mammut commit of last two monts
Tobias Arndt
4 years ago
parent
74113d5060
commit
46031fcd5d
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import tensorflow as tf
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from tensorflow.keras.callbacks import CSVLogger
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mnist = tf.keras.datasets.mnist
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(x_train, y_train), (x_test, y_test) = mnist.load_data()
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x_train = x_train.reshape(x_train.shape[0], 28, 28, 1)
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x_test = x_test.reshape(x_test.shape[0], 28, 28, 1)
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x_train, x_test = x_train / 255.0, x_test / 255.0
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y_train = tf.keras.utils.to_categorical(y_train)
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y_test = tf.keras.utils.to_categorical(y_test)
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model = tf.keras.models.Sequential()
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model.add(tf.keras.layers.Conv2D(24,kernel_size=5,padding='same',activation='relu',
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input_shape=(28,28,1)))
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model.add(tf.keras.layers.MaxPool2D())
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model.add(tf.keras.layers.Conv2D(64,kernel_size=5,padding='same',activation='relu'))
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model.add(tf.keras.layers.MaxPool2D(padding='same'))
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model.add(tf.keras.layers.Flatten())
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model.add(tf.keras.layers.Dense(256, activation='relu'))
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model.add(tf.keras.layers.Dense(10, activation='softmax'))
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model.compile(optimizer=tf.keras.optimizers.SGD(), loss="categorical_crossentropy", metrics=["accuracy"])
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csv_logger = CSVLogger('SGD_01_b32.log')
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history = model.fit(x_train, y_train, validation_data=(x_test, y_test), batch_size = 32, epochs=20, callbacks=[csv_logger])
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@ -0,0 +1,22 @@
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import tensorflow as tf
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mnist = tf.keras.datasets.mnist
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(x_train, y_train), (x_test, y_test) = mnist.load_data()
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x_train, x_test = x_train / 255.0, x_test / 255.0
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model = tf.keras.models.Sequential([
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tf.keras.layers.Flatten(input_shape=(28, 28)),
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tf.keras.layers.Dense(128, activation='relu'),
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tf.keras.layers.Dropout(0.2),
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tf.keras.layers.Dense(10)
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])
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loss_fn = tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True)
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model.compile(optimizer='adam',
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loss=loss_fn,
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metrics=['accuracy'])
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model.fit(x_train, y_train, epochs=10)
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@ -0,0 +1,26 @@
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import tensorflow as tf
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from tensorflow.keras.callbacks import CSVLogger
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mnist = tf.keras.datasets.mnist
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(x_train, y_train), (x_test, y_test) = mnist.load_data()
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x_train = x_train.reshape(x_train.shape[0], 28, 28, 1)
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x_test = x_test.reshape(x_test.shape[0], 28, 28, 1)
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x_train, x_test = x_train / 255.0, x_test / 255.0
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y_train = tf.keras.utils.to_categorical(y_train)
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y_test = tf.keras.utils.to_categorical(y_test)
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model = tf.keras.models.Sequential()
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model.add(tf.keras.layers.Conv2D(24,kernel_size=5,padding='same',activation='relu',
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input_shape=(28,28,1)))
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model.add(tf.keras.layers.MaxPool2D())
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model.add(tf.keras.layers.Conv2D(64,kernel_size=5,padding='same',activation='relu'))
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model.add(tf.keras.layers.MaxPool2D(padding='same'))
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model.add(tf.keras.layers.Flatten())
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model.add(tf.keras.layers.Dense(256, activation='relu'))
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model.add(tf.keras.layers.Dense(10, activation='softmax'))
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model.compile(optimizer=tf.keras.optimizers.SGD(learning_rate=0.1), loss="categorical_crossentropy", metrics=["accuracy"])
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csv_logger = CSVLogger('GD_1.log')
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history = model.fit(x_train, y_train, validation_data=(x_test, y_test), batch_size = x_train.shape[0], epochs=20, callbacks=[csv_logger])
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@ -0,0 +1,22 @@
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import tensorflow as tf
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mnist = tf.keras.datasets.mnist
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(x_train, y_train), (x_test, y_test) = mnist.load_data()
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x_train, x_test = x_train / 255.0, x_test / 255.0
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model = tf.keras.models.Sequential([
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tf.keras.layers.Flatten(input_shape=(28, 28)),
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tf.keras.layers.Dense(128, activation='relu'),
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tf.keras.layers.Dropout(0.2),
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tf.keras.layers.Dense(10)
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])
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loss_fn = tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True)
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model.compile(optimizer='adam',
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loss=loss_fn,
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metrics=['accuracy'])
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model.fit(x_train, y_train, epochs=10)
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#!/bin/bash -l
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#SBATCH --job-name="Keras MNIST"
