{=}
-      \int_{\min\left\{C_{g_{\xi}}^l,
+    & 
-      T\right\}}^{min\left\{C_{g_{\xi}}^u,T\right\}}(f^{*,\lambda}_w)''(x)
+      \int_{\max\left\{C_{g_{\xi}}^l,
-      dx. 
+      x\right\}}^{\min\left\{C_{g_{\xi}}^u,x\right\}}(f^{*,\lambda}_g)''(z)
+      dz. 
   \end{align*}
-  By the fundamental theorem of calculus and $\supp(f') \subset
+  With the fundamental theorem of calculus we get
-  \supp(f)$, (\ref{eq:s0}) follows with Lemma~\ref{lem:pieq}.
+  \[
+    \plimn \mathcal{RN}_{\tilde{w}}'(x) = f_g^{*,\lambda
+      '}(\min\left\{C_{g_{\xi}}^u, x\right\}) - f_g^{*,\lambda
+      '}(\max\left\{C_{g_{\xi}}^l, x\right\})
+  \]
+  As $f_g^{*,\lambda '}$ is constant on $\left[C_{g_\xi}^l,
+    C_{g_\xi}^u\right]^C$ because $\supp(f_g^{*,\lambda ''}) \subseteq
+  \supp(g) \subseteq \supp(g_\xi)$ we get
+  \[
+    \plimn \mathcal{RN}_{\tilde{w}}'(x) =  f_g^{*,\lambda
+      '},
+  \]
+  thus  (\ref{eq:s0}) follows with Lemma~\ref{lem:pieq}.
   \qed
   \label{lem:s0}
 \end{Lemma}
 
-\begin{Lemma}[Step 2]
+\begin{Lemma}
-  For any $\lambda > 0$ and training data $(x_i^{\text{train}},
+  For any $\lambda > 0$, $N \in \mathbb{N}$, and training data $(x_i^{\text{train}},
-  y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
+  y_i^{\text{train}}) \in \mathbb{R}^2$, with $i \in
   \left\{1,\dots,N\right\}$, we have
   \[
     \plimn F^{\tilde{\lambda}}_n(\mathcal{RN}_{\tilde{w}}) =
     F^{\lambda, g}(f^{*, \lambda}_g) = 0.
   \]
-  \proof
+  \proof Notes on the proof are given in Proof~\ref{proof:lem14}.
-  The proof is given in the appendix...
   \label{lem:s2}
 \end{Lemma}
 
-\begin{Lemma}[Step 3]
+\begin{Lemma}
-  For any $\lambda > 0$ and training data $(x_i^{\text{train}},
+  For any $\lambda > 0$, $N \in \mathbb{N}$, and training data $(x_i^{\text{train}},
-  y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
+  y_i^{\text{train}}) \in \mathbb{R}^2$, with $i \in
   \left\{1,\dots,N\right\}$, with $w^*$ as
   defined in Definition~\ref{def:rpnn} and $\tilde{\lambda}$ as
-  defined in Theroem~\ref{theo:main1}, it holds
+  defined in Theorem~\ref{theo:main1}, it holds
   \[
     \plimn \norm{\mathcal{RN}^{*,\tilde{\lambda}} -
       f^{w*, \tilde{\lambda}}}_{W^{1,\infty}(K)} = 0.
   \]
-  \proof The proof is given in Appendix ..
+  \proof Notes on the proof are given in Proof~\ref{proof:lem15}.
   \label{lem:s3}
 \end{Lemma}
 
-\begin{Lemma}[Step 4]
+\begin{Lemma}
-  For any $\lambda > 0$ and training data $(x_i^{\text{train}},
+  For any $\lambda > 0$, $N \in \mathbb{N}$, and training data $(x_i^{\text{train}},
-  y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
+  y_i^{\text{train}}) \in \mathbb{R}^2$, with $i \in
   \left\{1,\dots,N\right\}$, with $w^*$ and $\tilde{\lambda}$ as
-  defined in Definition~\ref{def:rpnn} and Theroem~\ref{theo:main1}
+  defined in Definition~\ref{def:rpnn} and Theorem~\ref{theo:main1}
   respectively, it holds
   \[
     \plimn \abs{F_n^{\tilde{\lambda}}(\mathcal{RN}^{*,\tilde{\lambda}}) -
       F^{\lambda, g}(f^{w*, \tilde{\lambda}})} = 0.
   \]
-  \proof The proof is given in appendix...
+  \proof Notes on the proof are given in Proof~\ref{proof:lem16}.
   \label{lem:s4}
 \end{Lemma}
 
