FYS-STK4155/doc/LectureNotes/_build/jupyter_execute/week45.ipynb

Week 45, Convolutional Neural Networks (CCNs) and Recurrent Neural Networks (RNNs)

Morten Hjorth-Jensen, Department of Physics, University of Oslo

Date: November 4-8

Plans for week 45

Material for the lecture on Monday November 4, 2024.

1. Convolutional Neural Networks, codes and examples (own code and TensorFlow and Pytorch implementations)

2. Recurrent Neural Networks (RNNs)

3. Readings and Videos:

a. These lecture notes at https://github.com/CompPhysics/MachineLearning/blob/master/doc/pub/week45/ipynb/week45.ipynb

b. For a more in depth discussion on CNNs and recurrent neural networks we recommend Goodfellow et al chapters 9 and 10. See also chapter 11 and 12 on practicalities and applications

c. Reading suggestions for implementation of CNNs and RNNs, see Raschka et al chapters 14-15 at https://github.com/rasbt/machine-learning-book.

d. Video on Recurrent Neural Networks from MIT at https://www.youtube.com/watch?v=SEnXr6v2ifU&ab_channel=AlexanderAmini

e. Video on Deep Learning at https://www.youtube.com/playlist?list=PLZHQObOWTQDNU6R1_67000Dx_ZCJB-3pi

Material for the lab sessions, additional ways to present classification results and other practicalities

Material for the active learning sessions on Tuesday and Wednesday.

1. Discussion of and work on project 3, available from Monday November 4, late evening

Material for Lecture Monday November 4

Convolutional Neural Networks (recognizing images)

Convolutional neural networks (CNNs) were developed during the last decade of the previous century, with a focus on character recognition tasks. Nowadays, CNNs are a central element in the spectacular success of deep learning methods. The success in for example image classifications have made them a central tool for most machine learning practitioners.

CNNs are very similar to ordinary Neural Networks. They are made up of neurons that have learnable weights and biases. Each neuron receives some inputs, performs a dot product and optionally follows it with a non-linearity. The whole network still expresses a single differentiable score function: from the raw image pixels on one end to class scores at the other. And they still have a loss function (for example Softmax) on the last (fully-connected) layer and all the tips/tricks we developed for learning regular Neural Networks still apply (back propagation, gradient descent etc etc).

What is the Difference

CNN architectures make the explicit assumption that the inputs are images, which allows us to encode certain properties into the architecture. These then make the forward function more efficient to implement and vastly reduce the amount of parameters in the network.

Neural Networks vs CNNs

Neural networks are defined as affine transformations, that is a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an output (to which a bias vector is usually added before passing the result through a nonlinear activation function). This is applicable to any type of input, be it an image, a sound clip or an unordered collection of features: whatever their dimensionality, their representation can always be flattened into a vector before the transformation.

Why CNNS for images, sound files, medical images from CT scans etc?

However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic structure. More formally, they share these important properties:

• They are stored as multi-dimensional arrays (think of the pixels of a figure) .

• They feature one or more axes for which ordering matters (e.g., width and height axes for an image, time axis for a sound clip).

• One axis, called the channel axis, is used to access different views of the data (e.g., the red, green and blue channels of a color image, or the left and right channels of a stereo audio track).

These properties are not exploited when an affine transformation is applied; in fact, all the axes are treated in the same way and the topological information is not taken into account. Still, taking advantage of the implicit structure of the data may prove very handy in solving some tasks, like computer vision and speech recognition, and in these cases it would be best to preserve it. This is where discrete convolutions come into play.

A discrete convolution is a linear transformation that preserves this notion of ordering. It is sparse (only a few input units contribute to a given output unit) and reuses parameters (the same weights are applied to multiple locations in the input).

Regular NNs don’t scale well to full images

As an example, consider an image of size 32\times 32\times 3 (32 wide, 32 high, 3 color channels), so a single fully-connected neuron in a first hidden layer of a regular Neural Network would have 32\times 32\times 3 = 3072 weights. This amount still seems manageable, but clearly this fully-connected structure does not scale to larger images. For example, an image of more respectable size, say 200\times 200\times 3, would lead to neurons that have 200\times 200\times 3 = 120,000 weights.

We could have several such neurons, and the parameters would add up quickly! Clearly, this full connectivity is wasteful and the huge number of parameters would quickly lead to possible overfitting.

Figure 1: A regular 3-layer Neural Network.

3D volumes of neurons

Convolutional Neural Networks take advantage of the fact that the input consists of images and they constrain the architecture in a more sensible way.

In particular, unlike a regular Neural Network, the layers of a CNN have neurons arranged in 3 dimensions: width, height, depth. (Note that the word depth here refers to the third dimension of an activation volume, not to the depth of a full Neural Network, which can refer to the total number of layers in a network.)

To understand it better, the above example of an image with an input volume of activations has dimensions 32\times 32\times 3 (width, height, depth respectively).

The neurons in a layer will only be connected to a small region of the layer before it, instead of all of the neurons in a fully-connected manner. Moreover, the final output layer could for this specific image have dimensions 1\times 1 \times 10, because by the end of the CNN architecture we will reduce the full image into a single vector of class scores, arranged along the depth dimension.

Figure 1: A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

Layers used to build CNNs

A simple CNN is a sequence of layers, and every layer of a CNN transforms one volume of activations to another through a differentiable function. We use three main types of layers to build CNN architectures: Convolutional Layer, Pooling Layer, and Fully-Connected Layer (exactly as seen in regular Neural Networks). We will stack these layers to form a full CNN architecture.

A simple CNN for image classification could have the architecture:

• 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.

• 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.

• 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]).

• POOL (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as [16\times 16\times 12].

• 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.

CNNs in brief

In summary:

• 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)

• There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)

• Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function

• Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)

• Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)

A deep CNN model (From Raschka et al)

Figure 1: A deep CNN

Key Idea

A dense neural network is representd by an affine operation (like matrix-matrix multiplication) where all parameters are included.

The key idea in CNNs for say imaging is that in images neighbor pixels tend to be related! So we connect only neighboring neurons in the input instead of connecting all with the first hidden layer.

We say we perform a filtering (convolution is the mathematical operation).

Building convolutional neural networks in Tensorflow and Keras

As discussed above, CNNs are neural networks built from the assumption that the inputs to the network are 2D images. This is important because the number of features or pixels in images grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network.

As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks are the convolutional and pooling layers stacked in pairs between the input and the hidden layer. In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D matrices, typically 1 for each color dimension (Red, Green, Blue).

Setting it up

It means that to represent the entire dataset of images, we require a 4D matrix or tensor. This tensor has the dimensions:

(n_{inputs},\, n_{pixels, width},\, n_{pixels, height},\, depth) .

The MNIST dataset again

The MNIST dataset consists of grayscale images with a pixel size of 28\times 28, meaning we require 28 \times 28 = 724 weights to each neuron in the first hidden layer.

If we were to analyze images of size 128\times 128 we would require 128 \times 128 = 16384 weights to each neuron. Even worse if we were dealing with color images, as most images are, we have an image matrix of size 128\times 128 for each color dimension (Red, Green, Blue), meaning 3 times the number of weights = 49152 are required for every single neuron in the first hidden layer.

Strong correlations

Images typically have strong local correlations, meaning that a small part of the image varies little from its neighboring regions. If for example we have an image of a blue car, we can roughly assume that a small blue part of the image is surrounded by other blue regions.

Therefore, instead of connecting every single pixel to a neuron in the first hidden layer, as we have previously done with deep neural networks, we can instead connect each neuron to a small part of the image (in all 3 RGB depth dimensions). The size of each small area is fixed, and known as a receptive.

Layers of a CNN

The layers of a convolutional neural network arrange neurons in 3D: width, height and depth.
The input image is typically a square matrix of depth 3.

