Morten Hjorth-Jensen
567 lines
22 KiB
Plaintext
567 lines
22 KiB
Plaintext
======= Convolutional Neural Networks =======
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Convolutional neural networks (CNNs) were developed during the last
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decade of the previous century, with a focus on character recognition
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tasks. Nowadays, CNNs are a central element in the spectacular success
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of deep learning methods. The success in for example image
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classifications have made them a central tool for most machine
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learning practitioners.
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CNNs are very similar to ordinary Neural Networks.
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They are made up of neurons that have learnable weights and
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biases. Each neuron receives some inputs, performs a dot product and
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optionally follows it with a non-linearity. The whole network still
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expresses a single differentiable score function: from the raw image
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pixels on one end to class scores at the other. And they still have a
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loss function (for example Softmax) on the last (fully-connected) layer
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and all the tips/tricks we developed for learning regular Neural
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Networks still apply (back propagation, gradient descent etc etc).
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What is the difference? _CNN architectures make the explicit assumption that
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the inputs are images, which allows us to encode certain properties
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into the architecture. These then make the forward function more
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efficient to implement and vastly reduce the amount of parameters in
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the network._
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As an example, consider
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an image of size $32\times 32\times 3$ (32 wide, 32 high, 3 color channels), so a
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single fully-connected neuron in a first hidden layer of a regular
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Neural Network would have $32\times 32\times 3 = 3072$ weights. This amount still
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seems manageable, but clearly this fully-connected structure does not
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scale to larger images. For example, an image of more respectable
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size, say $200\times 200\times 3$, would lead to neurons that have
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$200\times 200\times 3 = 120,000$ weights.
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We could have
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several such neurons, and the parameters would add up quickly! Clearly,
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this full connectivity is wasteful and the huge number of parameters
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would quickly lead to possible overfitting.
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FIGURE: [figslides/nn.jpeg, width=500 frac=0.6] A regular 3-layer Neural Network.
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Convolutional Neural Networks take advantage of the fact that the
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input consists of images and they constrain the architecture in a more
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sensible way.
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In particular, unlike a regular Neural Network, the
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layers of a CNN have neurons arranged in 3 dimensions: width,
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height, depth. (Note that the word depth here refers to the third
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dimension of an activation volume, not to the depth of a full Neural
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Network, which can refer to the total number of layers in a network.)
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To understand it better, the above example of an image
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with an input volume of
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activations has dimensions $32\times 32\times 3$ (width, height,
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depth respectively).
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The neurons in a layer will
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only be connected to a small region of the layer before it, instead of
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all of the neurons in a fully-connected manner. Moreover, the final
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output layer could for this specific image have dimensions $1\times 1 \times 10$,
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because by the
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end of the CNN architecture we will reduce the full image into a
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single vector of class scores, arranged along the depth
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dimension.
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FIGURE: [figslides/cnn.jpeg, width=500 frac=0.6] A CNN arranges its neurons in three dimensions (width, heigh#t, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D out#put volume of neuron activations. In this example, the red input layer holds the image, so its width and heigh#t would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).
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A simple CNN is a sequence of layers, and every layer of a CNN
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transforms one volume of activations to another through a
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differentiable function. We use three main types of layers to build
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CNN architectures: Convolutional Layer, Pooling Layer, and
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Fully-Connected Layer (exactly as seen in regular Neural Networks). We
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will stack these layers to form a full CNN architecture.
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A simple CNN for image classification could have the architecture:
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* _INPUT_ ($32\times 32 \times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.
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* _CONV_ (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\times 32\times 12]$ if we decided to use 12 filters.
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* _RELU_ layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\times 32\times 12]$).
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* _POOL_ (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\times 16\times 12]$.
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* _FC_ (i.e. fully-connected) layer will compute the class scores, resulting in volume of size $[1\times 1\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.
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CNNs transform the original image layer by layer from the original
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pixel values to the final class scores.
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Observe that some layers contain
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parameters and other don’t. In particular, the CNN layers perform
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transformations that are a function of not only the activations in the
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input volume, but also of the parameters (the weights and biases of
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the neurons). On the other hand, the RELU/POOL layers will implement a
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fixed function. The parameters in the CONV/FC layers will be trained
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with gradient descent so that the class scores that the CNN computes
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are consistent with the labels in the training set for each image.
