# Install the necessary dependencies
import os
import sys
!{sys.executable} -m pip install --quiet pandas scikit-learn numpy matplotlib jupyterlab_myst ipython tensorflow_addons opencv-python requests
8.7. Variational Autoencoder#
We’re trying to build a generative model here, not just a fuzzy data structure that can “memorize” images. We can’t generate anything yet, since we don’t know how to create latent vectors other than encoding them from images.
There’s a simple solution here. We add a constraint on the encoding network, that forces it to generate latent vectors that roughly follow a unit gaussian distribution. It is this constraint that separates a variational autoencoder from a standard one.
Generating new images is now easy: all we need to do is sample a latent vector from the unit gaussian and pass it into the decoder.
In practice, there’s a tradeoff between how accurate our network can be and how close its latent variables can match the unit gaussian distribution.
We let the network decide this itself. For our loss term, we sum up two separate losses: the generative loss, which is a mean squared error that measures how accurately the network reconstructed the images, and a latent loss, which is the KL divergence that measures how closely the latent variables match a unit gaussian. ref
8.7.1. Code#
import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
import tensorflow.keras as kr
%matplotlib inline
# MNIST Dataset parameters.
num_features = 784 # data features (img shape: 28*28).
# Training parameters.
batch_size = 128
epochs = 50
# Network Parameters
hidden_1 = 128 # 1st layer num features.
hidden_2 = 64 # 2nd layer num features (the latent dim).
from tensorflow.keras.datasets import mnist, fashion_mnist
def load_data(choice='mnist', labels=False):
if choice not in ['mnist', 'fashion_mnist']:
raise ('Choices are mnist and fashion_mnist')
if choice is 'mnist':
(X_train, y_train), (X_test, y_test) = mnist.load_data()
else:
(X_train, y_train), (X_test, y_test) = fashion_mnist.load_data()
X_train, X_test = X_train / 255., X_test / 255.
X_train, X_test = X_train.reshape([-1, 784]), X_test.reshape([-1, 784])
X_train = X_train.astype(np.float32, copy=False)
X_test = X_test.astype(np.float32, copy=False)
if labels:
return (X_train, y_train), (X_test, y_test)
return X_train, X_test
def plot_predictions(y_true, y_pred):
f, ax = plt.subplots(2, 10, figsize=(15, 4))
for i in range(10):
ax[0][i].imshow(np.reshape(y_true[i], (28, 28)), aspect='auto')
ax[1][i].imshow(np.reshape(y_pred[i], (28, 28)), aspect='auto')
plt.tight_layout()
<>:7: SyntaxWarning: "is" with a literal. Did you mean "=="?
<>:7: SyntaxWarning: "is" with a literal. Did you mean "=="?
<ipython-input-74-47545c9b4cc5>:7: SyntaxWarning: "is" with a literal. Did you mean "=="?
if choice is 'mnist':
def plot_digits(X, y, encoder, batch_size=128):
"""Plots labels and MNIST digits as function of 2D latent vector
Parameters:
----------
encoder: Model
A Keras Model instance
X: np.ndarray
Test data
y: np.ndarray
Test data labels
batch_size: int
Prediction batch size
"""
# display a 2D plot of the digit classes in the latent space
z_mean, _, _ = encoder.predict(X, batch_size=batch_size)
plt.figure(figsize=(12, 10))
plt.scatter(z_mean[:, 0], z_mean[:, 1], c=y)
plt.colorbar()
plt.xlabel("z[0] Latent Dimension")
plt.ylabel("z[1] Latent Dimension")
plt.show()
def generate_manifold(decoder):
"""Generates a manifold of MNIST digits from a random noisy data.
