# Chapter 5: Exploring Variational Autoencoders (VAEs)

## 5.5 Variations of VAEs (Beta-VAE, Conditional VAE)

Variational Autoencoders, or VAEs, have emerged as a groundbreaking and foundational framework in the world of machine learning. Numerous extensions and modifications have been developed from this base model, each with the goal of addressing specific challenges or enhancing certain aspects of the original VAE model.

Such continuous development and advancement in the field have made these models increasingly comprehensive and robust. In this section, we will delve into the specifics of two such popular variations: the Beta-VAE and the Conditional VAE.

These adaptations of the primary VAE model introduce an impressive degree of additional flexibility and control. This heightened level of adaptability further extends the range of applications for which VAE models can be utilized, making them an even more powerful tool in the field of machine learning and data analysis.

### 5.5.1 Beta-VAE

Beta-VAE is an innovative model that introduces a new hyperparameter, denoted as ( \beta ), to the objective function of a traditional Variational Autoencoder (VAE). This added element provides an improved level of control over the delicate balance between two key components of the function: the reconstruction loss and the Kullback-Leibler (KL) divergence.

The reconstruction loss pertains to the model's ability to recreate the input data, while the KL divergence measures the difference between the model's learned probability distribution and the true distribution.

By carefully adjusting the value of ( \beta ), the Beta-VAE model can more effectively encourage the learning of disentangled representations within the latent space. Disentangled representations can lead to improved interpretability and robustness in the model, making Beta-VAE a significant advancement in the field.

**Objective Function:**

Beta-VAE Loss=Reconstruction Loss+β×KL Divergence

A higher ( \beta ) places more emphasis on the KL divergence term, promoting disentanglement at the potential cost of reconstruction quality. Conversely, a lower ( \beta ) prioritizes reconstruction accuracy.

**Example: Beta-VAE Implementation**

`import tensorflow as tf`

from tensorflow.keras.layers import Input, Dense, Lambda, Layer

from tensorflow.keras.models import Model

from tensorflow.keras import backend as K

# Sampling layer using the reparameterization trick

class Sampling(Layer):

def call(self, inputs):

z_mean, z_log_var = inputs

batch = tf.shape(z_mean)[0]

dim = tf.shape(z_mean)[1]

epsilon = K.random_normal(shape=(batch, dim))

return z_mean + K.exp(0.5 * z_log_var) * epsilon

# Encoder network

def build_encoder(input_shape, latent_dim):

inputs = Input(shape=input_shape)

x = Dense(512, activation='relu')(inputs)

x = Dense(256, activation='relu')(x)

z_mean = Dense(latent_dim, name='z_mean')(x)

z_log_var = Dense(latent_dim, name='z_log_var')(x)

z = Sampling()([z_mean, z_log_var])

return Model(inputs, [z_mean, z_log_var, z], name='encoder')

# Decoder network

def build_decoder(latent_dim, output_shape):

latent_inputs = Input(shape=(latent_dim,))

x = Dense(256, activation='relu')(latent_inputs)

x = Dense(512, activation='relu')(x)

outputs = Dense(output_shape, activation='sigmoid')(x)

return Model(latent_inputs, outputs, name='decoder')

# Define the input shape and latent dimension

input_shape = (784,)

latent_dim = 2

# Build the encoder and decoder

encoder = build_encoder(input_shape, latent_dim)

decoder = build_decoder(latent_dim, input_shape[0])

# Define the Beta-VAE model

inputs = Input(shape=input_shape)

z_mean, z_log_var, z = encoder(inputs)

outputs = decoder(z)

beta_vae = Model(inputs, outputs, name='beta_vae')

# Define the Beta-VAE loss function

def beta_vae_loss(inputs, outputs, z_mean, z_log_var, beta=1.0):

reconstruction_loss = tf.keras.losses.binary_crossentropy(inputs, outputs)

reconstruction_loss *= input_shape[0]

kl_loss = 1 + z_log_var - K.square(z_mean) - K.exp(z_log_var)

kl_loss = K.sum(kl_loss, axis=-1)

kl_loss *= -0.5

return K.mean(reconstruction_loss + beta * kl_loss)

# Compile the Beta-VAE model

beta_vae.compile(optimizer='adam', loss=lambda x, y: beta_vae_loss(x, y, z_mean, z_log_var, beta=4.0))

# Train the Beta-VAE model

beta_vae.fit(x_train, x_train, epochs=50, batch_size=128, validation_data=(x_test, x_test))

In this example:

The script first imports necessary libraries and defines a Sampling layer, which is used for drawing random samples from the latent space using the reparameterization trick.

It then defines functions to build the encoder and decoder parts of the VAE, each of which is a deep neural network. The encoder transforms the input into a latent representation, and the decoder reconstructs the original input from the latent representation.

The input shape and latent dimension are then defined, and the encoder and decoder are built using these parameters.