-\begin{Lemma}[Step 7]
+\begin{Lemma}
-  For any $\lambda > 0$ and training data $(x_i^{\text{train}},
+  For any $\lambda > 0$, $N \in \mathbb{N}$, and training data $(x_i^{\text{train}},
-  y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in
+  y_i^{\text{train}}) \in \mathbb{R}^2$, with $i \in
   \left\{1,\dots,N\right\}$, for any sequence of functions $f^n \in
   W^{2,2}$ with
   \[
@@ -824,39 +865,45 @@ provided in the appendix.
   \[
     \plimn \norm{f^n - f^{*, \lambda}} = 0.
   \]
-  \proof The proof is given in appendix ...
+  \proof Notes on the proof are given in Proof~\ref{proof:lem19}.
   \label{lem:s7}
 \end{Lemma}
 Using these lemmata we can now proof Theorem~\ref{theo:main1}. We
 start by showing that the error measure of the smooth approximation of
 the ridge penalized randomized shallow neural network $F^{\lambda,
-  g}\left(f^{w^{*,\tilde{\lambda}}}\right)$ 
+  g}(f^{w^{*,\tilde{\lambda}}})$ 
 will converge in probability to the error measure of the adapted weighted regression
 spline $F^{\lambda, g}\left(f^{*,\lambda}\right)$ for the specified
 parameters.
 
 Using Lemma~\ref{lem:s4} we get that for every $P \in (0,1)$ and
 $\varepsilon > 0$ there exists a $n_1 \in \mathbb{N}$ such that
-\[
+\begin{equation}
   \mathbb{P}\left[F^{\lambda, g}\left(f^{w^{*,\tilde{\lambda}}}\right) \in
     F_n^{\tilde{\lambda}}\left(\mathcal{RN}^{*,\tilde{\lambda}}\right)
-    +[-\varepsilon, \varepsilon]\right] > P, \forall n \in \mathbb{N}_{> n_1}.
+    +[-\varepsilon, \varepsilon]\right] > P, \forall n \in
-\]
+  \mathbb{N}_{> n_1}.
+  \label{eq:squeeze_1}
+\end{equation}
 As $\mathcal{RN}^{*,\tilde{\lambda}}$ is the optimal network for
 $F_n^{\tilde{\lambda}}$ we know that
-\[
+\begin{equation}
   F_n^{\tilde{\lambda}}\left(\mathcal{RN}^{*,\tilde{\lambda}}\right)
   \leq F_n^{\tilde{\lambda}}\left(\mathcal{RN}_{\tilde{w}}\right).
-\]
+  \label{eq:squeeze_2}
+\end{equation}
 Using Lemma~\ref{lem:s2} we get that for every $P \in (0,1)$ and
-$\varepsilon > 0$ there exists a $n_2 \in \mathbb{N}$ such that
+$\varepsilon > 0$ a $n_2 \in \mathbb{N}$ exists such that
-\[
+\begin{equation}
   \mathbb{P}\left[F_n^{\tilde{\lambda}}\left(\mathcal{RN}_{\tilde{w}}\right)
     \in F^{\lambda, g}\left(f^{*,\lambda}_g\right)+[-\varepsilon,
     \varepsilon]\right] > P, \forall n \in \mathbb{N}_{> n_2}.
-\]
+  \label{eq:squeeze_3}
-If we combine these ... we get that for every $P \in (0,1)$ and
+\end{equation}
-$\varepsilon > 0$ and $n_3 \geq
+Combining (\ref{eq:squeeze_1}), (\ref{eq:squeeze_2}), and
+(\ref{eq:squeeze_3}) we get that for every $P \in (0,1)$ and for \linebreak
+every
+$\varepsilon > 0$ with $n_3 \geq
 \max\left\{n_1,n_2\right\}$
 \[
   \mathbb{P}\left[F^{\lambda,
@@ -864,45 +911,52 @@ $\varepsilon > 0$ and $n_3 \geq
       g}\left(f^{*,\lambda}_g\right)+2\varepsilon\right] > P, \forall
   n \in \mathbb{N}_{> n_3}.
 \]
-As ... is in ... and ... is optimal we know that
+As $\supp(f^{w^{*,\tilde{\lambda}}}) \subseteq \supp(g_\xi)$ and $f^{*,\lambda}_g$ is optimal we know that
 \[
-  F^{\lambda, g}\left(f^{*,\lambda}_g\right) \leq F^{\lambda, g}\left(f^{w^{*,\tilde{\lambda}}}\right)
+  F^{\lambda, g}\left(f^{*,\lambda}_g\right) \leq F^{\lambda,
+    g}\left(f^{w^{*,\tilde{\lambda}}}\right)
 \]
 and thus get with the squeeze theorem
 \[
   \plimn F^{\lambda, g}\left(f^{w^{*,\tilde{\lambda}}}\right) = F^{\lambda, g}\left(f^{*,\lambda}_g\right).
 \]
-We can now use Lemma~\ref{lem:s7} to follow that
+With Lemma~\ref{lem:s7} it follows that
 \begin{equation}
   \plimn \norm{f^{w^{*,\tilde{\lambda}}} - f^{*,\lambda}_g}
   _{W^{1,\infty}} = 0.
-  \label{eq:main2}
+  \label{eq:main4}
 \end{equation}
-Now by using the triangle inequality with Lemma~\ref{lem:s3} and
+By using the triangle inequality with Lemma~\ref{lem:s3} and
-(\ref{eq:main2}) we get
+(\ref{eq:main4}) we get
-\begin{align*}
+\begin{multline}
-  \plimn \norm{\mathcal{RN}^{*, \tilde{\lambda}} - f_g^{*,\lambda}}
+  \plimn \norm{\mathcal{RN}^{*, \tilde{\lambda}} - f_g^{*,\lambda}}\\
-  \leq& \plimn \bigg(\norm{\mathcal{RN}^{*, \tilde{\lambda}} -
+  \leq \plimn \bigg(\norm{\mathcal{RN}^{*, \tilde{\lambda}} -
-        f_g^{w^{*,\tilde{\lambda}}}}_{W^{1,\infty}}\\
+        f_g^{w^{*,\tilde{\lambda}}}}_{W^{1,\infty}}
-  &+ \norm{f^{w^{*,\tilde{\lambda}}} - f^{*,\lambda}_g} 
+  + \norm{f^{w^{*,\tilde{\lambda}}} - f^{*,\lambda}_g} 
     _{W^{1,\infty}}\bigg) = 0
-\end{align*}
+\end{multline}
 and thus have proven Theorem~\ref{theo:main1}.
+
 We now know that randomized shallow neural networks behave similar to
 spline regression if we regularize the size of the weights during
 training.
+
 \textcite{heiss2019} further explore a connection between ridge penalized
-networks and randomized shallow neural networks which are trained
+networks and randomized shallow neural networks trained using gradient
-which are only trained for a certain amount of epoch using gradient
 descent.
-And ... that the effect of weight regularization can be achieved by
+They infer that the effect of weight regularization
-training for a certain amount of iterations this ... between adapted
+can be achieved by stopping the training of the randomized shallow
-weighted regression splines and randomized shallow neural networks
+neural network early, with the number of iterations being proportional to
-where training is stopped early.
+the tuning parameter penalizing the size of the weights.
+They use this to further conclude that for a large number of training epochs and number of
+neurons shallow neural networks trained with gradient descent are
+very close to spline interpolations. Alternatively if the training
+is stopped early, they are close to adapted weighted cubic smoothing splines.
 