Parameters:
----------
decoder: Model
A Keras Model instance
"""
# display a 30x30 2D manifold of digits
n = 30
digit_size = 28
figure = np.zeros((digit_size * n, digit_size * n))
# linearly spaced coordinates corresponding to the 2D plot
# of digit classes in the latent space
grid_x = np.linspace(-4, 4, n)
grid_y = np.linspace(-4, 4, n)[::-1]
for i, yi in enumerate(grid_y):
for j, xi in enumerate(grid_x):
z_sample = np.array([[xi, yi]])
x_decoded = decoder.predict(z_sample)
digit = x_decoded[0].reshape(digit_size, digit_size)
figure[i * digit_size: (i + 1) * digit_size,
j * digit_size: (j + 1) * digit_size] = digit
plt.figure(figsize=(10, 10))
start_range = digit_size // 2
end_range = n * digit_size + start_range + 1
pixel_range = np.linspace(start_range, end_range, digit_size)
sample_range_x = np.linspace(start_range, end_range, digit_size)
sample_range_y = np.linspace(start_range, end_range, digit_size)
plt.xticks(pixel_range, sample_range_x)
plt.yticks(pixel_range, sample_range_y)
plt.xlabel("z[0] Latent Dimension")
plt.ylabel("z[1] Latent Dimension")
plt.imshow(figure, cmap='Greys_r')
plt.show()
'''
pixel_range = np.arange(start_range, end_range, digit_size)
sample_range_x = np.round(grid_x, 1)
sample_range_y = np.round(grid_y, 1)
plt.xticks(pixel_range, sample_range_x)
plt.yticks(pixel_range, sample_range_y)
plt.xlabel("z[0] Latent Dimension")
plt.ylabel("z[1] Latent Dimension")
plt.imshow(figure, cmap='Greys_r')
plt.show()
'''
'\n pixel_range = np.arange(start_range, end_range, digit_size)\n sample_range_x = np.round(grid_x, 1)\n sample_range_y = np.round(grid_y, 1)\n\n plt.xticks(pixel_range, sample_range_x)\n plt.yticks(pixel_range, sample_range_y)\n plt.xlabel("z[0] Latent Dimension")\n plt.ylabel("z[1] Latent Dimension")\n plt.imshow(figure, cmap=\'Greys_r\')\n plt.show()\n'
This code contains two functions, plot_digits and generate_manifold, for visualizing the performance of an autoencoder in the latent space. plot_digits function transforms test data into mean vectors in the latent space through an encoder and displays the distribution of these vectors on a two-dimensional plane, allowing observation of the distribution of different classes of data in the latent space. generate_manifold function generates a two-dimensional plane, maps it back to the data space, and generates corresponding MNIST digits for visualization, showing the digit features corresponding to different positions in the latent space. These functions help understand how the autoencoder learns data representations and effectively represents and generates data in the latent space.
def sampling(args):
"""Reparameterization trick. Instead of sampling from Q(z|X),
sample eps = N(0,I) z = z_mean + sqrt(var)*eps.
Parameters:
-----------
args: list of Tensors
Mean and log of variance of Q(z|X)
Returns
-------
z: Tensor
Sampled latent vector
"""
z_mean, z_log_var = args
eps = tf.random.normal(tf.shape(z_log_var), dtype=tf.float32, mean=0., stddev=1.0, name='epsilon')
z = z_mean + tf.exp(z_log_var / 2) * eps
return z
hidden_dim = 512
latent_dim = 2 # The bigger this is, more accurate the network is but 2 is for illustration purposes.