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#SBATCH --ntasks=1
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#SBATCH --ntasks-per-core=1
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#SBATCH --time=0-00:10:00
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#SBATCH --nodelist=node18
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srun python3 mnist.py
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#!/bin/bash -l
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#SBATCH --job-name="Keras MNIST"
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#SBATCH --ntasks=1
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#SBATCH --ntasks-per-core=1
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#SBATCH --time=0-00:10:00
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#SBATCH --nodelist=node18
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srun python3 mnist.py
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@ -0,0 +1,59 @@
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x=seq(0, 2*pi,0.1)
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y=sin(x)
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plot(x,y)
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x_i = x
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y_i = y
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x = x+rnorm(63,0,0.15)
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y = y+rnorm(63,0,0.15)
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plot(x, y)
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x_d = x
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y_d = y
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for(i in 5:63){
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x[i] = (sum(x_d[(i-4):i] * c(1/20,1/6,1/5,1/4,1/3)))
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}
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for(i in 5:63){
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y[i] = (sum(y_d[(i-4):i] * c(1/20,1/6,1/5,1/4,1/3)))
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}
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#x[1:4] = NA
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#y[1:4] = NA
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plot(x[-(1:4)],y[-(1:4)])
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image = image_read(path = "~/Masterarbeit/TeX/Plots/Data/klammern60_80.jpg")
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kernel <- matrix(0, ncol = 3, nrow = 3)
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kernel[c(1,3),1] = -1
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kernel[c(1,3),3] = 1
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kernel[2,1] = -2
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kernel[2,3] = 2
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kernel
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kernel <- matrix(data = c(1,4,7,4,1,4,16,26,16,4,7,26,41,26,7,4,16,26,16,4,1,4,7,4,1),
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ncol = 5, nrow=5)
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kernel = kernel/273
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n=11
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s=4
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kernel = matrix(0,nrow = n, ncol = n)
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for(i in 1:n){
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for(j in 1:n){
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kernel[i,j] = 1/(2*pi*s) * exp(-(i+j)/(2*s))
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}
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}
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image_con <- image_convolve(image, (kernel))
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image_con
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image_write(image_con, "~/Masterarbeit/TeX/Plots/Data/image_conv11.png", format="png")
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img <- readPNG("~/Masterarbeit/TeX/Plots/Data/image_conv11.png")
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out <- matrix(0, ncol = 15, nrow=20)
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for(j in 1:15){
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for(i in 1:20){
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out[i,j] = max(img[((i-1)*4 +1):((i-1)*4+4), ((j-1)*4 +1):((j-1)*4+4)])
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}
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}
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writePNG(out, target = "~/Masterarbeit/TeX/Plots/Data/image_conv12.png")
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import tensorflow as tf
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mnist = tf.keras.datasets.mnist
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(x_train, y_train), (x_test, y_test) = mnist.load_data()
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x_train, x_test = x_train / 255.0, x_test / 255.0
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model = tf.keras.models.Sequential([
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tf.keras.layers.Flatten(input_shape=(28, 28)),
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tf.keras.layers.Dense(128, activation='relu'),
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tf.keras.layers.Dropout(0.2),
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tf.keras.layers.Dense(10)
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])
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loss_fn = tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True)
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model.compile(optimizer='adam',
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loss=loss_fn,
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metrics=['accuracy'])
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model.fit(x_train, y_train, epochs=5)
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@ -0,0 +1,17 @@
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x,y
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-3.141592653589793,0.0802212608585366
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-2.722713633111154,-0.3759376368887911
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-2.303834612632515,-1.3264180339054117
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-1.8849555921538759,-0.8971334213504949
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-1.4660765716752369,-0.7724344034354425
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-1.0471975511965979,-0.9501497164520739
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-0.6283185307179586,-0.6224628757084738
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-0.2094395102393194,-0.35622668982623207
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0.2094395102393194,-0.18377660088356823
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0.6283185307179586,0.7836770998126841
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1.0471975511965974,0.5874762732054489
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1.4660765716752362,1.0696991264956026
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1.8849555921538759,1.1297065441952743
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2.3038346126325155,0.7587275382323738
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2.7227136331111543,-0.030547103790458163
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3.1415926535897922,0.044327111895927106