A convolution is performed on the image which outputs a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as filters.

Each filter slides along the input image, taking the dot product between each small part of the image and the filter, in all depth dimensions. This is then passed through a non-linear function, typically the Rectified Linear (ReLu) function, which serves as the activation of the neurons in the first convolutional layer. This is further passed through a pooling layer, which reduces the size of the convolutional layer, e.g. by taking the maximum or average across some small regions, and this serves as input to the next convolutional layer.

Systematic reduction

By systematically reducing the size of the input volume, through convolution and pooling, the network should create representations of small parts of the input, and then from them assemble representations of larger areas. The final pooling layer is flattened to serve as input to a hidden layer, such that each neuron in the final pooling layer is connected to every single neuron in the hidden layer. This then serves as input to the output layer, e.g. a softmax output for classification.

Prerequisites: Collect and pre-process data

In [1]:

%matplotlib inline

# import necessary packages
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets


# ensure the same random numbers appear every time
np.random.seed(0)

# display images in notebook
%matplotlib inline
plt.rcParams['figure.figsize'] = (12,12)


# download MNIST dataset
digits = datasets.load_digits()

# define inputs and labels
inputs = digits.images
labels = digits.target

# RGB images have a depth of 3
# our images are grayscale so they should have a depth of 1
inputs = inputs[:,:,:,np.newaxis]

print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape))
print("labels = (n_inputs) = " + str(labels.shape))


# choose some random images to display
n_inputs = len(inputs)
indices = np.arange(n_inputs)
random_indices = np.random.choice(indices, size=5)

for i, image in enumerate(digits.images[random_indices]):
    plt.subplot(1, 5, i+1)
    plt.axis('off')
    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
    plt.title("Label: %d" % digits.target[random_indices[i]])
plt.show()

inputs = (n_inputs, pixel_width, pixel_height, depth) = (1797, 8, 8, 1)
labels = (n_inputs) = (1797,)

Importing Keras and Tensorflow

In [2]:

from tensorflow.keras import datasets, layers, models
from tensorflow.keras.layers import Input
from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
#from tensorflow.keras import Conv2D
#from tensorflow.keras import MaxPooling2D
#from tensorflow.keras import Flatten

from sklearn.model_selection import train_test_split

# representation of labels
labels = to_categorical(labels)

# split into train and test data
# one-liner from scikit-learn library
train_size = 0.8
test_size = 1 - train_size
X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
                                                    test_size=test_size)

Running with Keras

In [3]:

def create_convolutional_neural_network_keras(input_shape, receptive_field,
                                              n_filters, n_neurons_connected, n_categories,
                                              eta, lmbd):
    model = Sequential()
    model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',
              activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
    model.add(layers.MaxPooling2D(pool_size=(2, 2)))
    model.add(layers.Flatten())
    model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))
    model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))
    
    sgd = optimizers.SGD(learning_rate=eta)
    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
    
    return model

epochs = 100
batch_size = 100
input_shape = X_train.shape[1:4]
receptive_field = 3
n_filters = 10
n_neurons_connected = 50
n_categories = 10

eta_vals = np.logspace(-5, 1, 7)
lmbd_vals = np.logspace(-5, 1, 7)

Final part

In [4]:

CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
        
for i, eta in enumerate(eta_vals):
    for j, lmbd in enumerate(lmbd_vals):
        CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
                                              n_filters, n_neurons_connected, n_categories,
                                              eta, lmbd)
        CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
        scores = CNN.evaluate(X_test, Y_test)
        
        CNN_keras[i][j] = CNN
        
        print("Learning rate = ", eta)
        print("Lambda = ", lmbd)
        print("Test accuracy: %.3f" % scores[1])
        print()

/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/keras/src/layers/convolutional/base_conv.py:107: UserWarning: Do not pass an `input_shape`/`input_dim` argument to a layer. When using Sequential models, prefer using an `Input(shape)` object as the first layer in the model instead.
  super().__init__(activity_regularizer=activity_regularizer, **kwargs)

 1/12 ━━━━━━━━━━━━━━━━━━━━ 2s 203ms/step - accuracy: 0.2812 - loss: 2.7253


 9/12 ━━━━━━━━━━━━━━━━━━━━ 0s 7ms/step - accuracy: 0.1957 - loss: 2.8906  


12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 9ms/step - accuracy: 0.1826 - loss: 2.9261

Learning rate =  1e-05
Lambda =  1e-05
Test accuracy: 0.150

 1/12 ━━━━━━━━━━━━━━━━━━━━ 1s 126ms/step - accuracy: 0.0938 - loss: 2.8780


 9/12 ━━━━━━━━━━━━━━━━━━━━ 0s 7ms/step - accuracy: 0.1082 - loss: 2.8294  


12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 9ms/step - accuracy: 0.1134 - loss: 2.8359

Learning rate =  1e-05
Lambda =  0.0001
Test accuracy: 0.125

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
Cell In[4], line 8
      4 for j, lmbd in enumerate(lmbd_vals):
      5     CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,
      6                                           n_filters, n_neurons_connected, n_categories,
      7                                           eta, lmbd)
----> 8     CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
      9     scores = CNN.evaluate(X_test, Y_test)
     11     CNN_keras[i][j] = CNN

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/keras/src/utils/traceback_utils.py:117, in filter_traceback.<locals>.error_handler(*args, **kwargs)
    115 filtered_tb = None
    116 try:
--> 117     return fn(*args, **kwargs)
    118 except Exception as e:
    119     filtered_tb = _process_traceback_frames(e.__traceback__)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/keras/src/backend/tensorflow/trainer.py:320, in TensorFlowTrainer.fit(self, x, y, batch_size, epochs, verbose, callbacks, validation_split, validation_data, shuffle, class_weight, sample_weight, initial_epoch, steps_per_epoch, validation_steps, validation_batch_size, validation_freq)
    318 for step, iterator in epoch_iterator.enumerate_epoch():
    319     callbacks.on_train_batch_begin(step)
--> 320     logs = self.train_function(iterator)
    321     callbacks.on_train_batch_end(step, logs)
    322     if self.stop_training:

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/util/traceback_utils.py:150, in filter_traceback.<locals>.error_handler(*args, **kwargs)
    148 filtered_tb = None
    149 try:
--> 150   return fn(*args, **kwargs)
    151 except Exception as e:
    152   filtered_tb = _process_traceback_frames(e.__traceback__)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/polymorphic_function/polymorphic_function.py:833, in Function.__call__(self, *args, **kwds)
    830 compiler = "xla" if self._jit_compile else "nonXla"
    832 with OptionalXlaContext(self._jit_compile):
--> 833   result = self._call(*args, **kwds)
    835 new_tracing_count = self.experimental_get_tracing_count()
    836 without_tracing = (tracing_count == new_tracing_count)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/polymorphic_function/polymorphic_function.py:878, in Function._call(self, *args, **kwds)
    875 self._lock.release()
    876 # In this case we have not created variables on the first call. So we can
    877 # run the first trace but we should fail if variables are created.
--> 878 results = tracing_compilation.call_function(
    879     args, kwds, self._variable_creation_config
    880 )
    881 if self._created_variables:
    882   raise ValueError("Creating variables on a non-first call to a function"
    883                    " decorated with tf.function.")