The Beta-VAE model is then defined, connecting the encoder and decoder networks.

A custom loss function for the Beta-VAE is then defined, which includes both the reconstruction loss and the Kullback-Leibler (KL) divergence. The KL divergence measures how much the learned latent distribution deviates from the prior distribution. The 'beta' factor controls the balance between the reconstruction loss and the KL divergence.

Finally, the Beta-VAE model is compiled and trained using the defined loss function, an 'adam' optimizer, and training and testing data.

### 5.5.2 Conditional VAE (CVAE)

The Conditional Variational Autoencoder (CVAE) is an extension of the standard Variational Autoencoder (VAE), a popular generative model. The CVAE enhances the functionality of VAE by conditioning both the encoder, which is responsible for compressing the input data into a latent representation, and the decoder, which reconstructs the original data from this latent representation, on additional information such as class labels.

This additional conditioning allows the model to generate data that adheres to specific attributes. Therefore, if you're looking to generate data that follows a certain criterion or want to have more control over the characteristics of the generated data, the CVAE is particularly useful.

This makes it an excellent choice for tasks requiring controlled generation where you need to have a certain degree of influence over the output.

**Objective Function:**

CVAE Loss=E

q(z∣x,y)

[−logp(x∣z,y)]+D

KL

(q(z∣x,y)∥p(z∣y))

In this formulation, ( y ) represents the additional conditioning information (e.g., class labels).

**Example: Conditional VAE Implementation**

`# Encoder network for CVAE`

def build_cvae_encoder(input_shape, num_classes, latent_dim):

inputs = Input(shape=input_shape)

labels = Input(shape=(num_classes,))

x = Dense(512, activation='relu')(inputs)

x = tf.keras.layers.concatenate([x, labels])

x = Dense(256, activation='relu')(x)

z_mean = Dense(latent_dim, name='z_mean')(x)

z_log_var = Dense(latent_dim, name='z_log_var')(x)

z = Sampling()([z_mean, z_log_var])

return Model([inputs, labels], [z_mean, z_log_var, z], name='cvae_encoder')

# Decoder network for CVAE

def build_cvae_decoder(latent_dim, num_classes, output_shape):

latent_inputs = Input(shape=(latent_dim,))

labels = Input(shape=(num_classes,))

x = Dense(256, activation='relu')(latent_inputs)

x = tf.keras.layers.concatenate([x, labels])

x = Dense(512, activation='relu')(x)

outputs = Dense(output_shape, activation='sigmoid')(x)

return Model([latent_inputs, labels], outputs, name='cvae_decoder')

# Define the input shape, number of classes, and latent dimension

input_shape = (784,)

num_classes = 10

latent_dim = 2

# Build the encoder and decoder for CVAE

cvae_encoder = build_cvae_encoder(input_shape, num_classes, latent_dim)

cvae_decoder = build_cvae_decoder(latent_dim, num_classes, input_shape[0])

# Define the Conditional VAE model

inputs = Input(shape=input_shape)

labels = Input(shape=(num_classes,))

z_mean, z_log_var, z = cvae_encoder([inputs, labels])

outputs = cvae_decoder([z, labels])

cvae = Model([inputs, labels], outputs, name='cvae')

# Define the CVAE loss function

def cvae_loss(inputs, outputs, z_mean, z_log_var):

reconstruction_loss = tf.keras.losses.binary_crossentropy(inputs, outputs)

reconstruction_loss *= input_shape[0]

kl_loss = 1 + z_log_var - K.square(z_mean) - K.exp(z_log_var)

kl_loss = K.sum(kl_loss, axis=-1)

kl_loss *= -0.5

return K.mean(reconstruction_loss + kl_loss)

# Compile the CVAE model

cvae.compile(optimizer='adam', loss=lambda x, y: cvae_loss(x, y, z_mean, z_log_var))

# Prepare the labels for training

y_train = tf.keras.utils.to_categorical(y_train, num_classes)

y_test = tf.keras.utils.to_categorical(y_test, num_classes)

# Train the CVAE model

cvae.fit([x_train, y_train], x_train, epochs=50, batch_size=128, validation_data=([x_test, y_test], x_test))

In this example:

The script first begins by defining a function to build the encoder part of the CVAE. The encoder's role in a CVAE is to take in the input data and encode it into a lower-dimensional latent space. This is done using dense (fully connected) layers and the 'relu' activation function. The encoder function takes the input shape, the number of classes, and the latent dimension as arguments and returns a model that performs this encoding.

The script then defines a function to build the decoder part of the CVAE. The decoder's role is to take a point in the latent space and decode it back into the original data space. Like the encoder, the decoder is built using dense layers and the 'relu' activation function. It takes the latent dimension, the number of classes, and the output shape as arguments and returns a model that performs this decoding.

