Deconvoluting Convolutional Neural Networks

Posted in Machine Learning

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Table of Contents

Introduction: A Simple CNN Example

As part of our weekly Deep Learning for Genomics reading group here in the Lab for Data Intensive Biology (DIB Lab), we are applying convolutional neural networks (deep learning) to various problems in genomics and biology.

For the most recent meeting, we prepared some notes on how convolutional neural networks work. The notes are in the form of a Jupyter notebook. This blog post summarizes some of the important conclusions from the notebook and links to relevant sections in the notebook.

In the notebook covered in this blog post, we set up a simple convolutional neural network from an example on the keras blog. This example is used to classify input images as being either a cat or a dog.

All materials covered in this blog post are in the charlesreid1/deconvoluting-convolutions repository on Github.

Exploring the Data

TL;DR: When developing a deep learning model for a problem, it is important to start by exploring the data and understanding it thoroughly.

Link to "Image Data" section of notebook

Create CNN

TL;DR: Our convolutional neural network consists of the following architecture:

  • Convolutional Stage #1
    • Convolution (3 x 3 kernel, 32 filters)
    • Activation (ReLU)
    • Max Pooling (2x2)
  • Convolutional Stage #2
    • Convolution (3 x 3 kernel, 32 filters)
    • Activation (ReLU)
    • Max Pooling (2x2)
  • Convolutional Stage #3
    • Convolution (3 x 3 kernel, 64 filters)
    • Activation (ReLU)
    • Max Pooling (2x2)
  • Flatten
  • Dense (64 nodes)
  • Activation (ReLU)
  • Dropout (0.5)
  • Dense (1 node)
  • Activation (ReLU)

Link to "Create Convolutional Neural Network" section of notebook

Analyzing Network Architecture and Tensor Shapes

TL;DR: Each step of the neural network transforms an input tensor of a given shape into an output tensor of a (potentially different) shape.

In this section of the notebook, we step through each of the neural network's layers to explain how the size of each layer's inputs and outputs are determined.

Link to "Network Architecture/Shapes" section of notebook

Input Image Layer

TL;DR: The size of the cat and dog images is 150 x 150 pixels. Each image is a color image, so it consists of 3 channels. Therefore, the input to the very first layer has a shape of

$$ (\mbox{None}, w_0, h_0, c_0) = (\mbox{None}, 150, 150, 3) $$

(where "None" indicates a variable-size dimension that is equal to the number of total input images, or alternatively, the number of images per batch, if we are using batch learning).

Link to "Input Image Layer" section of notebook

First Convolution Layer

TL;DR: A convolutional layer with a kernel size of \(k_1 \times k_1\) and a number of filters \(c_1\) will transform the shape of the input image to:

$$ (\mbox{None}, w_1, h_1, c_1) = (\mbox{None}, 148, 148, 32) $$

where

$$ w_1 = w_0 - k_1 + 1 \\ h_1 = h_0 - k_1 + 1 $$

Importantly, each of the three input channels are added together to determine their contribution to the final convolution filters - the number of input channels does not affect the number of output channels.

The total number of output channels is equal to the number of filters in the convolution layer.

Link to "First Convolutional Layer" section of notebook

First Activation Layer

TL;DR: The activation layer is a straightforward one-to-one mapping - each individual value from the output of the convolution layer is fed through the rectified linear unit (ReLU) function and the resulting output value becomes the input to the next layer. The ReLU function is given by:

$$ \mbox{ReLU}(x) = \max(0,x) $$

The activation layer does not change the shape of the input tensor.

Link to "First Activation Layer" section of notebook

First MaxPooling Layer

TL;DR: The max pooling layer is a way of making the final convolutional filters (the "feature-detectors" of the convolutional neural network) less sensitive to the exact placement of features. The pooling layer only affects the size of the filter, not the number of channels.

If we use a max pooling window of \(p_1 \times p_1\), we will reduce the image size by \(\mbox{ceil}(w_1/p_1)\) and \(\mbox{ceil}(h_1/p_1)\). This reduces the input tensor shape to:

$$ (\mbox{None}, \mbox{ceil}(w_1/p_1), \mbox{ceil}(h_1/p_1), c_1) = (\mbox{None}, 74, 74, 32) $$

Link to "First Max Pooling Layer" section of notebook

Second Convolution Layer

TL;DR: The second convolutional layer has a kernel size of \(k_2 \times k_2\) and a number of filters \(c_2\), which will transform the shape of the input image in the same way as described for the first convolutional layer.

Note that just as the number of channels (3) in each input to the first convolutional layer did not affect the final number of channels in the output of the convolutional layer (number of channels was fixed by specifying number of output filters for the convolutional layer), so the number of input channels to the second convolutional layer does not affect the number of output channels from the second convolutional layer.