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@ -0,0 +1,101 @@
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x_n_5000_tl_0.1,y_n_5000_tl_0.1,x_n_5000_tl_1.0,y_n_5000_tl_1.0,x_n_5000_tl_3.0,y_n_5000_tl_3.0
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-5.0,1.794615305950707,-5.0,0.3982406589003759,-5.0,-0.4811539502118497
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-4.898989898989899,1.6984389486364895,-4.898989898989899,0.35719218031912614,-4.898989898989899,-0.48887996302459025
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-4.797979797979798,1.6014200743009022,-4.797979797979798,0.3160182633093358,-4.797979797979798,-0.4966732473871599
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-4.696969696969697,1.5040575427157106,-4.696969696969697,0.27464978660531225,-4.696969696969697,-0.5045073579233731
|
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-4.595959595959596,1.4061194142774731,-4.595959595959596,0.23293440418365288,-4.595959595959596,-0.5123589845230747
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-4.494949494949495,1.3072651356075136,-4.494949494949495,0.19100397829173557,-4.494949494949495,-0.5202738824510786
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-4.393939393939394,1.2078259346207492,-4.393939393939394,0.1488314515422353,-4.393939393939394,-0.5282281154332915
|
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-4.292929292929293,1.1079271590765678,-4.292929292929293,0.10646618526238515,-4.292929292929293,-0.536250283913464
|
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-4.191919191919192,1.0073183089866045,-4.191919191919192,0.0637511521454329,-4.191919191919192,-0.5443068679044686
|
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|
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|
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|
||||
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|
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|
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|
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-3.484848484848485,0.29091175104318323,-3.484848484848485,-0.24139115547918963,-3.484848484848485,-0.6019914655156898
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|
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|
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|
||||
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|
||||
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
-1.1616161616161618,-0.8903321986477033,-1.1616161616161618,-0.7150493507052204,-1.1616161616161618,-0.5295933185437713
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
2.1717171717171713,0.46186578627525116,2.1717171717171713,0.8081407617658106,2.1717171717171713,0.8224404974364862,2.1717171717171713,0.38441776442662906,2.1717171717171713,0.5655375705620478,2.1717171717171713,0.5300324428024472,2.1717171717171713,0.49554940844796147,2.1717171717171713,0.4101839304627971,2.1717171717171713,0.4155357725301964
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||||
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||||
2.3737373737373737,0.43609986761848685,2.3737373737373737,0.675405575107383,2.3737373737373737,0.6874741372997285,2.3737373737373737,0.39951994286687353,2.3737373737373737,0.5335539998256553,2.3737373737373737,0.49865541506871236,2.3737373737373737,0.4655571015656922,2.3737373737373737,0.4173906236056948,2.3737373737373737,0.42027249977934045
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2.474747474747475,0.4066895271847391,2.474747474747475,0.5978840366507735,2.474747474747475,0.6073682995880296,2.474747474747475,0.40692119452733155,2.474747474747475,0.5117177142842388,2.474747474747475,0.4784532511364369,2.474747474747475,0.4501649451434452,2.474747474747475,0.4206585025597512,2.474747474747475,0.4213399238172195
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||||
2.5757575757575752,0.3749622763477891,2.5757575757575752,0.5099585586540418,2.5757575757575752,0.5223271133442401,2.5757575757575752,0.41415264022012394,2.5757575757575752,0.4850415148130571,2.5757575757575752,0.4567094947730761,2.5757575757575752,0.43458555601387144,2.5757575757575752,0.42158324745022285,2.5757575757575752,0.42181632222498416
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2.6767676767676765,0.3432350255108388,2.6767676767676765,0.4205365946887392,2.6767676767676765,0.432906236858961,2.6767676767676765,0.4199131836378292,2.6767676767676765,0.45218830888592937,2.6767676767676765,0.4332394825941561,2.6767676767676765,0.41774264448225407,2.6767676767676765,0.42145613907090707,2.6767676767676765,0.4215504924390677
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|
||||
2.878787878787879,0.27978052383693824,2.878787878787879,0.23807041272811588,2.878787878787879,0.24760314946640188,2.878787878787879,0.42557960962014507,2.878787878787879,0.3802049595409251,2.878787878787879,0.383057999391408,2.878787878787879,0.3840568214190192,2.878787878787879,0.41938009129458526,2.878787878787879,0.41854626446476473
|
||||
2.9797979797979792,0.24805327299998842,2.9797979797979792,0.14646854757187647,2.9797979797979792,0.15264712621771054,2.9797979797979792,0.428104678899817,2.9797979797979792,0.3432577786602793,2.9797979797979792,0.35694448241628624,2.9797979797979792,0.367213909887402,2.9797979797979792,0.41773298189050795,2.9797979797979792,0.4163510447804036
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||||
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||||
3.1818181818181817,0.18459877132608776,3.1818181818181817,-0.03733538955626138,3.1818181818181817,-0.03779843888287274,3.1818181818181817,0.4300994055256468,3.1818181818181817,0.26873960102765904,3.1818181818181817,0.30419224859801247,3.1818181818181817,0.3335280868241671,3.1818181818181817,0.41409475876758717,3.1818181818181817,0.41152646064562604
|
||||
3.282828282828282,0.15287152048913782,3.282828282828282,-0.12920906194738088,3.282828282828282,-0.13249853932321157,3.282828282828282,0.43099899837317435,3.282828282828282,0.2314874157056526,3.282828282828282,0.27788417508140784,3.282828282828282,0.3164995410780566,3.282828282828282,0.4122620364061852,3.282828282828282,0.40912247673587887
|
||||
3.383838383838384,0.12114426965218736,3.383838383838384,-0.22108273433850145,3.383838383838384,-0.22672866959540386,3.383838383838384,0.4318917322435721,3.383838383838384,0.19424068277399548,3.383838383838384,0.25176947991950477,3.383838383838384,0.2992528546417876,3.383838383838384,0.41043205422405316,3.383838383838384,0.40674183306733336
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3.4848484848484844,0.08941701881523752,3.4848484848484844,-0.3129564067296208,3.4848484848484844,-0.3204339220693533,3.4848484848484844,0.43278446611396965,3.4848484848484844,0.15713787053146627,3.4848484848484844,0.22587592408322044,3.4848484848484844,0.2820061682055188,3.4848484848484844,0.4086021011097265,3.4848484848484844,0.4043698847877142
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3.686868686868687,0.027654025225499562,3.686868686868687,-0.49422269067564845,3.686868686868687,-0.505293720158625,3.686868686868687,0.43456993385476517,3.686868686868687,0.08338176166505175,3.686868686868687,0.17451220690194294,3.686868686868687,0.24694025624429472,3.686868686868687,0.40494401437700783,3.686868686868687,0.39972779600606
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3.787878787878787,-0.0028542247424208616,3.787878787878787,-0.5825355853286744,3.787878787878787,-0.5971159649192432,3.787878787878787,0.4354626677251625,3.787878787878787,0.04665044899957155,3.787878787878787,0.14916273839002891,3.787878787878787,0.22899283485249716,3.787878787878787,0.40312179798093106,3.787878787878787,0.39746202764807126
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||||
4.09090909090909,-0.0943789746461824,4.09090909090909,-0.8443788481067765,4.09090909090909,-0.8681309126980375,4.09090909090909,0.43814086933635565,4.09090909090909,-0.06308596529257729,4.09090909090909,0.07491765400345742,4.09090909090909,0.1750164743635475,4.09090909090909,0.39766754707663343,4.09090909090909,0.3908577509521082
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||||
4.191919191919192,-0.12488722461410334,4.191919191919192,-0.9297917533069101,4.191919191919192,-0.9573364023412008,4.191919191919192,0.4390336032067535,4.191919191919192,-0.09929539509789244,4.191919191919192,0.05074971564267564,4.191919191919192,0.1568727764842795,4.191919191919192,0.3958543351530404,4.191919191919192,0.38872432233841003
|
||||
4.292929292929292,-0.15539547458202363,4.292929292929292,-1.0140884125491687,4.292929292929292,-1.0459165238042567,4.292929292929292,0.4399263370771512,4.292929292929292,-0.1349334585206603,4.292929292929292,0.02675516616820918,4.292929292929292,0.13872907860501169,4.292929292929292,0.3940418892740997,4.292929292929292,0.38661923148208605
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||||
4.3939393939393945,-0.18590372454994458,4.3939393939393945,-1.0972974392893766,4.3939393939393945,-1.1342383379633272,4.3939393939393945,0.4408190709475487,4.3939393939393945,-0.16982980680843562,4.3939393939393945,0.002964652994963484,4.3939393939393945,0.11796054958424437,4.3939393939393945,0.3922298874756054,4.3939393939393945,0.3845302650106349
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||||
4.494949494949495,-0.216411974517865,4.494949494949495,-1.179182894055243,4.494949494949495,-1.2221355458185688,4.494949494949495,0.44032091498508585,4.494949494949495,-0.20469748939648835,4.494949494949495,-0.0206002794035424,4.494949494949495,0.09701325884395126,4.494949494949495,0.39041788567711144,4.494949494949495,0.38248614430609396
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||||
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||||
4.696969696969697,-0.2774284744537062,4.696969696969697,-1.3408190143954206,4.696969696969697,-1.395667382198044,4.696969696969697,0.4377029248030613,4.696969696969697,-0.2744311888398445,4.696969696969697,-0.06739710896332894,4.696969696969697,0.05511867736336504,4.696969696969697,0.38683625018149875,4.696969696969697,0.37848669218529357
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||||
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||||
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||||
5.0,-0.3689532243574676,5.0,-1.5820215750973248,5.0,-1.6508596672714462,5.0,0.43307940950570034,5.0,-0.37879161071248096,5.0,-0.13636462992911846,5.0,-0.007723194857514326,5.0,0.38149127984729847,5.0,0.37272620912380855
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||||
|
@ -0,0 +1,7 @@
|
||||
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
|
||||
|
@ -0,0 +1,64 @@
|
||||
,x_i,y_i,x_d,y_d,x,y
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||||
"22",2.1,0.863209366648874,2.00475777514698,0.86568986134637,1.9826265174693,0.924298444442167
|
||||
"23",2.2,0.80849640381959,2.40773948766051,0.667018023975934,2.15807575978944,0.826761739840873
|
||||
"24",2.3,0.74570521217672,2.14892522112975,0.872704236332415,2.17485332420928,0.839957045849706
|
||||
"25",2.4,0.675463180551151,2.41696701330131,0.253955021611832,2.26412064248401,0.631186439537074
|
||||
"26",2.5,0.598472144103957,2.4087686184711,0.49450592290142,2.33847747374241,0.557319074033222
|
||||
"27",2.6,0.515501371821464,2.55312145187913,0.343944677655963,2.4151672191424,0.467867318187242
|
||||
"28",2.7,0.42737988023383,2.6585492172135,0.528990826178838,2.51649125567521,0.447178678139147
|
||||
"29",2.8,0.334988150155905,2.86281283456189,0.311400289332401,2.65184232661008,0.399952143417531
|
||||
"30",2.9,0.239249329213982,2.74379162744449,0.501282616227342,2.70796893413474,0.432791852065713
|
||||
"31",3,0.141120008059867,2.95951338295806,0.241385538727577,2.81576254355573,0.373424929745113
|
||||
"32",3.1,0.0415806624332905,2.87268165585702,0.0764217470113609,2.85626015646841,0.264426413128825
|
||||
"33",3.2,-0.0583741434275801,3.29898326143096,-0.272500742891131,3.0101734240017,0.0756660807058224
|
||||
"34",3.3,-0.157745694143249,3.64473302259565,-0.24394459655987,3.24463496592626,-0.0688606479078372
|
||||
"35",3.4,-0.255541102026832,3.46698556586598,-0.184272732807665,3.35339770834784,-0.15210430721581
|
||||
"36",3.5,-0.35078322768962,3.67208160089566,-0.119933071489115,3.51318482264886,-0.176430496141549
|
||||
"37",3.6,-0.442520443294852,3.73738883546162,-0.486197268315415,3.62961845872181,-0.283186040443485
|
||||
"38",3.7,-0.529836140908493,3.77209072631297,-0.70275845349803,3.68619468325631,-0.422698101171958
|
||||
"39",3.8,-0.611857890942719,3.66424718733509,-0.482410535792735,3.69727905622484,-0.462935060857071
|
||||
"40",3.9,-0.687766159183974,3.72257849834575,-0.58477261395861,3.71784166083333,-0.543108060927685
|
||||
"41",4,-0.756802495307928,3.85906293918747,-0.703015362823377,3.76539960460785,-0.618449987254768
|
||||
"42",4.1,-0.818277111064411,4.0131961543859,-0.900410257326814,3.84632588679948,-0.708384794580195
|
||||
"43",4.2,-0.871575772413588,4.0263131749378,-0.906044808231391,3.92085812717095,-0.789303202089581
|
||||
"44",4.3,-0.916165936749455,4.77220075671212,-0.530827398816399,4.22925719163087,-0.729943577630504
|
||||
"45",4.4,-0.951602073889516,4.4795636311648,-1.26672674728111,4.35331987391088,-0.921377204806384
|
||||
"46",4.5,-0.977530117665097,4.5088210845027,-0.886168448505782,4.44898342417679,-0.914264630323723
|
||||
"47",4.6,-0.993691003633465,4.70645816063034,-1.1082213336257,4.58861983576766,-0.97806804633887
|
||||
"48",4.7,-0.999923257564101,4.48408312008838,-0.98352521226689,4.55827710678399,-1.01979325501755
|
||||
"49",4.8,-0.996164608835841,4.97817348334347,-1.03043977928678,4.69715193557134,-1.02203657500247
|
||||
"50",4.9,-0.982452612624332,5.09171179984929,-0.948912592308037,4.8484480091335,-0.999631162740658
|
||||
"51",5,-0.958924274663138,4.87710566000798,-0.825224506141761,4.87693462801326,-0.937722874707385
|
||||
"52",5.1,-0.925814682327732,5.04139294635392,-0.718936957124138,4.97198282698482,-0.856650521199568
|
||||
"53",5.2,-0.883454655720153,4.94893136398377,-0.992753696742329,4.98294046406006,-0.885371127105841
|
||||
"54",5.3,-0.832267442223901,5.38128555915899,-0.717434652733088,5.10670981664685,-0.816103747160468
|
||||
"55",5.4,-0.772764487555987,5.46192736637355,-0.724060934669406,5.2398375587704,-0.780347098915984
|
||||
"56",5.5,-0.705540325570392,5.30834840605735,-0.721772537926303,5.28807996342596,-0.766498807502665
|
||||
"57",5.6,-0.631266637872321,5.53199687756185,-0.583133415115471,5.40779902870202,-0.688843253413245
|
||||
"58",5.7,-0.550685542597638,5.9238064899769,-0.541063721566544,5.59865656961444,-0.627040990301198
|
||||
"59",5.8,-0.464602179413757,5.8067999294844,-0.43156566524513,5.68077207716296,-0.552246304884294
|
||||
"60",5.9,-0.373876664830236,5.93089453525347,-0.604056792592816,5.80084302534748,-0.550733954237757
|
||||
"61",6,-0.279415498198926,6.02965160059402,-0.234452930170458,5.91786841211583,-0.434812265604247
|
||||
"62",6.1,-0.182162504272095,5.88697419016579,-0.135764844759742,5.91990685000071,-0.323660336266941
|
||||
"63",6.2,-0.0830894028174964,5.91445270773648,-0.0073552500992853,5.92798052258888,-0.205537962618181
|
||||
|
@ -0,0 +1,138 @@
|
||||
\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{% 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.
|
||||
}
|
||||
\end{figure}
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master:
|
||||
%%% End:
|
||||
@ -0,0 +1,91 @@
|
||||
\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 = 0.65\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 = 0.65\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}]
|
||||
\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{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}
|
||||
\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}
|
||||
1&1&1&1&&1&1&1&1
|
||||
\end{tabu}
|
||||
\caption{Performace metrics of the networks trained in
|
||||
Figure~\ref{ref:sdg_vs_gd} after 20 training epochs.}
|
||||
\end{table}
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "../main"
|
||||
%%% End:
|
||||
@ -0,0 +1,71 @@
|
||||
\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:
|
||||
@ -0,0 +1,146 @@
|
||||
\documentclass{article}
|
||||
\usepackage{pgfplots}
|
||||
\usepackage{filecontents}
|
||||
\usepackage{subcaption}
|
||||
\usepackage{adjustbox}
|
||||
\usepackage{xcolor}
|
||||
\usepackage{tabu}
|
||||
\usepackage{graphicx}
|
||||
\usetikzlibrary{calc, 3d}
|
||||
|
||||
\begin{document}
|
||||
\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}[b]{\textwidth}
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[tick style = {draw = none}, width = \textwidth,
|
||||
height = 0.7\textwidth,
|
||||
xtick = {1, 3, 5,7,9,11,13,15,17,19},
|
||||
xticklabels = {$2$, $4$, $6$, $8$,
|
||||
$10$,$12$,$14$,$16$,$18$,$20$},
|
||||
xlabel = {epoch}, ylabel = {Classification Accuracy}]
|
||||
\addplot table
|
||||
[x=epoch, y=val_accuracy, col sep=comma] {Data/GD_01.log};
|
||||
\addplot table
|
||||
[x=epoch, y=val_accuracy, col sep=comma] {Data/GD_05.log};
|
||||
\addplot table
|
||||
[x=epoch, y=val_accuracy, col sep=comma] {Data/GD_1.log};
|
||||
\addplot table
|
||||
[x=epoch, y=val_accuracy, col sep=comma]
|
||||
{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 = 0.7\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 = {epoch}, ylabel = {Error Measure}]
|
||||
\addplot table
|
||||
[x=epoch, y=val_loss, col sep=comma] {Data/GD_01.log};
|
||||
\addplot table
|
||||
[x=epoch, y=val_loss, col sep=comma] {Data/GD_05.log};
|
||||
\addplot table
|
||||
[x=epoch, y=val_loss, col sep=comma] {Data/GD_1.log};
|
||||
\addplot table
|
||||
[x=epoch, y=val_loss, col sep=comma] {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}
|
||||
\\~\\
|
||||
\begin{subfigure}[b]{1.0\linewidth}
|
||||
\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}
|
||||
1&1&1&1&&1&1&1&1
|
||||
\end{tabu}
|
||||
\caption{Performace metrics after 20 epochs}
|
||||
\end{subfigure}
|
||||
\caption{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$_{\text{rate}}$). For
|
||||
stochastic gradient descend a batch size of 32 and learning rate
|
||||
of 0.01 is used (SDG$_{0.01}$)}
|
||||
\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}
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: t
|
||||
%%% End:
|
||||
@ -0,0 +1,64 @@
|
||||
"","x_i","y_i","x_d","y_d","x","y"
|
||||
"1",0,0,0.0815633019993375,0.095134925029757,0.0815633019993375,0.095134925029757
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||||
"2",0.1,0.0998334166468282,-0.137539012603596,0.503920419784276,-0.137539012603596,0.503920419784276
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||||
"3",0.2,0.198669330795061,0.219868163218743,0.32022289024623,0.219868163218743,0.32022289024623
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||||
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||||
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"11",1,0.841470984807897,0.895913227983752,0.658288213778898,0.650886140047209,0.577618711891772
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||||
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||||
"20",1.9,0.946300087687414,1.835014641556,0.830477479204284,1.45242210409837,0.889715842048808
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||||
"21",2,0.909297426825682,1.98589997236352,0.887302138185342,1.56569111721857,0.901843632635883
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||||
"22",2.1,0.863209366648874,2.31436634488224,0.890096618924313,1.73810390755555,0.899632162941341
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||||
"23",2.2,0.80849640381959,2.14663445612581,0.697012453130415,1.77071083163663,0.831732978616874
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||||
"24",2.3,0.74570521217672,2.17162372560288,0.614243640399509,1.84774268936257,0.787400621584077
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||||
"25",2.4,0.675463180551151,2.2488591417345,0.447664288915269,1.93366609303299,0.707449056213168
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||||
"26",2.5,0.598472144103957,2.56271588872389,0.553368843490625,2.08922735802261,0.702402440783529
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||||
"27",2.6,0.515501371821464,2.60986205081511,0.503762006272682,2.17548673152621,0.657831176057599
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||||
"28",2.7,0.42737988023383,2.47840649766003,0.215060732402894,2.20251747034638,0.533903400086802
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||||
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||||
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||||
"31",3,0.141120008059867,2.86247369846558,0.0960140633436418,2.55274767368554,0.371740588261606
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||||
"32",3.1,0.0415806624332905,2.79458017090243,-0.187923650913249,2.59422388058738,0.234694070506915
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||||
"33",3.2,-0.0583741434275801,3.6498183243501,-0.186738431858275,2.9216851043241,0.173308072295566
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||||
"34",3.3,-0.157745694143249,3.19424275971809,-0.221908035274934,2.86681135711315,0.101325637659584
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
"48",4.7,-0.999923257564101,5.00475403211142,-0.922605880059771,4.41432228408005,-0.770625346502085
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||||
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||||
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||||
"51",5,-0.958924274663138,4.92057344952522,-0.872931430146499,4.61418118503201,-0.836318916150308
|
||||
"52",5.1,-0.925814682327732,5.37277893732831,-0.91444926304078,4.81555148166217,-0.864686555983682
|
||||
"53",5.2,-0.883454655720153,5.19524942845082,-1.41169784739596,4.84152902094499,-1.03768305406186
|
||||
"54",5.3,-0.832267442223901,5.4432222181271,-0.726481337519931,4.98565483155961,-0.856094353978009
|
||||
"55",5.4,-0.772764487555987,4.98285013865449,-0.692803346852181,4.90897053115903,-0.838425020062396
|
||||
"56",5.5,-0.705540325570392,5.33298025214155,-0.343702005257262,5.0497327607228,-0.711573964373115
|
||||
"57",5.6,-0.631266637872321,5.49935694796791,-0.828968673188174,5.15036520204232,-0.816467931201244
|
||||
"58",5.7,-0.550685542597638,5.69204187550805,-0.481580461165225,5.26232964126231,-0.689500817105975
|
||||
"59",5.8,-0.464602179413757,5.84391772412888,-0.20453899468884,5.38069867877875,-0.564365367144995
|
||||
"60",5.9,-0.373876664830236,5.48166674139637,-0.597796931577294,5.3357436834558,-0.649913835818738
|
||||
"61",6,-0.279415498198926,5.77474590863769,-0.280234463056808,5.46956415981143,-0.524503219480344
|
||||
"62",6.1,-0.182162504272095,6.36764321572312,-0.0996286988755344,5.7169871104113,-0.422854073705143
|
||||
"63",6.2,-0.0830894028174964,6.46175133910451,-0.025702847911482,5.83540227044819,-0.355719019286555
|
||||
|
@ -0,0 +1,45 @@
|
||||
\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{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:
|
||||
@ -0,0 +1,5 @@
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "../main"
|
||||
%%% End:
|
||||
@ -0,0 +1,33 @@
|
||||
|
||||
\newpage
|
||||
\begin{appendices}
|
||||
\section{Proofs for sone Lemmata in ...}
|
||||
In the following there will be proofs for some important Lemmata in
|
||||
Section~\ref{sec:theo38}. Further proofs not discussed here can be
|
||||
found in \textcite{heiss2019}
|
||||
\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}
|
||||
\end{appendices}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "main"
|
||||
%%% End:
|
||||
|
||||
@ -0,0 +1,329 @@
|
||||
\section{Application of NN to higher complexity Problems}
|
||||
|
||||
As neural networks are applied to problems of higher complexity often
|
||||
resulting in higher dimensionality of the input the amount of
|
||||
parameters in the network rises drastically. For example a network
|
||||
with ...
|
||||
A way to combat the
|
||||
|
||||
\subsection{Convolution}
|
||||
|
||||
Convolution is a mathematical operation, where the product of two
|
||||
functions is integrated after one has been reversed and shifted.
|
||||
|
||||
\[
|
||||
(f * g) (t) \coloneqq \int_{-\infty}^{\infty} f(t-s) g(s) ds.
|
||||
\]
|
||||
|
||||
This operation can be described as a filter-function $g$ being applied
|
||||
to $f$,
|
||||
as values $f(t)$ are being replaced by an average of values of $f$
|
||||
weighted by $g$ in position $t$.
|
||||
The convolution operation allows plentiful manipulation of data, with
|
||||
a simple example being smoothing of real-time data. Consider a sensor
|
||||
measuring the location of an object (e.g. via GPS). We expect the
|
||||
output of the sensor to be noisy as a result of a number of factors
|
||||
that will impact the accuracy. In order to get a better estimate of
|
||||
the actual location we want to smooth
|
||||
the data to reduce the noise. Using convolution for this task, we
|
||||
can control the significance we want to give each data-point. We
|
||||
might want to give a larger weight to more recent measurements than
|
||||
older ones. If we assume these measurements are taken on a discrete
|
||||
timescale, we need to introduce discrete convolution first. Let $f$,
|
||||
$g: \mathbb{Z} \to \mathbb{R}$ then
|
||||
|
||||
\[
|
||||
(f * g)(t) = \sum_{i \in \mathbb{Z}} f(t-i) g(i).
|
||||
\]
|
||||
Applying this on the data with the filter $g$ chosen accordingly we
|
||||
are
|
||||
able to improve the accuracy, which can be seen in
|
||||
Figure~\ref{fig:sin_conv}.
|
||||
\input{Plots/sin_conv.tex}
|
||||
This form of discrete convolution can also be applied to functions
|
||||
with inputs of higher dimensionality. Let $f$, $g: \mathbb{Z}^d \to
|
||||
\mathbb{R}$ then
|
||||
|
||||
\[
|
||||
(f * g)(x_1, \dots, x_d) = \sum_{i \in \mathbb{Z}^d} f(x_1 - i_1,
|
||||
\dots, x_d - i_d) g(i_1, \dots, i_d)
|
||||
\]
|
||||
This will prove to be a useful framework for image manipulation but
|
||||
in order to apply convolution to images we need to discuss
|
||||
representation of image data first. Most often images are represented
|
||||
by each pixel being a mixture of base colors these base colors define
|
||||
the color-space in which the image is encoded. Often used are
|
||||
color-spaces RGB (red,
|
||||
blue, green) or CMYK (cyan, magenta, yellow, black). An example of an
|
||||
image split in its red, green and blue channel is given in
|
||||
Figure~\ref{fig:rgb} Using this
|
||||
encoding of the image we can define a corresponding discrete function
|
||||
describing the image, by mapping the coordinates $(x,y)$ of an pixel
|
||||
and the
|
||||
channel (color) $c$ to the respective value $v$
|
||||
|
||||
\begin{align}
|
||||
\begin{split}
|
||||
I: \mathbb{N}^3 & \to \mathbb{R}, \\
|
||||
(x,y,c) & \mapsto v.
|
||||
\end{split}
|
||||
\label{def:I}
|
||||
\end{align}
|
||||
|
||||
\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]{Plots/Data/klammern_r.jpg}};
|
||||
\node[canvas is xy plane at z=2, transform shape] at (0,-0.2)
|
||||
{\includegraphics[width=5cm]{Plots/Data/klammern_g.jpg}};
|
||||
\node[canvas is xy plane at z=4, transform shape] at (0,-0.4)
|
||||
{\includegraphics[width=5cm]{Plots/Data/klammern_b.jpg}};
|
||||
\node[canvas is xy plane at z=4, transform shape] at (-8,-0.2)
|
||||
{\includegraphics[width=5.3cm]{Plots/Data/klammern_rgb.jpg}};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
\end{adjustbox}
|
||||
\caption{On the right the red, green and blue chances of the picture
|
||||
are displayed. In order to better visualize the color channels 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.}
|
||||
\label{fig:rgb}
|
||||
\end{figure}
|
||||
|
||||
With this representation of an image as a function, we can apply
|
||||
filters to the image using convolution for multidimensional functions
|
||||
as described above. In order to simplify the notation we will write
|
||||
the function $I$ given in (\ref{def:I}) as well as the filter-function $g$
|
||||
as a tensor from now on, resulting in the modified notation of
|
||||
convolution
|
||||
|
||||
\[
|
||||
(I * g)_{x,y,c} = \sum_{i,j,l \in \mathbb{Z}} I_{x-i,y-j,c-l} g_{i,j,l}.
|
||||
\]
|
||||
|
||||
Simple examples for image manipulation using
|
||||
convolution are smoothing operations or
|
||||
rudimentary detection of edges in grayscale images, meaning they only
|
||||
have one channel. A popular filter for smoothing images
|
||||
is the Gauss-filter which for a given $\sigma \in \mathbb{R}_+$ and
|
||||
size $s \in \mathbb{N}$ is
|
||||
defined as
|
||||
\[
|
||||
G_{x,y} = \frac{1}{2 \pi \sigma^2} e^{-\frac{x^2 + y^2}{2
|
||||
\sigma^2}}, ~ x,y \in \left\{1,\dots,s\right\}.
|
||||
\]
|
||||
|
||||
For edge detection purposes the Sobel operator is widespread. Here two
|
||||
filters are applied to the
|
||||
image $I$ and then combined. Edges in the $x$ direction are detected
|
||||
by convolution with
|
||||
\[
|
||||
G =\left[
|
||||
\begin{matrix}
|
||||
-1 & 0 & 1 \\
|
||||
-2 & 0 & 2 \\
|
||||
-1 & 0 & 1
|
||||
\end{matrix}\right],
|
||||
\]
|
||||
and edges is the y direction by convolution with $G^T$, the final
|
||||
output is given by
|
||||
|
||||
\[
|
||||
O = \sqrt{(I * G)^2 + (I*G^T)^2}
|
||||
\]
|
||||
where $\sqrt{\cdot}$ and $\cdot^2$ are applied component
|
||||
wise. Examples of convolution with both kernels are given in Figure~\ref{fig:img_conv}.
|
||||
|
||||
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{subfigure}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{Plots/Data/klammern.jpg}
|
||||
\caption{Original Picture}
|
||||
\label{subf:OrigPicGS}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{Plots/Data/image_conv9.png}
|
||||
\caption{Gaussian Blur $\sigma^2 = 1$}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{Plots/Data/image_conv10.png}
|
||||
\caption{Gaussian Blur $\sigma^2 = 4$}
|
||||
\end{subfigure}\\
|
||||
\begin{subfigure}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{Plots/Data/image_conv4.png}
|
||||
\caption{Sobel Operator $x$-direction}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{Plots/Data/image_conv5.png}
|
||||
\caption{Sobel Operator $y$-direction}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{Plots/Data/image_conv6.png}
|
||||
\caption{Sobel Operator combined}
|
||||
\end{subfigure}
|
||||
% \begin{subfigure}{0.24\textwidth}
|
||||
% \centering
|
||||
% \includegraphics[width=\textwidth]{Plots/Data/image_conv6.png}
|
||||
% \caption{test}
|
||||
% \end{subfigure}
|
||||
\caption{Convolution of original greyscale Image (a) with different
|
||||
kernels. In (b) and (c) Gaussian kernels of size 11 and stated
|
||||
$\sigma^2$ are used. In (d) - (f) the above defined Sobel Operator
|
||||
kernels are used.}
|
||||
\label{fig:img_conv}
|
||||
\end{figure}
|
||||
\clearpage
|
||||
\newpage
|
||||
\subsection{Convolutional NN}
|
||||
|
||||
In conventional neural networks as described in chapter ... all layers
|
||||
are fully connected, meaning each output node in a layer is influenced
|
||||
by all inputs. For $i$ inputs and $o$ output nodes this results in $i
|
||||
+ 1$ variables at each node (weights and bias) and a total $o(i + 1)$
|
||||
variables. For large inputs like image data the amount of variables
|
||||
that have to be trained in order to fit the model can get excessive
|
||||
and hinder the ability to train the model due to memory and
|
||||
computational restrictions. By using convolution we can extract
|
||||
meaningful information such as edges in an image with a kernel of a
|
||||
small size $k$ in the tens or hundreds independent of the size of the
|
||||
original image. Thus for a large image $k \cdot i$ can be several
|
||||
orders of magnitude smaller than $o\cdot i$ .
|
||||
|
||||
|
||||
As seen convolution lends itself for image manipulation. In this
|
||||
chapter we will explore how we can incorporate convolution in neural
|
||||
networks, and how that might be beneficial.
|
||||
|
||||
Convolutional Neural Networks as described by ... are made up of
|
||||
convolutional layers, pooling layers, and fully connected ones. The
|
||||
fully connected layers are layers in which each input node is
|
||||
connected to each output node which is the structure introduced in
|
||||
chapter ...
|
||||
|
||||
In a convolutional layer instead of combining all input nodes for each
|
||||
output node, the input nodes are interpreted as a tensor on which a
|
||||
kernel is applied via convolution, resulting in the output. Most often
|
||||
multiple kernels are used, resulting in multiple output tensors. These
|
||||
kernels are the variables, which can be altered in order to fit the
|
||||
model to the data. Using multiple kernels it is possible to extract
|
||||
different features from the image (e.g. edges -> sobel). As this
|
||||
increases dimensionality even further which is undesirable as it
|
||||
increases the amount of variables in later layers of the model, a convolutional layer
|
||||
is often followed by a pooling one. In a pooling layer the input is
|
||||
reduced in size by extracting a single value from a
|
||||
neighborhood \todo{moving...}... . The resulting output size is dependent on
|
||||
the offset of the neighborhoods used. Popular is max-pooling where the
|
||||
largest value in a neighborhood is used or.
|
||||
|
||||
This construct allows for extraction of features from the input while
|
||||
using far less input variables.
|
||||
|
||||
... \todo{Beispiel mit kleinem Bild, am besten das von oben}
|
||||
|
||||
\subsubsection{Parallels to the Visual Cortex in Mammals}
|
||||
|
||||
The choice of convolution for image classification tasks is not
|
||||
arbitrary. ... auge... bla bla
|
||||
|
||||
|
||||
\subsection{Limitations of the Gradient Descent Algorithm}
|
||||
|
||||
-Hyperparameter guesswork
|
||||
-Problems navigating valleys -> momentum
|
||||
-Different scale of gradients for vars in different layers -> ADAdelta
|
||||
|
||||
\subsection{Stochastic Training Algorithms}
|
||||
|
||||
For many applications in which neural networks are used such as
|
||||
image classification or segmentation, large training data sets become
|
||||
detrimental to capture the nuances of the
|
||||
data. However as training sets get larger the memory requirement
|
||||
during training grows with it.
|
||||
In order to update the weights with the gradient descent algorithm
|
||||
derivatives of the network with respect for each
|
||||
variable need to be calculated for all data points in order to get the
|
||||
full gradient of the error of the network.
|
||||
Thus the amount of memory and computing power available limits the
|
||||
size of the training data that can be efficiently used in fitting the
|
||||
network. A class of algorithms that augment the gradient descent
|
||||
algorithm in order to lessen this problem are stochastic gradient
|
||||
descent algorithms. Here the premise is that instead of using the whole
|
||||
dataset a (different) subset of data is chosen to
|
||||
compute the gradient in each iteration.
|
||||
The amount of iterations until each data point has been considered in
|
||||
updating the parameters is commonly called a ``epoch''.
|
||||
This reduces the amount of memory and computing power required for
|
||||
each iteration. This allows for use of very large training
|
||||
sets. Additionally the noise introduced on the gradient can improve
|
||||
the accuracy of the fit as stochastic gradient descent algorithms are
|
||||
less likely to get stuck on local extrema.
|
||||
|
||||
\input{Plots/SGD_vs_GD.tex}
|
||||
|
||||
Another benefit of using subsets even if enough memory is available to
|
||||
use the whole dataset is that depending on the size of the subsets the
|
||||
gradient can be calculated far quicker which allows to make more steps
|
||||
in the same time. If the approximated gradient is close enough to the
|
||||
``real'' one this can drastically cut down the time required for
|
||||
training the model.
|
||||
|
||||
\begin{itemize}
|
||||
\item ADAM
|
||||
\item momentum
|
||||
\item ADADETLA \textcite{ADADELTA}
|
||||
|
||||
|
||||
\end{itemize}
|
||||
|
||||
|
||||
|
||||
% \subsubsubsection{Stochastic Gradient Descent}
|
||||
|
||||
\subsection{Combating Overfitting}
|
||||
|
||||
% As in many machine learning applications if the model is overfit in
|
||||
% the data it can drastically reduce the generalization of the model. In
|
||||
% many machine learning approaches noise introduced in the learning
|
||||
% algorithm in order to reduce overfitting. This results in a higher
|
||||
% bias of the model but the trade off of lower variance of the model is
|
||||
% beneficial in many cases. For example the regression tree model
|
||||
% ... benefits greatly from restricting the training algorithm on
|
||||
% randomly selected features in every iteration and then averaging many
|
||||
% such trained trees inserted of just using a single one. \todo{noch
|
||||
% nicht sicher ob ich das nehmen will} For neural networks similar
|
||||
% strategies exist. A popular approach in regularizing convolutional neural network
|
||||
% is \textit{dropout} which has been first introduced in
|
||||
% \cite{Dropout}
|
||||
|
||||
Similarly to shallow networks overfitting still can impact the quality of
|
||||
convolutional neural networks. A popular way to combat this problem is
|
||||
by introducing noise into the training of the model. This is a
|
||||
successful strategy for ofter models as well, the a conglomerate of
|
||||
descision trees grown on bootstrapped trainig samples benefit greatly
|
||||
of randomizing the features available to use in each training
|
||||
iteration (Hastie, Bachelorarbeit??). The way noise is introduced into
|
||||
the model is by deactivating certain nodes (setting the output of the
|
||||
node to 0) in the fully connected layers of the convolutional neural
|
||||
networks. The nodes are chosen at random and change in every
|
||||
iteration, this practice is called Dropout and was introduced by
|
||||
\textcite{Dropout}.
|
||||
|
||||
\todo{Vergleich verschiedene dropout größen auf MNSIT o.ä.}
|
||||
|
||||
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "main"
|
||||
%%% End:
|
||||
@ -0,0 +1,10 @@
|
||||
\section{Introduction}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: "main"
|
||||
%%% End:
|
||||
@ -0,0 +1,25 @@
|
||||
\documentclass{article}
|
||||
\usepackage{pgfplots}
|
||||
\usepackage{filecontents}
|
||||
|
||||
|
||||
\begin{document}
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}
|
||||
\addplot+ [mark options={scale = 0.7}, mark = o] table [x=x,y=y, col sep = comma,
|
||||
only marks] {data_sin_d_t.csv};
|
||||
\addplot [black] table [x=x, y=y, col sep=comma, mark=none, color = 'black'] {matlab_sin_d_01.csv};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}
|
||||
\addplot table [x=x, y=y, col sep=comma, only marks] {data_sin_d_t.csv};
|
||||
\addplot table [black, x=x, y=y, col sep=comma, mark=none, color = 'black'] {matlab_sin_d_01.csv};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\end{document}
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: t
|
||||
%%% End:
|
||||
Loading…
Reference in New Issue