After defining the functions to build the encoder and decoder, the script then defines the specific parameters for this CVAE, including the input shape, the number of classes, and the latent dimension. It then uses these parameters and the previously defined functions to build the encoder and decoder.

The next part of the script defines the overall CVAE model. This is done by first defining input layers for the inputs and labels. These inputs and labels are then passed through the encoder to get the mean and log variance of the latent space and a sampled point in the latent space. This sampled point and the labels are then passed through the decoder to get the outputs. The CVAE model is then defined as taking the inputs and labels and outputting these outputs.

The script then defines a custom loss function for the CVAE. This loss function includes both a reconstruction loss (which measures how well the decoder can reconstruct the original input from the latent space) and a KL divergence (which measures how much the learned latent distribution deviates from the prior distribution). This loss function is then used to compile the CVAE model with the Adam optimizer.

The final part of the script prepares the labels for training by converting them to categorical format, and then trains the CVAE model using the training data, the prepared labels, and the previously defined loss function. The model is trained for 50 epochs with a batch size of 128, and the validation data is also provided for the model to evaluate its performance on unseen data.

**Summary**

Variations of VAEs, such as Beta-VAE and Conditional VAE, extend the capabilities of standard VAEs by introducing additional flexibility and control. Beta-VAE incorporates a hyperparameter ( \beta ) to balance the trade-off between reconstruction loss and KL divergence, encouraging disentangled representations. Conditional VAE (CVAE) allows for controlled data generation by conditioning the model on additional information, such as class labels.

By implementing and experimenting with these variations, you can tailor VAEs to better suit specific tasks and applications, enhancing the model's ability to learn meaningful latent representations and generate high-quality data. This comprehensive understanding of VAE variations opens up new possibilities for research and practical applications in generative modeling.

## 5.5 Variations of VAEs (Beta-VAE, Conditional VAE)

Variational Autoencoders, or VAEs, have emerged as a groundbreaking and foundational framework in the world of machine learning. Numerous extensions and modifications have been developed from this base model, each with the goal of addressing specific challenges or enhancing certain aspects of the original VAE model.

Such continuous development and advancement in the field have made these models increasingly comprehensive and robust. In this section, we will delve into the specifics of two such popular variations: the Beta-VAE and the Conditional VAE.

These adaptations of the primary VAE model introduce an impressive degree of additional flexibility and control. This heightened level of adaptability further extends the range of applications for which VAE models can be utilized, making them an even more powerful tool in the field of machine learning and data analysis.

### 5.5.1 Beta-VAE

Beta-VAE is an innovative model that introduces a new hyperparameter, denoted as ( \beta ), to the objective function of a traditional Variational Autoencoder (VAE). This added element provides an improved level of control over the delicate balance between two key components of the function: the reconstruction loss and the Kullback-Leibler (KL) divergence.

The reconstruction loss pertains to the model's ability to recreate the input data, while the KL divergence measures the difference between the model's learned probability distribution and the true distribution.

By carefully adjusting the value of ( \beta ), the Beta-VAE model can more effectively encourage the learning of disentangled representations within the latent space. Disentangled representations can lead to improved interpretability and robustness in the model, making Beta-VAE a significant advancement in the field.

**Objective Function:**

Beta-VAE Loss=Reconstruction Loss+β×KL Divergence

A higher ( \beta ) places more emphasis on the KL divergence term, promoting disentanglement at the potential cost of reconstruction quality. Conversely, a lower ( \beta ) prioritizes reconstruction accuracy.

**Example: Beta-VAE Implementation**

`import tensorflow as tf`

from tensorflow.keras.layers import Input, Dense, Lambda, Layer

from tensorflow.keras.models import Model

from tensorflow.keras import backend as K

# Sampling layer using the reparameterization trick

class Sampling(Layer):

def call(self, inputs):

z_mean, z_log_var = inputs

batch = tf.shape(z_mean)[0]

dim = tf.shape(z_mean)[1]

epsilon = K.random_normal(shape=(batch, dim))

return z_mean + K.exp(0.5 * z_log_var) * epsilon

# Encoder network

def build_encoder(input_shape, latent_dim):

inputs = Input(shape=input_shape)

x = Dense(512, activation='relu')(inputs)

x = Dense(256, activation='relu')(x)

z_mean = Dense(latent_dim, name='z_mean')(x)

z_log_var = Dense(latent_dim, name='z_log_var')(x)

z = Sampling()([z_mean, z_log_var])

return Model(inputs, [z_mean, z_log_var, z], name='encoder')

# Decoder network

def build_decoder(latent_dim, output_shape):

latent_inputs = Input(shape=(latent_dim,))

x = Dense(256, activation='relu')(latent_inputs)

x = Dense(512, activation='relu')(x)

outputs = Dense(output_shape, activation='sigmoid')(x)

return Model(latent_inputs, outputs, name='decoder')

# Define the input shape and latent dimension

input_shape = (784,)

latent_dim = 2

# Build the encoder and decoder

encoder = build_encoder(input_shape, latent_dim)

decoder = build_decoder(latent_dim, input_shape[0])

# Define the Beta-VAE model

inputs = Input(shape=input_shape)

z_mean, z_log_var, z = encoder(inputs)

outputs = decoder(z)

beta_vae = Model(inputs, outputs, name='beta_vae')

# Define the Beta-VAE loss function

def beta_vae_loss(inputs, outputs, z_mean, z_log_var, beta=1.0):

reconstruction_loss = tf.keras.losses.binary_crossentropy(inputs, outputs)

reconstruction_loss *= input_shape[0]

kl_loss = 1 + z_log_var - K.square(z_mean) - K.exp(z_log_var)

kl_loss = K.sum(kl_loss, axis=-1)

kl_loss *= -0.5

return K.mean(reconstruction_loss + beta * kl_loss)

# Compile the Beta-VAE model

beta_vae.compile(optimizer='adam', loss=lambda x, y: beta_vae_loss(x, y, z_mean, z_log_var, beta=4.0))

# Train the Beta-VAE model

beta_vae.fit(x_train, x_train, epochs=50, batch_size=128, validation_data=(x_test, x_test))

In this example:

The script first imports necessary libraries and defines a Sampling layer, which is used for drawing random samples from the latent space using the reparameterization trick.

It then defines functions to build the encoder and decoder parts of the VAE, each of which is a deep neural network. The encoder transforms the input into a latent representation, and the decoder reconstructs the original input from the latent representation.

The input shape and latent dimension are then defined, and the encoder and decoder are built using these parameters.

The Beta-VAE model is then defined, connecting the encoder and decoder networks.

A custom loss function for the Beta-VAE is then defined, which includes both the reconstruction loss and the Kullback-Leibler (KL) divergence. The KL divergence measures how much the learned latent distribution deviates from the prior distribution. The 'beta' factor controls the balance between the reconstruction loss and the KL divergence.

Finally, the Beta-VAE model is compiled and trained using the defined loss function, an 'adam' optimizer, and training and testing data.

### 5.5.2 Conditional VAE (CVAE)

The Conditional Variational Autoencoder (CVAE) is an extension of the standard Variational Autoencoder (VAE), a popular generative model. The CVAE enhances the functionality of VAE by conditioning both the encoder, which is responsible for compressing the input data into a latent representation, and the decoder, which reconstructs the original data from this latent representation, on additional information such as class labels.

This additional conditioning allows the model to generate data that adheres to specific attributes. Therefore, if you're looking to generate data that follows a certain criterion or want to have more control over the characteristics of the generated data, the CVAE is particularly useful.

This makes it an excellent choice for tasks requiring controlled generation where you need to have a certain degree of influence over the output.

**Objective Function:**

CVAE Loss=E

q(z∣x,y)

[−logp(x∣z,y)]+D

KL

(q(z∣x,y)∥p(z∣y))

In this formulation, ( y ) represents the additional conditioning information (e.g., class labels).

**Example: Conditional VAE Implementation**

`# Encoder network for CVAE`

def build_cvae_encoder(input_shape, num_classes, latent_dim):

inputs = Input(shape=input_shape)

labels = Input(shape=(num_classes,))

x = Dense(512, activation='relu')(inputs)

x = tf.keras.layers.concatenate([x, labels])

x = Dense(256, activation='relu')(x)

z_mean = Dense(latent_dim, name='z_mean')(x)

z_log_var = Dense(latent_dim, name='z_log_var')(x)

z = Sampling()([z_mean, z_log_var])

return Model([inputs, labels], [z_mean, z_log_var, z], name='cvae_encoder')

# Decoder network for CVAE

def build_cvae_decoder(latent_dim, num_classes, output_shape):

latent_inputs = Input(shape=(latent_dim,))

labels = Input(shape=(num_classes,))

x = Dense(256, activation='relu')(latent_inputs)

x = tf.keras.layers.concatenate([x, labels])

x = Dense(512, activation='relu')(x)

outputs = Dense(output_shape, activation='sigmoid')(x)

return Model([latent_inputs, labels], outputs, name='cvae_decoder')

# Define the input shape, number of classes, and latent dimension

input_shape = (784,)

num_classes = 10

latent_dim = 2

# Build the encoder and decoder for CVAE

cvae_encoder = build_cvae_encoder(input_shape, num_classes, latent_dim)

cvae_decoder = build_cvae_decoder(latent_dim, num_classes, input_shape[0])

# Define the Conditional VAE model

inputs = Input(shape=input_shape)

labels = Input(shape=(num_classes,))

z_mean, z_log_var, z = cvae_encoder([inputs, labels])

outputs = cvae_decoder([z, labels])

cvae = Model([inputs, labels], outputs, name='cvae')

# Define the CVAE loss function

def cvae_loss(inputs, outputs, z_mean, z_log_var):

reconstruction_loss = tf.keras.losses.binary_crossentropy(inputs, outputs)

reconstruction_loss *= input_shape[0]

kl_loss = 1 + z_log_var - K.square(z_mean) - K.exp(z_log_var)

kl_loss = K.sum(kl_loss, axis=-1)

kl_loss *= -0.5

return K.mean(reconstruction_loss + kl_loss)

# Compile the CVAE model

cvae.compile(optimizer='adam', loss=lambda x, y: cvae_loss(x, y, z_mean, z_log_var))

# Prepare the labels for training

y_train = tf.keras.utils.to_categorical(y_train, num_classes)

y_test = tf.keras.utils.to_categorical(y_test, num_classes)

# Train the CVAE model

cvae.fit([x_train, y_train], x_train, epochs=50, batch_size=128, validation_data=([x_test, y_test], x_test))

In this example:

The script first begins by defining a function to build the encoder part of the CVAE. The encoder's role in a CVAE is to take in the input data and encode it into a lower-dimensional latent space. This is done using dense (fully connected) layers and the 'relu' activation function. The encoder function takes the input shape, the number of classes, and the latent dimension as arguments and returns a model that performs this encoding.

The script then defines a function to build the decoder part of the CVAE. The decoder's role is to take a point in the latent space and decode it back into the original data space. Like the encoder, the decoder is built using dense layers and the 'relu' activation function. It takes the latent dimension, the number of classes, and the output shape as arguments and returns a model that performs this decoding.

After defining the functions to build the encoder and decoder, the script then defines the specific parameters for this CVAE, including the input shape, the number of classes, and the latent dimension. It then uses these parameters and the previously defined functions to build the encoder and decoder.

The next part of the script defines the overall CVAE model. This is done by first defining input layers for the inputs and labels. These inputs and labels are then passed through the encoder to get the mean and log variance of the latent space and a sampled point in the latent space. This sampled point and the labels are then passed through the decoder to get the outputs. The CVAE model is then defined as taking the inputs and labels and outputting these outputs.

The script then defines a custom loss function for the CVAE. This loss function includes both a reconstruction loss (which measures how well the decoder can reconstruct the original input from the latent space) and a KL divergence (which measures how much the learned latent distribution deviates from the prior distribution). This loss function is then used to compile the CVAE model with the Adam optimizer.

The final part of the script prepares the labels for training by converting them to categorical format, and then trains the CVAE model using the training data, the prepared labels, and the previously defined loss function. The model is trained for 50 epochs with a batch size of 128, and the validation data is also provided for the model to evaluate its performance on unseen data.

**Summary**

Variations of VAEs, such as Beta-VAE and Conditional VAE, extend the capabilities of standard VAEs by introducing additional flexibility and control. Beta-VAE incorporates a hyperparameter ( \beta ) to balance the trade-off between reconstruction loss and KL divergence, encouraging disentangled representations. Conditional VAE (CVAE) allows for controlled data generation by conditioning the model on additional information, such as class labels.

By implementing and experimenting with these variations, you can tailor VAEs to better suit specific tasks and applications, enhancing the model's ability to learn meaningful latent representations and generate high-quality data. This comprehensive understanding of VAE variations opens up new possibilities for research and practical applications in generative modeling.

## 5.5 Variations of VAEs (Beta-VAE, Conditional VAE)

Variational Autoencoders, or VAEs, have emerged as a groundbreaking and foundational framework in the world of machine learning. Numerous extensions and modifications have been developed from this base model, each with the goal of addressing specific challenges or enhancing certain aspects of the original VAE model.

Such continuous development and advancement in the field have made these models increasingly comprehensive and robust. In this section, we will delve into the specifics of two such popular variations: the Beta-VAE and the Conditional VAE.

These adaptations of the primary VAE model introduce an impressive degree of additional flexibility and control. This heightened level of adaptability further extends the range of applications for which VAE models can be utilized, making them an even more powerful tool in the field of machine learning and data analysis.

### 5.5.1 Beta-VAE

Beta-VAE is an innovative model that introduces a new hyperparameter, denoted as ( \beta ), to the objective function of a traditional Variational Autoencoder (VAE). This added element provides an improved level of control over the delicate balance between two key components of the function: the reconstruction loss and the Kullback-Leibler (KL) divergence.

The reconstruction loss pertains to the model's ability to recreate the input data, while the KL divergence measures the difference between the model's learned probability distribution and the true distribution.

By carefully adjusting the value of ( \beta ), the Beta-VAE model can more effectively encourage the learning of disentangled representations within the latent space. Disentangled representations can lead to improved interpretability and robustness in the model, making Beta-VAE a significant advancement in the field.

**Objective Function:**

Beta-VAE Loss=Reconstruction Loss+β×KL Divergence

A higher ( \beta ) places more emphasis on the KL divergence term, promoting disentanglement at the potential cost of reconstruction quality. Conversely, a lower ( \beta ) prioritizes reconstruction accuracy.

**Example: Beta-VAE Implementation**

`import tensorflow as tf`

from tensorflow.keras.layers import Input, Dense, Lambda, Layer

from tensorflow.keras.models import Model

from tensorflow.keras import backend as K

# Sampling layer using the reparameterization trick

class Sampling(Layer):

def call(self, inputs):

z_mean, z_log_var = inputs

batch = tf.shape(z_mean)[0]

dim = tf.shape(z_mean)[1]

epsilon = K.random_normal(shape=(batch, dim))

return z_mean + K.exp(0.5 * z_log_var) * epsilon

# Encoder network

def build_encoder(input_shape, latent_dim):

inputs = Input(shape=input_shape)

x = Dense(512, activation='relu')(inputs)

x = Dense(256, activation='relu')(x)

z_mean = Dense(latent_dim, name='z_mean')(x)

z_log_var = Dense(latent_dim, name='z_log_var')(x)

z = Sampling()([z_mean, z_log_var])

return Model(inputs, [z_mean, z_log_var, z], name='encoder')

# Decoder network

def build_decoder(latent_dim, output_shape):

latent_inputs = Input(shape=(latent_dim,))

x = Dense(256, activation='relu')(latent_inputs)

x = Dense(512, activation='relu')(x)

outputs = Dense(output_shape, activation='sigmoid')(x)

return Model(latent_inputs, outputs, name='decoder')

# Define the input shape and latent dimension

input_shape = (784,)

latent_dim = 2

# Build the encoder and decoder

encoder = build_encoder(input_shape, latent_dim)

decoder = build_decoder(latent_dim, input_shape[0])

# Define the Beta-VAE model

inputs = Input(shape=input_shape)

z_mean, z_log_var, z = encoder(inputs)

outputs = decoder(z)

beta_vae = Model(inputs, outputs, name='beta_vae')

# Define the Beta-VAE loss function

def beta_vae_loss(inputs, outputs, z_mean, z_log_var, beta=1.0):

reconstruction_loss = tf.keras.losses.binary_crossentropy(inputs, outputs)

reconstruction_loss *= input_shape[0]

kl_loss = 1 + z_log_var - K.square(z_mean) - K.exp(z_log_var)

kl_loss = K.sum(kl_loss, axis=-1)

kl_loss *= -0.5

return K.mean(reconstruction_loss + beta * kl_loss)

# Compile the Beta-VAE model

beta_vae.compile(optimizer='adam', loss=lambda x, y: beta_vae_loss(x, y, z_mean, z_log_var, beta=4.0))

# Train the Beta-VAE model

beta_vae.fit(x_train, x_train, epochs=50, batch_size=128, validation_data=(x_test, x_test))

In this example:

The script first imports necessary libraries and defines a Sampling layer, which is used for drawing random samples from the latent space using the reparameterization trick.

It then defines functions to build the encoder and decoder parts of the VAE, each of which is a deep neural network. The encoder transforms the input into a latent representation, and the decoder reconstructs the original input from the latent representation.

The input shape and latent dimension are then defined, and the encoder and decoder are built using these parameters.

The Beta-VAE model is then defined, connecting the encoder and decoder networks.

A custom loss function for the Beta-VAE is then defined, which includes both the reconstruction loss and the Kullback-Leibler (KL) divergence. The KL divergence measures how much the learned latent distribution deviates from the prior distribution. The 'beta' factor controls the balance between the reconstruction loss and the KL divergence.

Finally, the Beta-VAE model is compiled and trained using the defined loss function, an 'adam' optimizer, and training and testing data.

### 5.5.2 Conditional VAE (CVAE)

The Conditional Variational Autoencoder (CVAE) is an extension of the standard Variational Autoencoder (VAE), a popular generative model. The CVAE enhances the functionality of VAE by conditioning both the encoder, which is responsible for compressing the input data into a latent representation, and the decoder, which reconstructs the original data from this latent representation, on additional information such as class labels.

This additional conditioning allows the model to generate data that adheres to specific attributes. Therefore, if you're looking to generate data that follows a certain criterion or want to have more control over the characteristics of the generated data, the CVAE is particularly useful.

This makes it an excellent choice for tasks requiring controlled generation where you need to have a certain degree of influence over the output.

**Objective Function:**

CVAE Loss=E

q(z∣x,y)

[−logp(x∣z,y)]+D

KL

(q(z∣x,y)∥p(z∣y))

In this formulation, ( y ) represents the additional conditioning information (e.g., class labels).

**Example: Conditional VAE Implementation**

`# Encoder network for CVAE`

def build_cvae_encoder(input_shape, num_classes, latent_dim):

inputs = Input(shape=input_shape)

labels = Input(shape=(num_classes,))

x = Dense(512, activation='relu')(inputs)

x = tf.keras.layers.concatenate([x, labels])

x = Dense(256, activation='relu')(x)

z_mean = Dense(latent_dim, name='z_mean')(x)

z_log_var = Dense(latent_dim, name='z_log_var')(x)

z = Sampling()([z_mean, z_log_var])

return Model([inputs, labels], [z_mean, z_log_var, z], name='cvae_encoder')

# Decoder network for CVAE

def build_cvae_decoder(latent_dim, num_classes, output_shape):

latent_inputs = Input(shape=(latent_dim,))

labels = Input(shape=(num_classes,))

x = Dense(256, activation='relu')(latent_inputs)

x = tf.keras.layers.concatenate([x, labels])

x = Dense(512, activation='relu')(x)

outputs = Dense(output_shape, activation='sigmoid')(x)

return Model([latent_inputs, labels], outputs, name='cvae_decoder')

# Define the input shape, number of classes, and latent dimension

input_shape = (784,)

num_classes = 10

latent_dim = 2

# Build the encoder and decoder for CVAE

cvae_encoder = build_cvae_encoder(input_shape, num_classes, latent_dim)

cvae_decoder = build_cvae_decoder(latent_dim, num_classes, input_shape[0])

# Define the Conditional VAE model

inputs = Input(shape=input_shape)

labels = Input(shape=(num_classes,))

z_mean, z_log_var, z = cvae_encoder([inputs, labels])

outputs = cvae_decoder([z, labels])

cvae = Model([inputs, labels], outputs, name='cvae')

# Define the CVAE loss function

def cvae_loss(inputs, outputs, z_mean, z_log_var):

reconstruction_loss = tf.keras.losses.binary_crossentropy(inputs, outputs)

reconstruction_loss *= input_shape[0]

kl_loss = 1 + z_log_var - K.square(z_mean) - K.exp(z_log_var)

kl_loss = K.sum(kl_loss, axis=-1)

kl_loss *= -0.5

return K.mean(reconstruction_loss + kl_loss)

# Compile the CVAE model

cvae.compile(optimizer='adam', loss=lambda x, y: cvae_loss(x, y, z_mean, z_log_var))

# Prepare the labels for training

y_train = tf.keras.utils.to_categorical(y_train, num_classes)

y_test = tf.keras.utils.to_categorical(y_test, num_classes)

# Train the CVAE model

cvae.fit([x_train, y_train], x_train, epochs=50, batch_size=128, validation_data=([x_test, y_test], x_test))

In this example:

The script first begins by defining a function to build the encoder part of the CVAE. The encoder's role in a CVAE is to take in the input data and encode it into a lower-dimensional latent space. This is done using dense (fully connected) layers and the 'relu' activation function. The encoder function takes the input shape, the number of classes, and the latent dimension as arguments and returns a model that performs this encoding.

The script then defines a function to build the decoder part of the CVAE. The decoder's role is to take a point in the latent space and decode it back into the original data space. Like the encoder, the decoder is built using dense layers and the 'relu' activation function. It takes the latent dimension, the number of classes, and the output shape as arguments and returns a model that performs this decoding.

After defining the functions to build the encoder and decoder, the script then defines the specific parameters for this CVAE, including the input shape, the number of classes, and the latent dimension. It then uses these parameters and the previously defined functions to build the encoder and decoder.

The next part of the script defines the overall CVAE model. This is done by first defining input layers for the inputs and labels. These inputs and labels are then passed through the encoder to get the mean and log variance of the latent space and a sampled point in the latent space. This sampled point and the labels are then passed through the decoder to get the outputs. The CVAE model is then defined as taking the inputs and labels and outputting these outputs.

The script then defines a custom loss function for the CVAE. This loss function includes both a reconstruction loss (which measures how well the decoder can reconstruct the original input from the latent space) and a KL divergence (which measures how much the learned latent distribution deviates from the prior distribution). This loss function is then used to compile the CVAE model with the Adam optimizer.

The final part of the script prepares the labels for training by converting them to categorical format, and then trains the CVAE model using the training data, the prepared labels, and the previously defined loss function. The model is trained for 50 epochs with a batch size of 128, and the validation data is also provided for the model to evaluate its performance on unseen data.

**Summary**

Variations of VAEs, such as Beta-VAE and Conditional VAE, extend the capabilities of standard VAEs by introducing additional flexibility and control. Beta-VAE incorporates a hyperparameter ( \beta ) to balance the trade-off between reconstruction loss and KL divergence, encouraging disentangled representations. Conditional VAE (CVAE) allows for controlled data generation by conditioning the model on additional information, such as class labels.

By implementing and experimenting with these variations, you can tailor VAEs to better suit specific tasks and applications, enhancing the model's ability to learn meaningful latent representations and generate high-quality data. This comprehensive understanding of VAE variations opens up new possibilities for research and practical applications in generative modeling.