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/polymorphic_function/tracing_compilation.py:139, in call_function(args, kwargs, tracing_options)
    137 bound_args = function.function_type.bind(*args, **kwargs)
    138 flat_inputs = function.function_type.unpack_inputs(bound_args)
--> 139 return function._call_flat(  # pylint: disable=protected-access
    140     flat_inputs, captured_inputs=function.captured_inputs
    141 )

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/polymorphic_function/concrete_function.py:1322, in ConcreteFunction._call_flat(self, tensor_inputs, captured_inputs)
   1318 possible_gradient_type = gradients_util.PossibleTapeGradientTypes(args)
   1319 if (possible_gradient_type == gradients_util.POSSIBLE_GRADIENT_TYPES_NONE
   1320     and executing_eagerly):
   1321   # No tape is watching; skip to running the function.
-> 1322   return self._inference_function.call_preflattened(args)
   1323 forward_backward = self._select_forward_and_backward_functions(
   1324     args,
   1325     possible_gradient_type,
   1326     executing_eagerly)
   1327 forward_function, args_with_tangents = forward_backward.forward()

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/polymorphic_function/atomic_function.py:216, in AtomicFunction.call_preflattened(self, args)
    214 def call_preflattened(self, args: Sequence[core.Tensor]) -> Any:
    215   """Calls with flattened tensor inputs and returns the structured output."""
--> 216   flat_outputs = self.call_flat(*args)
    217   return self.function_type.pack_output(flat_outputs)

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/polymorphic_function/atomic_function.py:251, in AtomicFunction.call_flat(self, *args)
    249 with record.stop_recording():
    250   if self._bound_context.executing_eagerly():
--> 251     outputs = self._bound_context.call_function(
    252         self.name,
    253         list(args),
    254         len(self.function_type.flat_outputs),
    255     )
    256   else:
    257     outputs = make_call_op_in_graph(
    258         self,
    259         list(args),
    260         self._bound_context.function_call_options.as_attrs(),
    261     )

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/context.py:1500, in Context.call_function(self, name, tensor_inputs, num_outputs)
   1498 cancellation_context = cancellation.context()
   1499 if cancellation_context is None:
-> 1500   outputs = execute.execute(
   1501       name.decode("utf-8"),
   1502       num_outputs=num_outputs,
   1503       inputs=tensor_inputs,
   1504       attrs=attrs,
   1505       ctx=self,
   1506   )
   1507 else:
   1508   outputs = execute.execute_with_cancellation(
   1509       name.decode("utf-8"),
   1510       num_outputs=num_outputs,
   (...)
   1514       cancellation_manager=cancellation_context,
   1515   )

File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/execute.py:53, in quick_execute(op_name, num_outputs, inputs, attrs, ctx, name)
     51 try:
     52   ctx.ensure_initialized()
---> 53   tensors = pywrap_tfe.TFE_Py_Execute(ctx._handle, device_name, op_name,
     54                                       inputs, attrs, num_outputs)
     55 except core._NotOkStatusException as e:
     56   if name is not None:

KeyboardInterrupt: 

Final visualization

In [5]:

# visual representation of grid search
# uses seaborn heatmap, could probably do this in matplotlib
import seaborn as sns

sns.set()

train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))

for i in range(len(eta_vals)):
    for j in range(len(lmbd_vals)):
        CNN = CNN_keras[i][j]

        train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]
        test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]

        
fig, ax = plt.subplots(figsize = (10, 10))
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
ax.set_title("Training Accuracy")
ax.set_ylabel("$\eta$")
ax.set_xlabel("$\lambda$")
plt.show()

fig, ax = plt.subplots(figsize = (10, 10))
sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
ax.set_title("Test Accuracy")
ax.set_ylabel("$\eta$")
ax.set_xlabel("$\lambda$")
plt.show()

The CIFAR01 data set

The CIFAR10 dataset contains 60,000 color images in 10 classes, with 6,000 images in each class. The dataset is divided into 50,000 training images and 10,000 testing images. The classes are mutually exclusive and there is no overlap between them.

In [6]:

import tensorflow as tf

from tensorflow.keras import datasets, layers, models
import matplotlib.pyplot as plt

# We import the data set
(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()

# Normalize pixel values to be between 0 and 1 by dividing by 255. 
train_images, test_images = train_images / 255.0, test_images / 255.0

Verifying the data set

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.

In [7]:

class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',
               'dog', 'frog', 'horse', 'ship', 'truck']
plt.figure(figsize=(10,10))
for i in range(25):
    plt.subplot(5,5,i+1)
    plt.xticks([])
    plt.yticks([])
    plt.grid(False)
    plt.imshow(train_images[i], cmap=plt.cm.binary)
    # The CIFAR labels happen to be arrays, 
    # which is why you need the extra index
    plt.xlabel(class_names[train_labels[i][0]])
plt.show()

Set up the model

The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.

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.

In [8]:

model = models.Sequential()
model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(64, (3, 3), activation='relu'))
model.add(layers.MaxPooling2D((2, 2)))
model.add(layers.Conv2D(64, (3, 3), activation='relu'))

# Let's display the architecture of our model so far.

model.summary()

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.

Add Dense layers on top

To complete our model, you will feed the last output tensor from the convolutional base (of shape (4, 4, 64)) into one or more Dense layers to perform classification. Dense layers take vectors as input (which are 1D), while the current output is a 3D tensor. First, you will flatten (or unroll) the 3D output to 1D, then add one or more Dense layers on top. CIFAR has 10 output classes, so you use a final Dense layer with 10 outputs and a softmax activation.

In [9]:

model.add(layers.Flatten())
model.add(layers.Dense(64, activation='relu'))
model.add(layers.Dense(10))
# Here's the complete architecture of our model

model.summary()

As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.

Compile and train the model

In [10]:

model.compile(optimizer='adam',
              loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
              metrics=['accuracy'])

history = model.fit(train_images, train_labels, epochs=10, 
                    validation_data=(test_images, test_labels))

Finally, evaluate the model

In [11]:

plt.plot(history.history['accuracy'], label='accuracy')
plt.plot(history.history['val_accuracy'], label = 'val_accuracy')
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)

Building our own CNN code

Here we present a flexible and readable python code for a CNN implemented with NumPy. We will present the code, showcase how to use the codebase and fit a CNN that yields a 99% accuracy on the 28x28 MNIST dataset within reasonable time.

The codes here were developed by Eric Reber and Gregor Kajda during spring 2023.

The CNN is compatible with all schedulers, cost functions and activation functions discussed in constructing our neural network codes.

The CNN code consists of different types of Layer classes, including Convolution2DLayer, Pooling2DLayer, FlattenLayer, FullyConnectedLayer and OutputLayer, which can be added to the CNN object using the interface of the CNN class. This allows you to easily construct your own CNN, as well as allowing you to get used to an interface similar to that of TensorFlow which is used for real world applications.

Another important feature of this code is that it throws errors if unreasonable decisions are made (for example using a kernel that is larger than the image, not using a FlattenLayer, etc), and provides the user with an informative error message.

List of contents:

1. Schedulers

2. Activation Functions

3. Cost Functions

4. Convolution

5. Layers

6. CNN

7. Some final remarks

Schedulers

The code below shows object oriented implementations of the Constant, Momentum, Adagrad, AdagradMomentum, RMS prop and Adam schedulers. All of the classes belong to the shared abstract Scheduler class, and share the update_change() and reset() methods allowing for any of the schedulers to be seamlessly used during the training stage, as will later be shown in the fit() method of the neural network. Update_change() only has one parameter, the gradient (\delta^{l}_{j}a^{l-1}_k), and returns the change which will be subtracted from the weights. The reset() function takes no parameters, and resets the desired variables. For Constant and Momentum, reset does nothing.

In [12]:

import autograd.numpy as np

class Scheduler:
    """
    Abstract class for Schedulers
    """

    def __init__(self, eta):
        self.eta = eta

    # should be overwritten
    def update_change(self, gradient):
        raise NotImplementedError

    # overwritten if needed
    def reset(self):
        pass


class Constant(Scheduler):
    def __init__(self, eta):
        super().__init__(eta)

    def update_change(self, gradient):
        return self.eta * gradient
    
    def reset(self):
        pass


class Momentum(Scheduler):
    def __init__(self, eta: float, momentum: float):
        super().__init__(eta)
        self.momentum = momentum
        self.change = 0

    def update_change(self, gradient):
        self.change = self.momentum * self.change + self.eta * gradient
        return self.change

    def reset(self):
        pass


class Adagrad(Scheduler):
    def __init__(self, eta):
        super().__init__(eta)
        self.G_t = None

    def update_change(self, gradient):
        delta = 1e-8  # avoid division ny zero

        if self.G_t is None:
            self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))

        self.G_t += gradient @ gradient.T

        G_t_inverse = 1 / (
            delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))
        )
        return self.eta * gradient * G_t_inverse

    def reset(self):
        self.G_t = None


class AdagradMomentum(Scheduler):
    def __init__(self, eta, momentum):
        super().__init__(eta)
        self.G_t = None
        self.momentum = momentum
        self.change = 0

    def update_change(self, gradient):
        delta = 1e-8  # avoid division ny zero

        if self.G_t is None:
            self.G_t = np.zeros((gradient.shape[0], gradient.shape[0]))

        self.G_t += gradient @ gradient.T

        G_t_inverse = 1 / (
            delta + np.sqrt(np.reshape(np.diagonal(self.G_t), (self.G_t.shape[0], 1)))
        )
        self.change = self.change * self.momentum + self.eta * gradient * G_t_inverse
        return self.change

    def reset(self):
        self.G_t = None


class RMS_prop(Scheduler):
    def __init__(self, eta, rho):
        super().__init__(eta)
        self.rho = rho
        self.second = 0.0

    def update_change(self, gradient):
        delta = 1e-8  # avoid division ny zero
        self.second = self.rho * self.second + (1 - self.rho) * gradient * gradient
        return self.eta * gradient / (np.sqrt(self.second + delta))

    def reset(self):
        self.second = 0.0


class Adam(Scheduler):
    def __init__(self, eta, rho, rho2):
        super().__init__(eta)
        self.rho = rho
        self.rho2 = rho2
        self.moment = 0
        self.second = 0
        self.n_epochs = 1

    def update_change(self, gradient):
        delta = 1e-8  # avoid division ny zero

        self.moment = self.rho * self.moment + (1 - self.rho) * gradient
        self.second = self.rho2 * self.second + (1 - self.rho2) * gradient * gradient

        moment_corrected = self.moment / (1 - self.rho**self.n_epochs)
        second_corrected = self.second / (1 - self.rho2**self.n_epochs)

        return self.eta * moment_corrected / (np.sqrt(second_corrected + delta))

    def reset(self):
        self.n_epochs += 1
        self.moment = 0
        self.second = 0

Usage of schedulers

To initalize a scheduler, simply create the object and pass in the necessary parameters such as the learning rate and the momentum as shown below. As the Scheduler class is an abstract class it should not called directly, and will raise an error upon usage.

In [13]:

momentum_scheduler = Momentum(eta=1e-3, momentum=0.9)
adam_scheduler = Adam(eta=1e-3, rho=0.9, rho2=0.999)

Here is a small example for how a segment of code using schedulers could look. Switching out the schedulers is simple.

In [14]:

weights = np.ones((3,3))
print(f"Before scheduler:\n{weights=}")

epochs = 10
for e in range(epochs):
    gradient = np.random.rand(3, 3)
    change = adam_scheduler.update_change(gradient)
    weights = weights - change
    adam_scheduler.reset()

print(f"\nAfter scheduler:\n{weights=}")

Cost functions

In this section we will quickly look at cost functions that can be used when creating the neural network. Every cost function takes the target vector as its parameter, and returns a function valued only at X such that it may easily be differentiated.

In [15]:

def CostOLS(target):
    """
    Return OLS function valued only at X, so
    that it may be easily differentiated
    """

    def func(X):
        return (1.0 / target.shape[0]) * np.sum((target - X) ** 2)

    return func


def CostLogReg(target):
    """
    Return Logistic Regression cost function
    valued only at X, so that it may be easily differentiated
    """

    def func(X):
        return -(1.0 / target.shape[0]) * np.sum(
            (target * np.log(X + 10e-10)) + ((1 - target) * np.log(1 - X + 10e-10))
        )

    return func


def CostCrossEntropy(target):
    """
    Return cross entropy cost function valued only at X, so
    that it may be easily differentiated
    """
    
    def func(X):
        return -(1.0 / target.size) * np.sum(target * np.log(X + 10e-10))

    return func

Usage of cost functions

Below we will provide a short example of how these cost function may be used to obtain results if you wish to test them out on your own using AutoGrad's automatic differentiation.

In [16]:

from autograd import grad

target = np.array([[1, 2, 3]]).T
a = np.array([[4, 5, 6]]).T

cost_func = CostCrossEntropy
cost_func_derivative = grad(cost_func(target))

valued_at_a = cost_func_derivative(a)
print(f"Derivative of cost function {cost_func.__name__} valued at a:\n{valued_at_a}")

Activation functions

Finally, before we look at the layers that make up the neural network, we will look at the activation functions which can be specified between the hidden layers and as the output function. Each function can be valued for any given vector or matrix X, and can be differentiated via derivate().

In [17]:

import autograd.numpy as np
from autograd import elementwise_grad

def identity(X):
    return X


def sigmoid(X):
    try:
        return 1.0 / (1 + np.exp(-X))
    except FloatingPointError:
        return np.where(X > np.zeros(X.shape), np.ones(X.shape), np.zeros(X.shape))


def softmax(X):
    X = X - np.max(X, axis=-1, keepdims=True)
    delta = 10e-10
    return np.exp(X) / (np.sum(np.exp(X), axis=-1, keepdims=True) + delta)


def RELU(X):
    return np.where(X > np.zeros(X.shape), X, np.zeros(X.shape))


def LRELU(X):
    delta = 10e-4
    return np.where(X > np.zeros(X.shape), X, delta * X)


def derivate(func):
    if func.__name__ == "RELU":

        def func(X):
            return np.where(X > 0, 1, 0)

        return func

    elif func.__name__ == "LRELU":

        def func(X):
            delta = 10e-4
            return np.where(X > 0, 1, delta)

        return func

    else:
        return elementwise_grad(func)

Usage of activation functions

Below we present a short demonstration of how to use an activation function. The derivative of the activation function will be important when calculating the output delta term during backpropagation. Note that derivate() can also be used for cost functions for a more generalized approach.

In [18]:

z = np.array([[4, 5, 6]]).T
print(f"Input to activation function:\n{z}")

act_func = sigmoid
a = act_func(z)
print(f"\nOutput from {act_func.__name__} activation function:\n{a}")

act_func_derivative = derivate(act_func)
valued_at_z = act_func_derivative(a)
print(f"\nDerivative of {act_func.__name__} activation function valued at z:\n{valued_at_z}")

Convolution

In order to construct a convolutional neural network (CNN), it is crucial to comprehend the fundamental principles of convolution and how it aids in extracting information from images. Convolution, at its core, is merely a mathematical operation between two functions that yields another function. It is represented by an integral between two functions, which is typically expressed as:

(f \ast g)(t):=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau.

Here, f and g are the two functions on which we want to perform an operation. The outcome of the convolution operation is represented by (f \ast g), and it is derived by sliding the function g over f and computing the integral of their product at each position. If both functions are continuous, convolution takes the form shown above. However, if we discretize both f and g, the convolution operation will take the form of a sum between the elements of f and g:

(f \ast g)[n]=\sum_{m=0}^{n-1} f(m) g(n-m).

The key idea we utilize to extract the information contained in an image is to slide an m \times n matrix g over an m \times n matrix f. In our case, f represents the image, while g represents the kernel, oftentimes called a filter. However, since our convolution will be a two-dimensional variant, we need to extend our mathematical formula with an additional summation:

(f \ast g)(i, j)\sum_{m=0}^{M-1}\sum_{n=0}^{N-1} f(m,n) g(i-m, j-n).

It is imperative to note that the size of the kernel g is significantly smaller than the size of the input image f, thereby reducing the amount of computation necessary for feature extraction. Furthermore, the kernel is usually a trainable parameter in a convolutional neural network, allowing the network to learn appropriate kernels for specific tasks.

To give you an example of how 2D convolution works in practice, suppose we have an image f of dimension 6 \times 6

f = \begin{bmatrix}
4 & 1 & 2 & 9 & 8 & 6 \\
9 & 5 & 9 & 5 & 8 & 5 \\
1 & 5 & 9 & 7 & 6 & 4 \\
2 & 9 & 8 & 3 & 7 & 1 \\
8 & 1 & 6 & 4 & 2 & 2 \\
1 & 0 & 5 & 7 & 8 & 2 \\
\end{bmatrix}

and a 3 \times 3 kernel g called a low-pass filter. Note that the kernel is usually rotated by 180 degrees during convolution, however this has no effect on this kernel.

g = \frac{1}{9}
\begin{bmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1 \\
\end{bmatrix}

In order to filter the image, we have to extract a 3 \times 3 element from the upper left corner of f, and perform element-wise multiplication of the extracted image pixels with the elements of the kernel g:

\begin{bmatrix}
4 & 1 & 2 \\
9 & 5 & 9 \\
1 & 5 & 9 \\
\end{bmatrix}
\cdot
\begin{bmatrix}
\frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\
\frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\
\frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\
\end{bmatrix}
=
\begin{bmatrix}
\frac{4}{9} & \frac{1}{9} & \frac{2}{9} \\
\frac{9}{9} & \frac{5}{9} & \frac{9}{9} \\
\frac{1}{9} & \frac{5}{9} & \frac{9}{9} \\
\end {bmatrix}
= \boldsymbol{A}

Then, following the multiplication, we summarize all the elements of the resulting matrix \boldsymbol{A}:

(f \ast g)(0, 0)= \sum_{i=0}^{2} \sum_{j=0}^{2} a_{i,j} = 5,

which corresponds to the first element of the filtered image (f \ast g).

Here we use a stride of S=1, a parameter denoted S which describes how many indexes we move the kernel g to the right before repeating the calculations above for the next 3 \times 3 element of the image f. It is usually presumed that S=1, however, larger values for S can be used to reduce the dimentionality of the filtered image such that the convolution operation is more computationally efficient. In the context of a convolutional neural network, this will become very useful.

The full result of the convolution is:

(f \ast g) =
\begin{bmatrix}
5 & 5.78 & 7 & 6.44 \\
6.33 & 6.67 & 6.89 & 5.11 \\
5.44 & 5.78 & 5.78 & 4 \\
4.44 & 4.78 & 5.56 & 4 \\
\end{bmatrix}

The result is markedly smaller in shape than the original image. This occurs when using convolution without first padding the image with additional columns and rows, allowing us to keep the original image shape after sliding the kernel over the image. How many rows and columns we wish to pad the image with depends strictly on the shape of the kernel, as we wish to pad the image with r additional rows and c additional columns.

r =\lfloor \frac{\mathrm{kernel height}}{2} \rfloor \cdot 2 \\
c =\lfloor \frac{\mathrm{kernel width}}{2} \rfloor \cdot 2

Note the notation \lfloor \frac{\mathrm{kernel width}}{2} \rfloor means that we floor the result of the division, meaning we round down to a whole number in case \frac{\mathrm{kernel width}}{2} results in a floating point number.

Using those simple equations, we find out by how much we have to extend the dimensions of the original image. Before proceeding, however, we might ask what we shall fill the additional rows and columns with? One of the most common approaches to padding is zero-padding, which as the name suggest, involves filling the rows and columns with zeros. This is the approach that we will be using for this demonstration. If we apply this padding to out original 6 \times 6 image, the result will be an 8 \times 8 image as the kernel has a width and height of 3. Note that the original image is encapsuled by the zero-padded rows and columns:

\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 4 & 1 & 2 & 9 & 8 & 6 & 0 \\
0 & 9 & 5 & 9 & 5 & 8 & 5 & 0 \\
0 & 1 & 5 & 9 & 7 & 6 & 4 & 0 \\
0 & 2 & 9 & 8 & 3 & 7 & 1 & 0 \\
0 & 8 & 1 & 6 & 4 & 2 & 2 & 0 \\
0 & 1 & 0 & 5 & 7 & 8 & 2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{bmatrix}.

Below we have provided code that demonstrates padding and convolution. As you will see when we run the code, the size of the image will remain unchanged when using padding.~

In [19]:

import numpy as np

def padding(image, kernel):
    # calculate r and c
    r = (kernel.shape[0] // 2) * 2
    c = (kernel.shape[1] // 2) * 2
    
    # padded image dimensions
    padded_height = image.shape[0] + r
    padded_width = image.shape[1] + c
    
    # for more readable code
    k_half_height = kernel.shape[0] // 2
    k_half_width = kernel.shape[1] // 2

    # zero matrix with padded dimensions
    padded_img = np.zeros((padded_height, padded_width))

    # place image into zero matrix
    padded_img[k_half_height : padded_height - k_half_height,
               k_half_width : padded_width - k_half_width] = image[:, :]

    return padded_img

def convolve(original_image, padded_image, kernel, stride=1):
    # rotate kernel by 180 degrees
    kernel = np.rot90(np.rot90(kernel))

    # note that kernel height // 2 is written as 'm'
    # and kernel width // 2 as 'n' in the mathematical notation
    m = kernel.shape[0] // 2
    n = kernel.shape[1] // 2
    
    r = (kernel.shape[0] // 2) * 2
    c = (kernel.shape[1] // 2) * 2
    
    # initialize output array
    convolved_image = np.zeros(original_image.shape)
    image_height = original_image.shape[0]
    image_width = original_image.shape[1]

    # the convolution
    for i in range(m, image_height + m, stride):
        for j in range(n, image_width + n, stride):
            convolved_image[i-m, j-n] = np.sum(
                padded_image[i : i + m, j : j + n]
                * kernel
            )
            
    return convolved_image

def convolve(image, kernel, stride=1):
    for i in range(2):
        kernel = np.rot90(kernel)

    k_half_height = kernel.shape[0] // 2
    k_half_width = kernel.shape[0] // 2

    conv_image = np.zeros(image.shape)
    pad_image = padding(image, kernel)

    for i in range(k_half_height, conv_image.shape[0] + k_half_height, stride):
        for j in range(k_half_width, conv_image.shape[1] + k_half_width, stride):
            conv_image[i - k_half_height, j - k_half_width] = np.sum(
                pad_image[
                    i - k_half_height : i + k_half_height + 1, j - k_half_width : j + k_half_width + 1
                ]
                * kernel
            )

    return conv_image

Fun fact: When filtering images, you will see that convolution involves rotating the kernel by 180 degrees. However, this is not the case when applying convolution in a CNN, where the same operation that is not rotated by 180 degrees is called cross-correlation, which is normally implemented in most libraries.

In [20]:

original_image = np.array([[4, 1, 2, 9, 8, 6],
                 [9, 5, 9, 5, 8, 5],
                 [1, 5, 9, 7, 6, 4],
                 [2, 9, 8, 3, 7, 1],
                 [8, 1, 6, 4, 2, 2],
                 [1, 0, 5, 7, 8, 2]])

kernel = (1/9)*np.ones((3,3))

print(f"{original_image.shape=}")

# note that convolve() performs padding
convolved_image = convolve(original_image, kernel, stride=1)

print(f"{convolved_image.shape=}")

As you can see, the resulting image is of the same size as the original image. To round of our demonstration of convolution, we will present the results of convolution using commonly used kernels. In a CNN, the values of the kernels are randomly initialized, and then learned during training. These kernels will extract information regarding the picture, such as for example the edge detection filter demonstrated below extracts the edges present in the picture. Of course, there is no guarantee that the CNN will learn an edge detection filter, but this should provide some intuiton as to how the CNN is able to use kernels to make better predictions than a regular feed forward neural network.

In [21]:

# Now an example using a real image and first a gaussian low-pass filter and then a Sobel filter
import numpy as np
import imageio.v3 as imageio
import matplotlib.pyplot as plt
import time

def generate_gauss_mask(sigma, K=1):
    side = np.ceil(1 + 8 * sigma)
    y, x = np.mgrid[-side // 2 + 1 : (side // 2) + 1, -side // 2 + 1 : (side // 2) + 1]
    ker_coef = K / (2 * np.pi * sigma**2)
    g = np.exp(-((x**2 + y**2) / (2.0 * sigma**2)))

    return g, ker_coef


img_path = "data/IMG-2167.JPG"
image_of_cute_dog = imageio.imread(img_path, mode='L')

plt.imshow(image_of_cute_dog, cmap="gray", vmin=0, vmax=255, aspect="auto")
plt.title("Original image")
plt.show()

gauss, kernel = generate_gauss_mask(sigma=6)
gauss_kernel = gauss*kernel

filtered_image = convolve(image_of_cute_dog, gauss_kernel)
plt.imshow(filtered_image, cmap="gray", vmin=0, vmax=255, aspect="auto")
plt.title("Result of convolution with gauss kernel (blurring filter)")
plt.show()

sobel_kernel = np.array([[1, 2, 1],
                    [0, 0, 0], 
                    [-1, -2, -1]])

filtered_image = convolve(image_of_cute_dog, sobel_kernel)

plt.imshow(filtered_image, cmap="gray", vmin=0, vmax=255, aspect="auto")
plt.title("Result of convolution with sobel kernel (edge detection filter)")
plt.show()

Layers

The code below initialises global variables for readability and describes the abstract class Layers. This is not important in order to understand the CNN, but is benefitial for organizing the code neatly.

In [22]:

import math
import autograd.numpy as np
from copy import deepcopy, copy
from autograd import grad
from typing import Callable

# global variables for index readability
input_index = 0
node_index = 1
bias_index = 1
input_channel_index = 1
feature_maps_index = 1
height_index = 2
width_index = 3
kernel_feature_maps_index = 1
kernel_input_channels_index = 0


class Layer:
    def __init__(self, seed):
        self.seed = seed

    def _feedforward(self):
        raise NotImplementedError

    def _backpropagate(self):
        raise NotImplementedError

    def _reset_weights(self, previous_nodes):
        raise NotImplementedError

Convolution2DLayer: convolution in a hidden layer

After establishing the foundational understanding of applying convolution to spatial data, let us delve into the intricate workings of a convolutional layer in a Convolutional Neural Network (CNN). The primary function of convolution, as previously discussed, is to extract pertinent information from images while simultaneously decreasing the scale of our data. To initiate the image processing, we shall begin by partitioning the images into color channels (unless the image is grayscale), comprising three primary colors: red, green, and blue. We will subsequently utilize trainable kernels to construct a higher-dimensional encoding of each channel called feature maps. Successive layers will receive these feature maps as inputs, generating further encodings, albeit with reduced dimensions. The term trainable kernels denotes the initialization of pre-defined kernel-shaped weights, which we will then train via backpropagation, similar to how weights are trained in a Feedforward Neural Network.

To ensure seamless integration between our implementation of the convolutional layer and popular machine learning frameworks like Tensorflow (Keras) and PyTorch, we have adopted a design pattern that mirrors the construction of models using these APIs. This involves implementing our convolutional layer as a Python class or object, which allows for a more modular and flexible approach to building neural networks. By structuring our code in this way, users can easily incorporate our implementation into their existing machine learning pipelines without having to make significant changes to their codebase. Additionally, this design pattern promotes code reusability and makes it easier to maintain and update our convolutional layer implementation over time.

Note that the Convolution2DLayer takes in an activation function as a parameter, as it also performs non-linearity.

In [23]:

class Convolution2DLayer(Layer):
    def __init__(
        self,
        input_channels,
        feature_maps,
        kernel_height,
        kernel_width,
        v_stride,
        h_stride,
        pad,
        act_func: Callable,
        seed=None,
        reset_weights_independently=True,
    ):
        super().__init__(seed)
        self.input_channels = input_channels
        self.feature_maps = feature_maps
        self.kernel_height = kernel_height
        self.kernel_width = kernel_width
        self.v_stride = v_stride
        self.h_stride = h_stride
        self.pad = pad
        self.act_func = act_func

        # such that the layer can be used on its own
        # outside of the CNN module
        if reset_weights_independently == True:
            self._reset_weights_independently()

    def _feedforward(self, X_batch):
        # note that the shape of X_batch = [inputs, input_maps, img_height, img_width]

        # pad the input batch
        X_batch_padded = self._padding(X_batch)

        # calculate height_index and width_index after stride
        strided_height = int(np.ceil(X_batch.shape[height_index] / self.v_stride))
        strided_width = int(np.ceil(X_batch.shape[width_index] / self.h_stride))

        # create output array
        output = np.ndarray(
            (
                X_batch.shape[input_index],
                self.feature_maps,
                strided_height,
                strided_width,
            )
        )

        # save input and output for backpropagation
        self.X_batch_feedforward = X_batch
        self.output_shape = output.shape

        # checking for errors, no need to look here :)
        self._check_for_errors()

        # convolve input with kernel
        for img in range(X_batch.shape[input_index]):
            for chin in range(self.input_channels):
                for fmap in range(self.feature_maps):
                    out_h = 0
                    for h in range(0, X_batch.shape[height_index], self.v_stride):
                        out_w = 0
                        for w in range(0, X_batch.shape[width_index], self.h_stride):
                            output[img, fmap, out_h, out_w] = np.sum(
                                X_batch_padded[
                                    img,
                                    chin,
                                    h : h + self.kernel_height,
                                    w : w + self.kernel_width,
                                ]
                                * self.kernel[chin, fmap, :, :]
                            )
                            out_w += 1
                        out_h += 1

        # Pay attention to the fact that we're not rotating the kernel by 180 degrees when filtering the image in
        # the convolutional layer, as convolution in terms of Machine Learning is a procedure known as cross-correlation
        # in image processing and signal processing

        # return a
        return self.act_func(output / (self.kernel_height))

    def _backpropagate(self, delta_term_next):
        # intiate matrices
        delta_term = np.zeros((self.X_batch_feedforward.shape))
        gradient_kernel = np.zeros((self.kernel.shape))

        # pad input for convolution
        X_batch_padded = self._padding(self.X_batch_feedforward)

        # Since an activation function is used at the output of the convolution layer, its derivative
        # has to be accounted for in the backpropagation -> as if ReLU was a layer on its own.
        act_derivative = derivate(self.act_func)
        delta_term_next = act_derivative(delta_term_next)

        # fill in 0's for values removed by vertical stride in feedforward
        if self.v_stride > 1:
            v_ind = 1
            for i in range(delta_term_next.shape[height_index]):
                for j in range(self.v_stride - 1):
                    delta_term_next = np.insert(
                        delta_term_next, v_ind, 0, axis=height_index
                    )
                v_ind += self.v_stride

        # fill in 0's for values removed by horizontal stride in feedforward
        if self.h_stride > 1:
            h_ind = 1
            for i in range(delta_term_next.shape[width_index]):
                for k in range(self.h_stride - 1):
                    delta_term_next = np.insert(
                        delta_term_next, h_ind, 0, axis=width_index
                    )
                h_ind += self.h_stride

        # crops out 0-rows and 0-columns
        delta_term_next = delta_term_next[
            :,
            :,
            : self.X_batch_feedforward.shape[height_index],
            : self.X_batch_feedforward.shape[width_index],
        ]

        # the gradient received from the next layer also needs to be padded
        delta_term_next = self._padding(delta_term_next)

        # calculate delta term by convolving next delta term with kernel
        for img in range(self.X_batch_feedforward.shape[input_index]):
            for chin in range(self.input_channels):
                for fmap in range(self.feature_maps):
                    for h in range(self.X_batch_feedforward.shape[height_index]):
                        for w in range(self.X_batch_feedforward.shape[width_index]):
                            delta_term[img, chin, h, w] = np.sum(
                                delta_term_next[
                                    img,
                                    fmap,
                                    h : h + self.kernel_height,
                                    w : w + self.kernel_width,
                                ]
                                * np.rot90(np.rot90(self.kernel[chin, fmap, :, :]))
                            )

        # calculate gradient for kernel for weight update
        # also via convolution
        for chin in range(self.input_channels):
            for fmap in range(self.feature_maps):
                for k_x in range(self.kernel_height):
                    for k_y in range(self.kernel_width):
                        gradient_kernel[chin, fmap, k_x, k_y] = np.sum(
                            X_batch_padded[
                                img,
                                chin,
                                h : h + self.kernel_height,
                                w : w + self.kernel_width,
                            ]
                            * delta_term_next[
                                img,
                                fmap,
                                h : h + self.kernel_height,
                                w : w + self.kernel_width,
                            ]
                        )
        # all kernels are updated with weight gradient of kernel
        self.kernel -= gradient_kernel

        # return delta term
        return delta_term

    def _padding(self, X_batch, batch_type="image"):

        # same padding for images
        if self.pad == "same" and batch_type == "image":
            padded_height = X_batch.shape[height_index] + (self.kernel_height // 2) * 2
            padded_width = X_batch.shape[width_index] + (self.kernel_width // 2) * 2
            half_kernel_height = self.kernel_height // 2
            half_kernel_width = self.kernel_width // 2

            # initialize padded array
            X_batch_padded = np.ndarray(
                (
                    X_batch.shape[input_index],
                    X_batch.shape[feature_maps_index],
                    padded_height,
                    padded_width,
                )
            )

            # zero pad all images in X_batch
            for img in range(X_batch.shape[input_index]):
                padded_img = np.zeros(
                    (X_batch.shape[feature_maps_index], padded_height, padded_width)
                )
                padded_img[
                    :,
                    half_kernel_height : padded_height - half_kernel_height,
                    half_kernel_width : padded_width - half_kernel_width,
                ] = X_batch[img, :, :, :]
                X_batch_padded[img, :, :, :] = padded_img[:, :, :]

            return X_batch_padded

        # same padding for gradients
        elif self.pad == "same" and batch_type == "grad":
            padded_height = X_batch.shape[height_index] + (self.kernel_height // 2) * 2
            padded_width = X_batch.shape[width_index] + (self.kernel_width // 2) * 2
            half_kernel_height = self.kernel_height // 2
            half_kernel_width = self.kernel_width // 2

            # initialize padded array
            delta_term_padded = np.zeros(
                (
                    X_batch.shape[input_index],
                    X_batch.shape[feature_maps_index],
                    padded_height,
                    padded_width,
                )
            )

            # zero pad delta term
            delta_term_padded[
                :, :, : X_batch.shape[height_index], : X_batch.shape[width_index]
            ] = X_batch[:, :, :, :]

            return delta_term_padded

        else:
            return X_batch

    def _reset_weights_independently(self):
        # sets seed to remove randomness inbetween runs
        if self.seed is not None:
            np.random.seed(self.seed)

        # initializes kernel matrix
        self.kernel = np.ndarray(
            (
                self.input_channels,
                self.feature_maps,
                self.kernel_height,
                self.kernel_width,
            )
        )

        # randomly initializes weights
        for chin in range(self.kernel.shape[kernel_input_channels_index]):
            for fmap in range(self.kernel.shape[kernel_feature_maps_index]):
                self.kernel[chin, fmap, :, :] = np.random.rand(
                    self.kernel_height, self.kernel_width
                )

    def _reset_weights(self, previous_nodes):
        # sets weights
        self._reset_weights_independently()

        # returns shape of output used for subsequent layer's weight initiation
        strided_height = int(
            np.ceil(previous_nodes.shape[height_index] / self.v_stride)
        )
        strided_width = int(np.ceil(previous_nodes.shape[width_index] / self.h_stride))
        next_nodes = np.ones(
            (
                previous_nodes.shape[input_index],
                self.feature_maps,
                strided_height,
                strided_width,
            )
        )
        return next_nodes / self.kernel_height

    def _check_for_errors(self):
        if self.X_batch_feedforward.shape[input_channel_index] != self.input_channels:
            raise AssertionError(
                f"ERROR: Number of input channels in data ({self.X_batch_feedforward.shape[input_channel_index]}) is not equal to input channels in Convolution2DLayerOPT ({self.input_channels})! Please change the number of input channels of the Convolution2DLayer such that they are equal"
            )

Backpropagation in the convolutional layer

As you may have noticed, we have not yet explained how the backpropagation algorithm works in a convolutional layer. However, having covered all other major details about convolutional layers, we are now prepared to do so. It should come as no surprise that the calculation of delta terms at each convolutional layer takes the form of convolution. After the gradient has been propagated backwards through the flattening layer, where it was reshaped into an appropriate form, calculating the update value for the kernel is simply a matter of convolving the output gradient with the input of the layer for which we are updating the weights. For more detail, this article serves as an excellent resource, see https://pavisj.medium.com/convolutions-and-backpropagations-46026a8f5d2c

Demonstration

We can use the convolutional layer above to perform a simple convolution on an image of the now familiar cute dog.

In [24]:

import numpy as np
import imageio.v3 as imageio
import matplotlib.pyplot as plt

def plot_convolution_result(X, layer):
    plt.imshow(X[0, 0, :, :], vmin=0, vmax=255, cmap="gray")
    plt.title("Original image")
    plt.colorbar()
    plt.show()
    conv_result = layer._feedforward(X)
    plt.title("Result of convolutional layer")
    plt.imshow(conv_result[0, 0, :, :], vmin=0, vmax=255, cmap="gray")
    plt.colorbar()
    plt.show()

# create layer
layer = Convolution2DLayer(
    input_channels=3,
    feature_maps=1,
    kernel_height=4,
    kernel_width=4,
    v_stride=2,
    h_stride=2,
    pad="same",
    act_func=identity,
    seed=2023,
    )

# read in image path, make data correct format
img_path = img_path = "data/IMG-2167.JPG"
image_of_cute_dog = imageio.imread(img_path)
image_shape = image_of_cute_dog.shape
image_of_cute_dog = image_of_cute_dog.reshape(1, image_shape[0], image_shape[1], image_shape[2])
image_of_cute_dog = image_of_cute_dog.transpose(0, 3, 1, 2)

# plot the result of the convolution
plot_convolution_result(image_of_cute_dog, layer)

We cobserve that the result has half the pixels on each axis due to the fact that we've used a horizontal and vertical stride of 2. The result of this convolution is not very insightfull, as the kernel has completely random values for the first feedforward pass. However, as we perform multiple forward and backward passes, the results of the convolution should provide identifying features of the image it uses for classification.

Note that image data usually comes in many different shapes and sizes, but for our CNN we require the input data be formatted as [Number of inputs, input channels, input height, input width]. Occasionally, the data you come accross use will be formatted like this, but on many occasions reshaping and transposing the dimensions is sadly necessary.

Pooling Layer

The pooling layer is another widely used type of layer in convolutional neural networks that enables data downsampling to a more manageable size. Despite recent technological advancements that allow for convolution without excessive size reduction of the data, the pooling layer still remains a fundamental component of convolutional neural networks. It can be used before, after, or in between convolutional layers, although finding the optimal placement of layers and network depth requires experimentation to achieve the best performance for a given problem. The code we provide allows you to perform two types of pooling known as max pooling and average pooling.

In [25]:

class Pooling2DLayer(Layer):
    def __init__(
        self,
        kernel_height,
        kernel_width,
        v_stride,
        h_stride,
        pooling="max",
        seed=None,
    ):
        super().__init__(seed)
        self.kernel_height = kernel_height
        self.kernel_width = kernel_width
        self.v_stride = v_stride
        self.h_stride = h_stride
        self.pooling = pooling

    def _feedforward(self, X_batch):
        # Saving the input for use in the backwardpass
        self.X_batch_feedforward = X_batch

        # check if user is silly
        self._check_for_errors()

        # Computing the size of the feature maps based on kernel size and the stride parameter
        strided_height = (
            X_batch.shape[height_index] - self.kernel_height
        ) // self.v_stride + 1
        if X_batch.shape[height_index] == X_batch.shape[width_index]:
            strided_width = strided_height
        else:
            strided_width = (
                X_batch.shape[width_index] - self.kernel_width
            ) // self.h_stride + 1

        # initialize output array
        output = np.ndarray(
            (
                X_batch.shape[input_index],
                X_batch.shape[feature_maps_index],
                strided_height,
                strided_width,
            )
        )

        # select pooling action, either max or average pooling
        if self.pooling == "max":
            self.pooling_action = np.max
        elif self.pooling == "average":
            self.pooling_action = np.mean

        # pool based on kernel size and stride
        for img in range(output.shape[input_index]):
            for fmap in range(output.shape[feature_maps_index]):
                for h in range(strided_height):
                    for w in range(strided_width):
                        output[img, fmap, h, w] = self.pooling_action(
                            X_batch[
                                img,
                                fmap,
                                (h * self.v_stride) : (h * self.v_stride)
                                + self.kernel_height,
                                (w * self.h_stride) : (w * self.h_stride)
                                + self.kernel_width,
                            ]
                        )

        # output for feedforward in next layer
        return output

    def _backpropagate(self, delta_term_next):
        # initiate delta term array
        delta_term = np.zeros((self.X_batch_feedforward.shape))

        for img in range(delta_term_next.shape[input_index]):
            for fmap in range(delta_term_next.shape[feature_maps_index]):
                for h in range(0, delta_term_next.shape[height_index], self.v_stride):
                    for w in range(
                        0, delta_term_next.shape[width_index], self.h_stride
                    ):
                        # max pooling
                        if self.pooling == "max":
                            # get window
                            window = self.X_batch_feedforward[
                                img,
                                fmap,
                                h : h + self.kernel_height,
                                w : w + self.kernel_width,
                            ]

                            # find max values indices in window
                            max_h, max_w = np.unravel_index(
                                window.argmax(), window.shape
                            )

                            # set values in new, upsampled delta term
                            delta_term[
                                img,
                                fmap,
                                (h + max_h),
                                (w + max_w),
                            ] += delta_term_next[img, fmap, h, w]

                        # average pooling
                        if self.pooling == "average":
                            delta_term[
                                img,
                                fmap,
                                h : h + self.kernel_height,
                                w : w + self.kernel_width,
                            ] = (
                                delta_term_next[img, fmap, h, w]
                                / self.kernel_height
                                / self.kernel_width
                            )
        # returns input to backpropagation in previous layer
        return delta_term

    def _reset_weights(self, previous_nodes):
        # calculate strided height, strided width
        strided_height = (
            previous_nodes.shape[height_index] - self.kernel_height
        ) // self.v_stride + 1
        if previous_nodes.shape[height_index] == previous_nodes.shape[width_index]:
            strided_width = strided_height
        else:
            strided_width = (
                previous_nodes.shape[width_index] - self.kernel_width
            ) // self.h_stride + 1

        # initiate output array
        output = np.ones(
            (
                previous_nodes.shape[input_index],
                previous_nodes.shape[feature_maps_index],
                strided_height,
                strided_width,
            )
        )

        # returns output with shape used for reset weights in next layer
        return output

    def _check_for_errors(self):
        # check if input is smaller than kernel size -> error
        assert (
            self.X_batch_feedforward.shape[width_index] >= self.kernel_width
        ), f"ERROR: Pooling kernel width_index ({self.kernel_width}) larger than data width_index ({self.X_batch_feedforward.input.shape[2]}), please lower the kernel width_index of the Pooling2DLayer"
        assert (
            self.X_batch_feedforward.shape[height_index] >= self.kernel_height
        ), f"ERROR: Pooling kernel height_index ({self.kernel_height}) larger than data height_index ({self.X_batch_feedforward.input.shape[3]}), please lower the kernel height_index of the Pooling2DLayer"

Flattening Layer

Before we can begin building our first CNN model, we need to introduce the flattening layer. As its name suggests, the flattening layer transforms the data into a one-dimensional vector that can be fed into the feedforward layers of our network. This layer plays a crucial role in preparing the data for further processing in the network. Additionally, the flattening layer is responsible for reshaping the gradient to the proper shape during backpropagation. This ensures that the kernels are correctly updated, allowing for effective learning in the network.

In [26]:

class FlattenLayer(Layer):
    def __init__(self, act_func=LRELU, seed=None):
        super().__init__(seed)
        self.act_func = act_func

    def _feedforward(self, X_batch):
        # save input for backpropagation
        self.X_batch_feedforward_shape = X_batch.shape
        # Remember, the data has the following shape: (I, FM, H, W, ) in the convolutional layers
        # whilst the data has the shape (I, FM * H * W) in the fully connected layers
        # I = Inputs, FM = Feature Maps, H = Height and W = Width.
        X_batch = X_batch.reshape(
            X_batch.shape[input_index],
            X_batch.shape[feature_maps_index]
            * X_batch.shape[height_index]
            * X_batch.shape[width_index],
        )

        # add bias to a
        self.z_matrix = X_batch
        bias = np.ones((X_batch.shape[input_index], 1)) * 0.01
        self.a_matrix = np.hstack([bias, X_batch])

        # return a, the input to feedforward in next layer
        return self.a_matrix

    def _backpropagate(self, weights_next, delta_term_next):
        activation_derivative = derivate(self.act_func)

        # calculate delta term
        delta_term = (
            weights_next[bias_index:, :] @ delta_term_next.T
        ).T * activation_derivative(self.z_matrix)

        # FlattenLayer does not update weights
        # reshapes delta layer to convolutional layer data format [Input, Feature_Maps, Height, Width]
        return delta_term.reshape(self.X_batch_feedforward_shape)

    def _reset_weights(self, previous_nodes):
        # note that the previous nodes to the FlattenLayer are from the convolutional layers
        previous_nodes = previous_nodes.reshape(
            previous_nodes.shape[input_index],
            previous_nodes.shape[feature_maps_index]
            * previous_nodes.shape[height_index]
            * previous_nodes.shape[width_index],
        )

        # return shape used in reset_weights in next layer
        return previous_nodes.shape[node_index]

    def get_prev_a(self):
        return self.a_matrix

Warning:

Output truncated. This notebook contains too many cells to display efficiently.