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=== CNNs in brief ===
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In summary:
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* A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)
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* There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)
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* Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function
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* Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)
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* Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)
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===== CNNs in more detail, building convolutional neural networks in Tensorflow and Keras =====
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As discussed above, CNNs are neural networks built from the assumption that the inputs
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to the network are 2D images. This is important because the number of features or pixels in images
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grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network.
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As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks
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are the _convolutional_ and _pooling_ layers stacked in pairs between the input and the hidden layer.
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In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D
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matrices, typically 1 for each color dimension (Red, Green, Blue).
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It means that to represent the entire
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dataset of images, we require a 4D matrix or _tensor_. This tensor has the dimensions:
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!bt
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\[
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(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) .
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\]
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!et
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The MNIST dataset consists of grayscale images with a pixel size of
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$28\times 28$, meaning we require $28 \times 28 = 724$ weights to each
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neuron in the first hidden layer.
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If we were to analyze images of size $128\times 128$ we would require
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$128 \times 128 = 16384$ weights to each neuron. Even worse if we were
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dealing with color images, as most images are, we have an image matrix
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of size $128\times 128$ for each color dimension (Red, Green, Blue),
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meaning 3 times the number of weights $= 49152$ are required for every
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single neuron in the first hidden layer.
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Images typically have strong local correlations, meaning that a small
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part of the image varies little from its neighboring regions. If for
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example we have an image of a blue car, we can roughly assume that a
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small blue part of the image is surrounded by other blue regions.
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Therefore, instead of connecting every single pixel to a neuron in the
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first hidden layer, as we have previously done with deep neural
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networks, we can instead connect each neuron to a small part of the
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image (in all 3 RGB depth dimensions). The size of each small area is
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fixed, and known as a "receptive":"https://en.wikipedia.org/wiki/Receptive_field".
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The layers of a convolutional neural network arrange neurons in 3D: width, height and depth.
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The input image is typically a square matrix of depth 3.
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A _convolution_ is performed on the image which outputs
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a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as _filters_.
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Each filter slides along the input image, taking the dot product
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between each small part of the image and the filter, in all depth
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dimensions. This is then passed through a non-linear function,
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typically the _Rectified Linear (ReLu)_ function, which serves as the
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activation of the neurons in the first convolutional layer. This is
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further passed through a _pooling layer_, which reduces the size of the
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convolutional layer, e.g. by taking the maximum or average across some
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small regions, and this serves as input to the next convolutional
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layer.
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By systematically reducing the size of the input volume, through
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convolution and pooling, the network should create representations of
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small parts of the input, and then from them assemble representations
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of larger areas. The final pooling layer is flattened to serve as
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input to a hidden layer, such that each neuron in the final pooling
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layer is connected to every single neuron in the hidden layer. This
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then serves as input to the output layer, e.g. a softmax output for
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classification.
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=== Prerequisites: Collect and pre-process data ===
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!bc pycod
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# import necessary packages
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import numpy as np
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import matplotlib.pyplot as plt
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from sklearn import datasets
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# ensure the same random numbers appear every time
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np.random.seed(0)
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# display images in notebook
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%matplotlib inline
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plt.rcParams['figure.figsize'] = (12,12)
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# download MNIST dataset
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digits = datasets.load_digits()
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# define inputs and labels
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inputs = digits.images
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labels = digits.target
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# RGB images have a depth of 3
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# our images are grayscale so they should have a depth of 1
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inputs = inputs[:,:,:,np.newaxis]
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print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
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print("labels = (n_inputs) = " + str(labels.shape))
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# choose some random images to display
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n_inputs = len(inputs)
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indices = np.arange(n_inputs)
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random_indices = np.random.choice(indices, size=5)
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for i, image in enumerate(digits.images[random_indices]):
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plt.subplot(1, 5, i+1)
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plt.axis('off')
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plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
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plt.title("Label: %d" % digits.target[random_indices[i]])
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plt.show()
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!ec
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=== Importing Keras and Tensorflow ===
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!bc pycod
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from tensorflow.keras import datasets, layers, models
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from tensorflow.keras.layers import Input
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from tensorflow.keras.models import Sequential #This allows appending layers to existing models
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from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer
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from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)
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from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)
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from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function
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#from tensorflow.keras import Conv2D
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#from tensorflow.keras import MaxPooling2D
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#from tensorflow.keras import Flatten
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from sklearn.model_selection import train_test_split
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# representation of labels
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labels = to_categorical(labels)
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# split into train and test data
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# one-liner from scikit-learn library
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train_size = 0.8