Encoder
inputs = kr.layers.Input(shape=(num_features, ), name='input')
x = kr.layers.Dense(hidden_dim, activation='relu')(inputs)
z_mean = kr.layers.Dense(latent_dim, name='z_mean')(x)
z_log_var = kr.layers.Dense(latent_dim, name='z_log_var')(x)
Sampling Layer
Use reparameterization trick to push the sampling out as input
z = kr.layers.Lambda(sampling, name='z')([z_mean, z_log_var])
# instantiate encoder model
encoder = kr.Model(inputs, [z_mean, z_log_var, z], name='encoder')
encoder.summary()
Model: "encoder"
__________________________________________________________________________________________________
Layer (type) Output Shape Param # Connected to
==================================================================================================
input (InputLayer) [(None, 784)] 0 []
dense_12 (Dense) (None, 512) 401920 ['input[0][0]']
z_mean (Dense) (None, 2) 1026 ['dense_12[0][0]']
z_log_var (Dense) (None, 2) 1026 ['dense_12[0][0]']
z (Lambda) (None, 2) 0 ['z_mean[0][0]',
'z_log_var[0][0]']
==================================================================================================
Total params: 403972 (1.54 MB)
Trainable params: 403972 (1.54 MB)
Non-trainable params: 0 (0.00 Byte)
__________________________________________________________________________________________________
Decoder
latent_inputs = kr.layers.Input(shape=(latent_dim,), name='z_sampling')
x = kr.layers.Dense(hidden_dim, activation='relu')(latent_inputs)
outputs = kr.layers.Dense(num_features, activation='sigmoid')(x)
# instantiate decoder model
decoder = kr.Model(latent_inputs, outputs, name='decoder')
decoder.summary()
Model: "decoder"
_________________________________________________________________
Layer (type) Output Shape Param #
=================================================================
z_sampling (InputLayer) [(None, 2)] 0
dense_13 (Dense) (None, 512) 1536
dense_14 (Dense) (None, 784) 402192
=================================================================
Total params: 403728 (1.54 MB)
Trainable params: 403728 (1.54 MB)
Non-trainable params: 0 (0.00 Byte)
_________________________________________________________________
# # VAE model = encoder + decoder
outputs = decoder(encoder(inputs)[2]) # Select the Z value from outputs of the encoder
vae = kr.Model(inputs, outputs, name='vae')
Define VAE Loss
# Reconstruction loss
reconstruction_loss = tf.losses.mean_squared_error(inputs, outputs)
reconstruction_loss = reconstruction_loss * num_features
# KL Divergence loss
kl_loss = 1 + z_log_var - tf.square(z_mean) - tf.exp(z_log_var)
kl_loss = -0.5 * tf.reduce_sum(kl_loss, axis=-1)
vae_loss = tf.reduce_mean(reconstruction_loss + kl_loss)
vae.add_loss(vae_loss)
vae.compile(optimizer='adam')
vae.summary()
Model: "vae"
__________________________________________________________________________________________________
Layer (type) Output Shape Param # Connected to
==================================================================================================
input (InputLayer) [(None, 784)] 0 []
encoder (Functional) [(None, 2), 403972 ['input[0][0]']
(None, 2),
(None, 2)]
decoder (Functional) (None, 784) 403728 ['encoder[0][2]']
dense_12 (Dense) (None, 512) 401920 ['input[0][0]']
z_log_var (Dense) (None, 2) 1026 ['dense_12[0][0]']
z_mean (Dense) (None, 2) 1026 ['dense_12[0][0]']
tf.__operators__.add_8 (TF (None, 2) 0 ['z_log_var[0][0]']
OpLambda)
tf.math.square_4 (TFOpLamb (None, 2) 0 ['z_mean[0][0]']
da)
tf.convert_to_tensor_4 (TF (None, 784) 0 ['decoder[0][0]']
OpLambda)
tf.cast_4 (TFOpLambda) (None, 784) 0 ['input[0][0]']