 \newpage
 \subsection{Simulations}
-In the following the behaviour described in Theorem~\ref{theo:main1}
+\label{sec:rsnn_sim}
+In the following the behavior described in Theorem~\ref{theo:main1}
 is visualized in a simulated example. For this two sets of training
 data have been generated.
 \begin{itemize}
@@ -935,22 +989,28 @@ Theorem~\ref{theo:main1}
 would equate to $g(x) = \frac{\mathbb{E}[v_k^2|\xi_k = x]}{10}$. In
 order to utilize the
 smoothing spline implemented in Mathlab, $g$ has been simplified to $g
-\equiv \frac{1}{10}$ instead. For all figures $f_1^{*, \lambda}$ has
+\equiv \frac{1}{10}$ instead.
-been calculated with Matlab's ..... As ... minimizes
+
+For all figures $f_1^{*, \lambda}$ has
+been calculated with Matlab's {\sffamily{smoothingspline}}, as this minimizes
 \[
   \bar{\lambda} \sum_{i=1}^N(y_i^{train} - f(x_i^{train}))^2 + (1 -
   \bar{\lambda}) \int (f''(x))^2 dx
 \]
-the smoothing parameter used for fittment is $\bar{\lambda} =
+the smoothing parameter used for fitment is $\bar{\lambda} =
 \frac{1}{1 + \lambda}$. The parameter $\tilde{\lambda}$ for training
-the networks is chosen as defined in Theorem~\ref{theo:main1} and each
+the networks is chosen as defined in Theorem~\ref{theo:main1}.
-one is trained on the full training data for 5000 epoch using
+
+Each
+network contains 10.000 hidden nodes and is trained on the full
+training data for 100.000 epochs using 
 gradient descent. The
-results are given in Figure~\ref{fig:rs_vs_rs}, here it can be seen that in
+results are given in Figure~\ref{fig:rn_vs_rs}, where it can be seen
-the intervall of the traing data $[-\pi, \pi]$ the neural network and
+that the neural network and
-smoothing spline are nearly identical, coinciding with the proposition.
+smoothing spline are nearly identical, coinciding with the
+proposition.
 
-\input{Plots/RN_vs_RS}
+\input{Figures/RN_vs_RS}
 
 
 %%% Local Variables: