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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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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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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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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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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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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File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
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Load Diff
File diff suppressed because it is too large
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Load Diff
@ -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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|
-4.090909090909091,0.9064682044248323,-4.090909090909091,0.020965778107027506,-4.090909090909091,-0.5524049731989601
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-3.9898989898989896,0.805095064694333,-3.9898989898989896,-0.02200882631350869,-3.9898989898989896,-0.5605562335116703
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-3.888888888888889,0.7032463151196859,-3.888888888888889,-0.06548644224881082,-3.888888888888889,-0.5687680272492979
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-3.787878787878788,0.6007843964001714,-3.787878787878788,-0.10914135786185346,-3.787878787878788,-0.5770307386196555
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|
-3.686868686868687,0.4978572358270573,-3.686868686868687,-0.15292201515712506,-3.686868686868687,-0.5853131654059709
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|
-3.5858585858585856,0.39465522349482535,-3.5858585858585856,-0.19694472820060063,-3.5858585858585856,-0.593636189078738
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-3.484848484848485,0.29091175104318323,-3.484848484848485,-0.24139115547918963,-3.484848484848485,-0.6019914655156898
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|
-3.383838383838384,0.1868284306918275,-3.383838383838384,-0.28617728400089926,-3.383838383838384,-0.6103823599700093
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|
-3.282828282828283,0.0817944681090728,-3.282828282828283,-0.33119615483860937,-3.282828282828283,-0.6188088888423856
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|
-3.1818181818181817,-0.023670753859105602,-3.1818181818181817,-0.3764480559542342,-3.1818181818181817,-0.6272515625106694
|
||||||
|
-3.080808080808081,-0.1299349094939808,-3.080808080808081,-0.42202262988259276,-3.080808080808081,-0.6357221532633648
|
||||||
|
-2.9797979797979797,-0.2360705715363967,-2.9797979797979797,-0.467584017465408,-2.9797979797979797,-0.6440454918766952
|
||||||
|
-2.878787878787879,-0.34125419448980393,-2.878787878787879,-0.5126079284225549,-2.878787878787879,-0.65203614244987
|
||||||
|
-2.7777777777777777,-0.443504036212927,-2.7777777777777777,-0.5569084060463078,-2.7777777777777777,-0.6594896031012563
|
||||||
|
-2.676767676767677,-0.5411482698953787,-2.676767676767677,-0.6002683604183435,-2.676767676767677,-0.6661215834468585
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|
-2.5757575757575757,-0.6363089624800997,-2.5757575757575757,-0.6396725440402657,-2.5757575757575757,-0.6715398637661353
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|
-2.474747474747475,-0.725241414197713,-2.474747474747475,-0.6753456416248385,-2.474747474747475,-0.674565545688341
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|
-2.3737373737373737,-0.8010191169999671,-2.3737373737373737,-0.7066964605752718,-2.3737373737373737,-0.6765307025278043
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|
-2.272727272727273,-0.8626605255789729,-2.272727272727273,-0.7348121862404637,-2.272727272727273,-0.6766187567521622
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|
-2.1717171717171717,-0.911435840482434,-2.1717171717171717,-0.7592451818361001,-2.1717171717171717,-0.6747200340049733
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|
-2.070707070707071,-0.9518228090965052,-2.070707070707071,-0.7755022118880182,-2.070707070707071,-0.6711535886166349
|
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|
-1.9696969696969697,-0.9791642715505677,-1.9696969696969697,-0.7889078495544403,-1.9696969696969697,-0.6653309071624213
|
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|
-1.868686868686869,-0.9959505678135467,-1.868686868686869,-0.7978655263590677,-1.868686868686869,-0.6574048849245917
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|
-1.7676767676767677,-1.0042572630521163,-1.7676767676767677,-0.8024926242661324,-1.7676767676767677,-0.6465258005011485
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|
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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
|
||||||
|
@ -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
|
||||||
|
"1",0,0,-0.251688505259414,-0.109203329280437,-0.0838961684198045,-0.0364011097601456
|
||||||
|
"2",0.1,0.0998334166468282,0.216143831477992,0.112557051753147,0.00912581751114394,0.0102181849309398
|
||||||
|
"3",0.2,0.198669330795061,0.351879533708722,0.52138915851383,0.120991434720523,0.180094983253476
|
||||||
|
"4",0.3,0.29552020666134,-0.0169121548298757,0.0870956013269369,0.0836131805695847,0.163690012207993
|
||||||
|
"5",0.4,0.389418342308651,0.278503661037003,0.464752686490904,0.182421968363305,0.294268636359638
|
||||||
|
"6",0.5,0.479425538604203,0.241783494554983,0.521480762031938,0.216291763003623,0.399960258238722
|
||||||
|
"7",0.6,0.564642473395035,0.67288177436767,0.617435509386938,0.35521581484916,0.469717955748659
|
||||||
|
"8",0.7,0.644217687237691,0.692239292735764,0.395366561077235,0.492895242512842,0.472257444593698
|
||||||
|
"9",0.8,0.717356090899523,0.779946606884677,0.830045203984444,0.621840812496715,0.609161571471379
|
||||||
|
"10",0.9,0.783326909627483,0.796987424421658,0.801263132114778,0.723333122197902,0.682652280249237
|
||||||
|
"11",1,0.841470984807897,1.06821012817873,0.869642838589798,0.860323524382936,0.752971972337735
|
||||||
|
"12",1.1,0.891207360061435,1.50128637982775,0.899079529605641,1.09148187598916,0.835465707990221
|
||||||
|
"13",1.2,0.932039085967226,1.1194263347154,0.906626360727432,1.13393429991233,0.875953352580199
|
||||||
|
"14",1.3,0.963558185417193,1.24675170552299,1.07848030956084,1.2135821540696,0.950969562327306
|
||||||
|
"15",1.4,0.98544972998846,1.32784804980202,0.76685418220594,1.2818141129714,0.899892140468108
|
||||||
|
"16",1.5,0.997494986604054,1.23565831982523,1.07310713979952,1.2548338349408,0.961170357331681
|
||||||
|
"17",1.6,0.999573603041505,1.90289281875567,0.88003153305018,1.47254506382487,0.94006950203764
|
||||||
|
"18",1.7,0.991664810452469,1.68871194985252,1.01829329437246,1.56940444551462,0.955793455192302
|
||||||
|
"19",1.8,0.973847630878195,1.72179983981017,1.02268013575533,1.64902528694529,0.988666907865147
|
||||||
|
"20",1.9,0.946300087687414,2.0758716236832,0.805032560816536,1.83908127693465,0.928000158917177
|
||||||
|
"21",2,0.909297426825682,2.11118945422405,1.0134691646089,1.94365432453739,0.957334347939419
|
||||||
|
"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
|
||||||
|
"2",0.1,0.0998334166468282,-0.137539012603596,0.503920419784276,-0.137539012603596,0.503920419784276
|
||||||
|
"3",0.2,0.198669330795061,0.219868163218743,0.32022289024623,0.219868163218743,0.32022289024623
|
||||||
|
"4",0.3,0.29552020666134,0.378332723534869,0.474906286765401,0.378332723534869,0.474906286765401
|
||||||
|
"5",0.4,0.389418342308651,0.286034335293811,0.422891394375764,0.215056588291437,0.412478430748051
|
||||||
|
"6",0.5,0.479425538604203,-0.109871707385461,0.229661026779107,0.122574532557623,0.353221043330047
|
||||||
|
"7",0.6,0.564642473395035,0.91036951450573,0.56079130435097,0.451160317716352,0.452893574072324
|
||||||
|
"8",0.7,0.644217687237691,0.899001194675409,0.714355793051917,0.491731451724399,0.514477919331008
|
||||||
|
"9",0.8,0.717356090899523,0.733791390723896,0.694085383523086,0.488943974889845,0.530054084580656
|
||||||
|
"10",0.9,0.783326909627483,0.893642943873427,0.739792642916928,0.599785378272423,0.575149967162231
|
||||||
|
"11",1,0.841470984807897,0.895913227983752,0.658288213778898,0.650886140047209,0.577618711891772
|
||||||
|
"12",1.1,0.891207360061435,1.01252219752013,0.808981437684505,0.726263244907525,0.643161394030218
|
||||||
|
"13",1.2,0.932039085967226,1.30930912337975,1.04111824066026,0.872590842152803,0.745714536528734
|
||||||
|
"14",1.3,0.963558185417193,1.0448292335495,0.741250429230841,0.850147062957694,0.687171673021914
|
||||||
|
"15",1.4,0.98544972998846,1.57369086195552,1.17277927321094,1.06520673597544,0.847936751231165
|
||||||
|
"16",1.5,0.997494986604054,1.61427415976939,1.3908361301708,1.15616745244604,0.969474391592075
|
||||||
|
"17",1.6,0.999573603041505,1.34409615749122,0.976992098566069,1.13543598207093,0.889434319996364
|
||||||
|
"18",1.7,0.991664810452469,1.79278028030419,1.02939764179765,1.33272772191879,0.935067381106346
|
||||||
|
"19",1.8,0.973847630878195,1.50721559744085,0.903076361857071,1.30862923824728,0.91665506605512
|
||||||
|
"20",1.9,0.946300087687414,1.835014641556,0.830477479204284,1.45242210409837,0.889715842048808
|
||||||
|
"21",2,0.909297426825682,1.98589997236352,0.887302138185342,1.56569111721857,0.901843632635883
|
||||||
|
"22",2.1,0.863209366648874,2.31436634488224,0.890096618924313,1.73810390755555,0.899632162941341
|
||||||
|
"23",2.2,0.80849640381959,2.14663445612581,0.697012453130415,1.77071083163663,0.831732978616874
|
||||||
|
"24",2.3,0.74570521217672,2.17162372560288,0.614243640399509,1.84774268936257,0.787400621584077
|
||||||
|
"25",2.4,0.675463180551151,2.2488591417345,0.447664288915269,1.93366609303299,0.707449056213168
|
||||||
|
"26",2.5,0.598472144103957,2.56271588872389,0.553368843490625,2.08922735802261,0.702402440783529
|
||||||
|
"27",2.6,0.515501371821464,2.60986205081511,0.503762006272682,2.17548673152621,0.657831176057599
|
||||||
|
"28",2.7,0.42737988023383,2.47840649766003,0.215060732402894,2.20251747034638,0.533903400086802
|
||||||
|
"29",2.8,0.334988150155905,2.99861119922542,0.28503285049582,2.43015164462239,0.512492561673074
|
||||||
|
"30",2.9,0.239249329213982,3.09513467852082,0.245355736487949,2.54679545455398,0.461447717313721
|
||||||
|
"31",3,0.141120008059867,2.86247369846558,0.0960140633436418,2.55274767368554,0.371740588261606
|
||||||
|
"32",3.1,0.0415806624332905,2.79458017090243,-0.187923650913249,2.59422388058738,0.234694070506915
|
||||||
|
"33",3.2,-0.0583741434275801,3.6498183243501,-0.186738431858275,2.9216851043241,0.173308072295566
|
||||||
|
"34",3.3,-0.157745694143249,3.19424275971809,-0.221908035274934,2.86681135711315,0.101325637659584
|
||||||
|
"35",3.4,-0.255541102026832,3.53166785156005,-0.295496842654793,3.03827050777863,0.0191967841533109
|
||||||
|
"36",3.5,-0.35078322768962,3.53250700922714,-0.364585027403596,3.12709094619305,-0.0558446366563474
|
||||||
|
"37",3.6,-0.442520443294852,3.52114271616751,-0.363845774016092,3.18702722489489,-0.10585071711408
|
||||||
|
"38",3.7,-0.529836140908493,3.72033580551176,-0.386489608468821,3.31200591645168,-0.158195730190865
|
||||||
|
"39",3.8,-0.611857890942719,4.0803717995796,-0.64779795182054,3.49862620703954,-0.284999326812438
|
||||||
|
"40",3.9,-0.687766159183974,3.88351729419721,-0.604406622894426,3.51908925124143,-0.324791870057922
|
||||||
|
"41",4,-0.756802495307928,3.9941257036697,-0.8061112437715,3.62222513609486,-0.438560071688316
|
||||||
|
"42",4.1,-0.818277111064411,3.81674488816054,-0.548538951165239,3.63032709398802,-0.41285438330036
|
||||||
|
"43",4.2,-0.871575772413588,4.47703348424544,-0.998992385231986,3.88581748102334,-0.592305016590357
|
||||||
|
"44",4.3,-0.916165936749455,4.46179199544059,-0.969288921090897,3.96444243944485,-0.643076376622242
|
||||||
|
"45",4.4,-0.951602073889516,4.15184730382548,-1.11987501275525,3.93838897981045,-0.743258835859858
|
||||||
|
"46",4.5,-0.977530117665097,4.64522916494355,-0.772872365801468,4.15504805602606,-0.691414328153313
|
||||||
|
"47",4.6,-0.993691003633465,4.68087925098283,-0.650422764094352,4.24176417425486,-0.675107584174976
|
||||||
|
"48",4.7,-0.999923257564101,5.00475403211142,-0.922605880059771,4.41432228408005,-0.770625346502085
|
||||||
|
"49",4.8,-0.996164608835841,4.71428836112322,-1.14280193223997,4.41279031790692,-0.861010494025717
|
||||||
|
"50",4.9,-0.982452612624332,5.02115518218406,-0.9819618243158,4.57449352886454,-0.843786948015608
|
||||||
|
"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