The final shape coming out of the second convolutional layer is:

$$ (\mbox{None}, w_2, h_2, c_2) = (\mbox{None}, 72, 72, 32) $$

where

$$ w_2 = w_1 - k_2 + 1 \\ h_2 = h_1 - k_2 + 1 \\ $$

Link to "Second Convolutional Layer" section of notebook

Second Activation Layer

TL;DR: The activation layer again uses a function to map input values to output values in a one-to-one mapping, so the activation layer does not change the shape of the input tensor.

Link to "Second Activation Layer" section of notebook

Second MaxPooling Layer

TL;DR: The second max pooling layer uses a pooling window of size \(p_2 \times p_2\). This will reduce the input size to \(\mbox{ceil}(w_2/p_2) \times \mbox{ceil}(h_2/p_2)\). This reduces the input tensor shape to:

$$ (\mbox{None}, \mbox{ceil}(w_2/p), \mbox{ceil}(h_2/p), c_2) = (\mbox{None}, 36, 36, 32) $$

Link to "Second Max Pooling Layer" section of notebook

Third Convolution Layer

TL;DR: The third convolution layer with a kernel size of \(k_3 \times k_3\) and \(c_3\) output filters will transform the input tensor shape in the following way (note that the third convolutional layer has 64 filters, not 32):

$$ (\mbox{None}, w_3, h_3, c_3) = (\mbox{None}, 34, 34, 64) $$

where

$$ w_3 = w_2 - k_3 + 1 \\ h_3 = h_2 - k_3 + 1 $$

Link to "Third Convolutional Layer" section of notebook

Third Activation Layer

TL;DR: The activation layer again uses a function to map input values to output values in a one-to-one mapping, so the activation layer does not change the shape of the input tensor.

Link to "Third Activation Layer" section of notebook

Third MaxPooling Layer

TL;DR: The thid max pooling layer uses a pooling window of size \(p_3 \times p_3\). This will reduce the input size to \(\mbox{ceil}(w_3/p_3) \times \mbox{ceil}(h_3/p_3)\). This reduces the input tensor shape to:

$$ (\mbox{None}, \mbox{ceil}(w_3/p_3), \mbox{ceil}(h_3/p_3), c_2) = (\mbox{None}, 17, 17, 64) $$

Link to "Third Max Pooling Layer" section of notebook

Flatten and Dense Layers

TL;DR: The flatten layer converts a tensor of dimension \((\mbox{None}, 17, 17, 64)\) into a 1D vector of \(17 \times 17 \times 64 = 18,496\) neural network nodes. This does not change any of the values, it simply reshapes the input tensor.

The first dense layer reduces the flattened \(18,496\) nodes to \(64\) nodes, using a fully connected layer of nodes. These values are then passed through an activation function (as with the above activation layers, this is a one-to-one mapping and does not change the shape of the input tensor). The dense layer is followed by a dropout layer to help prevent overfitting; this pattern is common in convolutional neural networks.

The second dense layer further reduces the \(64\) nodes to a single node, whose output will determine whether the input image is a cat or a dog.

Link to "Flatten Layer" section of notebook

Link to "Dense (64) Layers" section of notebook

Link to "Dense (1) Layers" section of notebook

Categorical Output

TL;DR: Normally when classifying cats and dogs, we would have two output neurons, one to output a binary yes/no to answer "is this a cat?" and another output a binary yes/no to answer "is this a dog?". However, in this example, we assume that all inputs contain either only cats or only dogs, so the single-output binary classifier is determining whether an image is a dog (0) or a cat (1).

Image Transformer

TL;DR: The ImageDataGenerator class is a class provided by keras for loading image data from a directory and (optionally) applying various transformations to the images in order to generate additional training data from a set of images. For example, the following code block from the notebook creates an ImageDataGenerator class that will load images from a folder on disk, and applies various transformations (shearing, zooming, and horizontally flipping) to each image during the training process.

train_datagen = ImageDataGenerator(
    rescale=1. / 255,
    shear_range=0.2,
    zoom_range=0.2,
    horizontal_flip=True)

This can then be used to generate test image data:

train_generator = train_datagen.flow_from_directory(
    'train',
    target_size=(img_width, img_height),
    batch_size=batch_size,
    class_mode='binary')

This will look for images in the relative path train/data/ (note the implicit data/ directory tacked on the end). Note that this image data generator allows us to use images that do not have size \(150 \times 150\), as they will be re-sized to target_size.

Link to "Image Transformer" section of notebook

Next Steps

Now that we have walked through a sample convolutional neural network and covered how each layer transforms the size of the input tensor, we are ready to start applying convolutional neural networks to real problems.

Our next blog post will cover the materials in the charlesreid1/deep-learning-genomics repository on Github, which applies the convolutional neural network concept in a 1D context (applying convolutions to 1D sequences, instead of 2D images) to learn about (and predict) DNA transcription factor binding sites.