tf.math.subtract_8 (TFOpLa (None, 2) 0 ['tf.__operators__.add_8[0][0]
mbda) ',
'tf.math.square_4[0][0]']
tf.math.exp_4 (TFOpLambda) (None, 2) 0 ['z_log_var[0][0]']
tf.math.squared_difference (None, 784) 0 ['tf.convert_to_tensor_4[0][0]
_4 (TFOpLambda) ',
'tf.cast_4[0][0]']
tf.math.subtract_9 (TFOpLa (None, 2) 0 ['tf.math.subtract_8[0][0]',
mbda) 'tf.math.exp_4[0][0]']
tf.math.reduce_mean_8 (TFO (None,) 0 ['tf.math.squared_difference_4
pLambda) [0][0]']
tf.math.reduce_sum_4 (TFOp (None,) 0 ['tf.math.subtract_9[0][0]']
Lambda)
tf.math.multiply_8 (TFOpLa (None,) 0 ['tf.math.reduce_mean_8[0][0]'
mbda) ]
tf.math.multiply_9 (TFOpLa (None,) 0 ['tf.math.reduce_sum_4[0][0]']
mbda)
tf.__operators__.add_9 (TF (None,) 0 ['tf.math.multiply_8[0][0]',
OpLambda) 'tf.math.multiply_9[0][0]']
tf.math.reduce_mean_9 (TFO () 0 ['tf.__operators__.add_9[0][0]
pLambda) ']
add_loss_4 (AddLoss) () 0 ['tf.math.reduce_mean_9[0][0]'
]
==================================================================================================
Total params: 807700 (3.08 MB)
Trainable params: 807700 (3.08 MB)
Non-trainable params: 0 (0.00 Byte)
__________________________________________________________________________________________________
(X_train, _), (X_test, y) = load_data('mnist', labels=True)
vae.fit(X_train, epochs=epochs, batch_size=batch_size, validation_data=(X_test, None))
Epoch 1/50
469/469 [==============================] - 27s 54ms/step - loss: 52.2818 - val_loss: 44.4062
Epoch 2/50
469/469 [==============================] - 21s 44ms/step - loss: 43.5608 - val_loss: 42.5819
Epoch 3/50
469/469 [==============================] - 15s 32ms/step - loss: 42.2779 - val_loss: 41.6887
Epoch 4/50
469/469 [==============================] - 14s 30ms/step - loss: 41.5440 - val_loss: 41.1011
Epoch 5/50
469/469 [==============================] - 14s 30ms/step - loss: 41.0153 - val_loss: 40.6011
Epoch 6/50
469/469 [==============================] - 13s 28ms/step - loss: 40.6053 - val_loss: 40.2510
Epoch 7/50
469/469 [==============================] - 14s 29ms/step - loss: 40.2667 - val_loss: 40.0506
Epoch 8/50
469/469 [==============================] - 14s 29ms/step - loss: 40.0024 - val_loss: 39.8003
Epoch 9/50
469/469 [==============================] - 14s 30ms/step - loss: 39.7573 - val_loss: 39.4700
Epoch 10/50
469/469 [==============================] - 15s 33ms/step - loss: 39.5194 - val_loss: 39.3395
Epoch 11/50
469/469 [==============================] - 13s 28ms/step - loss: 39.3101 - val_loss: 39.2497
Epoch 12/50
469/469 [==============================] - 13s 28ms/step - loss: 39.1259 - val_loss: 39.0037
Epoch 13/50
469/469 [==============================] - 14s 30ms/step - loss: 38.9862 - val_loss: 38.8456
Epoch 14/50
469/469 [==============================] - 14s 29ms/step - loss: 38.8217 - val_loss: 38.7503
Epoch 15/50
469/469 [==============================] - 13s 28ms/step - loss: 38.6687 - val_loss: 38.7065
Epoch 16/50
469/469 [==============================] - 15s 32ms/step - loss: 38.5580 - val_loss: 38.5613
Epoch 17/50
469/469 [==============================] - 14s 30ms/step - loss: 38.4199 - val_loss: 38.5088
Epoch 18/50
469/469 [==============================] - 14s 30ms/step - loss: 38.3110 - val_loss: 38.4251
Epoch 19/50
469/469 [==============================] - 13s 28ms/step - loss: 38.2156 - val_loss: 38.2711
Epoch 20/50
469/469 [==============================] - 13s 28ms/step - loss: 38.1195 - val_loss: 38.2555
Epoch 21/50
469/469 [==============================] - 14s 29ms/step - loss: 38.0060 - val_loss: 38.1528
Epoch 22/50
469/469 [==============================] - 13s 28ms/step - loss: 37.9203 - val_loss: 38.1048
Epoch 23/50
469/469 [==============================] - 14s 29ms/step - loss: 37.8317 - val_loss: 37.9606
Epoch 24/50
469/469 [==============================] - 14s 29ms/step - loss: 37.7610 - val_loss: 38.0686
Epoch 25/50
469/469 [==============================] - 13s 28ms/step - loss: 37.6570 - val_loss: 37.8181
Epoch 26/50
469/469 [==============================] - 14s 29ms/step - loss: 37.5858 - val_loss: 37.7498
Epoch 27/50
469/469 [==============================] - 15s 31ms/step - loss: 37.4975 - val_loss: 37.8305
Epoch 28/50
469/469 [==============================] - 13s 28ms/step - loss: 37.4453 - val_loss: 37.8925
Epoch 29/50
469/469 [==============================] - 13s 28ms/step - loss: 37.3894 - val_loss: 37.7599
Epoch 30/50
469/469 [==============================] - 14s 29ms/step - loss: 37.2989 - val_loss: 37.7335
Epoch 31/50
469/469 [==============================] - 14s 29ms/step - loss: 37.2311 - val_loss: 37.5889
Epoch 32/50
469/469 [==============================] - 14s 29ms/step - loss: 37.1723 - val_loss: 37.6826
Epoch 33/50
469/469 [==============================] - 13s 28ms/step - loss: 37.1153 - val_loss: 37.5796
Epoch 34/50
469/469 [==============================] - 14s 30ms/step - loss: 37.0726 - val_loss: 37.4669
Epoch 35/50
469/469 [==============================] - 14s 31ms/step - loss: 37.0142 - val_loss: 37.5379
Epoch 36/50
469/469 [==============================] - 16s 34ms/step - loss: 36.9736 - val_loss: 37.4775
Epoch 37/50
469/469 [==============================] - 16s 34ms/step - loss: 36.9053 - val_loss: 37.3929
Epoch 38/50
469/469 [==============================] - 16s 34ms/step - loss: 36.8330 - val_loss: 37.4538
Epoch 39/50
469/469 [==============================] - 16s 34ms/step - loss: 36.8007 - val_loss: 37.4065
Epoch 40/50
469/469 [==============================] - 16s 34ms/step - loss: 36.7870 - val_loss: 37.3811
Epoch 41/50
469/469 [==============================] - 16s 35ms/step - loss: 36.7256 - val_loss: 37.4299
Epoch 42/50
469/469 [==============================] - 17s 37ms/step - loss: 36.6767 - val_loss: 37.3244
Epoch 43/50
469/469 [==============================] - 16s 34ms/step - loss: 36.6228 - val_loss: 37.3011
Epoch 44/50
469/469 [==============================] - 15s 33ms/step - loss: 36.6010 - val_loss: 37.3018
Epoch 45/50
469/469 [==============================] - 16s 34ms/step - loss: 36.5258 - val_loss: 37.2985
Epoch 46/50
469/469 [==============================] - 16s 34ms/step - loss: 36.5276 - val_loss: 37.2850
Epoch 47/50
469/469 [==============================] - 16s 34ms/step - loss: 36.4927 - val_loss: 37.2672
Epoch 48/50
469/469 [==============================] - 15s 33ms/step - loss: 36.4331 - val_loss: 37.2286
Epoch 49/50
469/469 [==============================] - 16s 34ms/step - loss: 36.4083 - val_loss: 37.1546
Epoch 50/50
469/469 [==============================] - 16s 34ms/step - loss: 36.4009 - val_loss: 37.1385
<keras.src.callbacks.History at 0x7e9e2f7f3970>
generate_manifold(decoder)
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plot_digits(X_test, y, encoder) # y for label coloring
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Looks like by using the decoder on noisy data we can generate a completely new data!
8.7.2. Your turn! 🚀#
Assignment - Denoising difussion model
8.7.3. Acknowledgments#
Thanks to Fazil T for creating the open-source project kaggle for creating the open-source courses Variantional Autoencoders(VAE). They inspire the majority of the content in this chapter.