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test_size = 1 - train_size
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X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
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test_size=test_size)
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!ec
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!bc pycod
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def create_convolutional_neural_network_keras(input_shape, receptive_field,
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n_filters, n_neurons_connected, n_categories,
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eta, lmbd):
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model = Sequential()
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model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
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activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
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model.add(layers.MaxPooling2D(pool_size=(2, 2)))
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model.add(layers.Flatten())
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model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
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model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
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sgd = optimizers.SGD(lr=eta)
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model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
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return model
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epochs = 100
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batch_size = 100
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input_shape = X_train.shape[1:4]
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receptive_field = 3
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n_filters = 10
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n_neurons_connected = 50
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n_categories = 10
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eta_vals = np.logspace(-5, 1, 7)
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lmbd_vals = np.logspace(-5, 1, 7)
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!ec
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!bc pycod
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CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
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for i, eta in enumerate(eta_vals):
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for j, lmbd in enumerate(lmbd_vals):
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CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
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n_filters, n_neurons_connected, n_categories,
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eta, lmbd)
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CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
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scores = CNN.evaluate(X_test, Y_test)
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CNN_keras[i][j] = CNN
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print("Learning rate = ", eta)
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print("Lambda = ", lmbd)
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print("Test accuracy: %.3f" % scores[1])
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print()
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!ec
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=== Final visualization ===
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!bc pycod
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# visual representation of grid search
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# uses seaborn heatmap, could probably do this in matplotlib
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import seaborn as sns
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sns.set()
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train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
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test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
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for i in range(len(eta_vals)):
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for j in range(len(lmbd_vals)):
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CNN = CNN_keras[i][j]
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train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
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test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]
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fig, ax = plt.subplots(figsize = (10, 10))
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sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
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ax.set_title("Training Accuracy")
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ax.set_ylabel("$\eta$")
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ax.set_xlabel("$\lambda$")
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plt.show()
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fig, ax = plt.subplots(figsize = (10, 10))
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sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
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ax.set_title("Test Accuracy")
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ax.set_ylabel("$\eta$")
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ax.set_xlabel("$\lambda$")
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plt.show()
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!ec
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===== The CIFAR01 data set =====
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The CIFAR10 dataset contains 60,000 color images in 10 classes, with
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6,000 images in each class. The dataset is divided into 50,000
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training images and 10,000 testing images. The classes are mutually
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exclusive and there is no overlap between them.
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!bc pycod
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import tensorflow as tf
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from tensorflow.keras import datasets, layers, models
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import matplotlib.pyplot as plt
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# We import the data set
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(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()
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# Normalize pixel values to be between 0 and 1 by dividing by 255.
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train_images, test_images = train_images / 255.0, test_images / 255.0
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!ec
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To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image.
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!bc pycod
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class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
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'dog', 'frog', 'horse', 'ship', 'truck']
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plt.figure(figsize=(10,10))
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for i in range(25):
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plt.subplot(5,5,i+1)
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plt.xticks([])
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plt.yticks([])
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plt.grid(False)
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plt.imshow(train_images[i], cmap=plt.cm.binary)
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# The CIFAR labels happen to be arrays,
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# which is why you need the extra index
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plt.xlabel(class_names[train_labels[i][0]])
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plt.show()
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!ec
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The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.
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As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer.
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!bc pycod
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model = models.Sequential()
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model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
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model.add(layers.MaxPooling2D((2, 2)))
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model.add(layers.Conv2D(64, (3, 3), activation='relu'))
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model.add(layers.MaxPooling2D((2, 2)))
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model.add(layers.Conv2D(64, (3, 3), activation='relu'))
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# Let's display the architecture of our model so far.
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model.summary()
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!ec
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You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.
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To complete our model, you will feed the last output tensor from the
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convolutional base (of shape (4, 4, 64)) into one or more Dense layers
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to perform classification. Dense layers take vectors as input (which
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are 1D), while the current output is a 3D tensor. First, you will
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flatten (or unroll) the 3D output to 1D, then add one or more Dense
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layers on top. CIFAR has 10 output classes, so you use a final Dense
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layer with 10 outputs and a softmax activation.
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!bc pycod
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model.add(layers.Flatten())
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model.add(layers.Dense(64, activation='relu'))
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model.add(layers.Dense(10))
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Here's the complete architecture of our model.
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model.summary()
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!ec
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As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.
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Compile and train the model.
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!bc pycod
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model.compile(optimizer='adam',
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loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
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metrics=['accuracy'])
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history = model.fit(train_images, train_labels, epochs=10,
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validation_data=(test_images, test_labels))
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!ec
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Finally, we evaluate the model.
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!bc pycod
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plt.plot(history.history['accuracy'], label='accuracy')
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plt.plot(history.history['val_accuracy'], label = 'val_accuracy')
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plt.xlabel('Epoch')
|
||
plt.ylabel('Accuracy')
|
||
plt.ylim([0.5, 1])
|
||
plt.legend(loc='lower right')
|
||
|
||
test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2)
|
||
|
||
print(test_acc)
|
||
|
||
!ec
|
||
|
||
|
||
|
||
===== Recurrent neural networks: Overarching view =====
|
||
|
||
Till now our focus has been, including convolutional neural networks
|
||
as well, on feedforward neural networks. The output or the activations
|
||
flow only in one direction, from the input layer to the output layer.
|
||
|
||
A recurrent neural network (RNN) looks very much like a feedforward
|
||
neural network, except that it also has connections pointing
|
||
backward.
|
||
|
||
RNNs are used to analyze time series data such as stock prices, and
|
||
tell you when to buy or sell. In autonomous driving systems, they can
|
||
anticipate car trajectories and help avoid accidents. More generally,
|
||
they can work on sequences of arbitrary lengths, rather than on
|
||
fixed-sized inputs like all the nets we have discussed so far. For
|
||
example, they can take sentences, documents, or audio samples as
|
||
input, making them extremely useful for natural language processing
|
||
systems such as automatic translation and speech-to-text.
|
||
|
||
|
||
|
||
|
||
=== A simple example ===
|
||
|
||
!bc pycod
|
||
# Start importing packages
|
||
import pandas as pd
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
import tensorflow as tf
|
||
from tensorflow.keras import datasets, layers, models
|
||
from tensorflow.keras.layers import Input
|
||
from tensorflow.keras.models import Model, Sequential
|
||
from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU
|
||
from tensorflow.keras import optimizers
|
||
from tensorflow.keras import regularizers
|
||
from tensorflow.keras.utils import to_categorical
|
||
|
||
|
||
|
||
# convert into dataset matrix
|
||
def convertToMatrix(data, step):
|
||
X, Y =[], []
|
||
for i in range(len(data)-step):
|
||
d=i+step
|
||
X.append(data[i:d,])
|
||
Y.append(data[d,])
|
||
return np.array(X), np.array(Y)
|
||
|
||
step = 4
|
||
N = 1000
|
||
Tp = 800
|
||
|
||
t=np.arange(0,N)
|
||
x=np.sin(0.02*t)+2*np.random.rand(N)
|
||
df = pd.DataFrame(x)
|
||
df.head()
|
||
|
||
plt.plot(df)
|
||
plt.show()
|
||
|
||
values=df.values
|
||
train,test = values[0:Tp,:], values[Tp:N,:]
|
||
|
||
# add step elements into train and test
|
||
test = np.append(test,np.repeat(test[-1,],step))
|
||
train = np.append(train,np.repeat(train[-1,],step))
|
||
|
||
trainX,trainY =convertToMatrix(train,step)
|
||
testX,testY =convertToMatrix(test,step)
|
||
trainX = np.reshape(trainX, (trainX.shape[0], 1, trainX.shape[1]))
|
||
testX = np.reshape(testX, (testX.shape[0], 1, testX.shape[1]))
|
||
|
||
model = Sequential()
|
||
model.add(SimpleRNN(units=32, input_shape=(1,step), activation="relu"))
|
||
model.add(Dense(8, activation="relu"))
|
||
model.add(Dense(1))
|
||
model.compile(loss='mean_squared_error', optimizer='rmsprop')
|
||
model.summary()
|
||
|
||
model.fit(trainX,trainY, epochs=100, batch_size=16, verbose=2)
|
||
trainPredict = model.predict(trainX)
|
||
testPredict= model.predict(testX)
|
||
predicted=np.concatenate((trainPredict,testPredict),axis=0)
|
||
|
||
trainScore = model.evaluate(trainX, trainY, verbose=0)
|
||
print(trainScore)
|
||
|
||
index = df.index.values
|
||
plt.plot(index,df)
|
||
plt.plot(index,predicted)
|
||
plt.axvline(df.index[Tp], c="r")
|
||
plt.show()
|
||
!ec
|
||
|
||
|