## 5.5 Variations of VAEs (Beta-VAE, Conditional VAE)

### 5.5.1 Beta-VAE

**Objective Function:**

Beta-VAE Loss=Reconstruction Loss+β×KL Divergence

**Example: Beta-VAE Implementation**

`import tensorflow as tf`

from tensorflow.keras.layers import Input, Dense, Lambda, Layer

from tensorflow.keras.models import Model

from tensorflow.keras import backend as K

# Sampling layer using the reparameterization trick

class Sampling(Layer):

def call(self, inputs):

z_mean, z_log_var = inputs

batch = tf.shape(z_mean)[0]

dim = tf.shape(z_mean)[1]

epsilon = K.random_normal(shape=(batch, dim))

return z_mean + K.exp(0.5 * z_log_var) * epsilon

# Encoder network

def build_encoder(input_shape, latent_dim):

inputs = Input(shape=input_shape)

x = Dense(512, activation='relu')(inputs)

x = Dense(256, activation='relu')(x)

z_mean = Dense(latent_dim, name='z_mean')(x)

z_log_var = Dense(latent_dim, name='z_log_var')(x)

z = Sampling()([z_mean, z_log_var])

return Model(inputs, [z_mean, z_log_var, z], name='encoder')

# Decoder network

def build_decoder(latent_dim, output_shape):

latent_inputs = Input(shape=(latent_dim,))

x = Dense(256, activation='relu')(latent_inputs)

x = Dense(512, activation='relu')(x)

outputs = Dense(output_shape, activation='sigmoid')(x)

return Model(latent_inputs, outputs, name='decoder')

# Define the input shape and latent dimension

input_shape = (784,)

latent_dim = 2

# Build the encoder and decoder

encoder = build_encoder(input_shape, latent_dim)

decoder = build_decoder(latent_dim, input_shape[0])

# Define the Beta-VAE model

inputs = Input(shape=input_shape)

z_mean, z_log_var, z = encoder(inputs)

outputs = decoder(z)

beta_vae = Model(inputs, outputs, name='beta_vae')

# Define the Beta-VAE loss function

def beta_vae_loss(inputs, outputs, z_mean, z_log_var, beta=1.0):

reconstruction_loss = tf.keras.losses.binary_crossentropy(inputs, outputs)

reconstruction_loss *= input_shape[0]

kl_loss = 1 + z_log_var - K.square(z_mean) - K.exp(z_log_var)

kl_loss = K.sum(kl_loss, axis=-1)

kl_loss *= -0.5

return K.mean(reconstruction_loss + beta * kl_loss)

# Compile the Beta-VAE model

beta_vae.compile(optimizer='adam', loss=lambda x, y: beta_vae_loss(x, y, z_mean, z_log_var, beta=4.0))

# Train the Beta-VAE model

beta_vae.fit(x_train, x_train, epochs=50, batch_size=128, validation_data=(x_test, x_test))

In this example:

The Beta-VAE model is then defined, connecting the encoder and decoder networks.

### 5.5.2 Conditional VAE (CVAE)

**Objective Function:**

CVAE Loss=E

q(z∣x,y)

[−logp(x∣z,y)]+D

KL

(q(z∣x,y)∥p(z∣y))

In this formulation, ( y ) represents the additional conditioning information (e.g., class labels).

**Example: Conditional VAE Implementation**

`# Encoder network for CVAE`

def build_cvae_encoder(input_shape, num_classes, latent_dim):

inputs = Input(shape=input_shape)

labels = Input(shape=(num_classes,))

x = Dense(512, activation='relu')(inputs)

x = tf.keras.layers.concatenate([x, labels])

x = Dense(256, activation='relu')(x)

z_mean = Dense(latent_dim, name='z_mean')(x)

z_log_var = Dense(latent_dim, name='z_log_var')(x)

z = Sampling()([z_mean, z_log_var])

return Model([inputs, labels], [z_mean, z_log_var, z], name='cvae_encoder')

# Decoder network for CVAE

def build_cvae_decoder(latent_dim, num_classes, output_shape):

latent_inputs = Input(shape=(latent_dim,))

labels = Input(shape=(num_classes,))

x = Dense(256, activation='relu')(latent_inputs)

x = tf.keras.layers.concatenate([x, labels])

x = Dense(512, activation='relu')(x)

outputs = Dense(output_shape, activation='sigmoid')(x)

return Model([latent_inputs, labels], outputs, name='cvae_decoder')

# Define the input shape, number of classes, and latent dimension

input_shape = (784,)

num_classes = 10

latent_dim = 2

# Build the encoder and decoder for CVAE

cvae_encoder = build_cvae_encoder(input_shape, num_classes, latent_dim)

cvae_decoder = build_cvae_decoder(latent_dim, num_classes, input_shape[0])

# Define the Conditional VAE model

inputs = Input(shape=input_shape)

labels = Input(shape=(num_classes,))

z_mean, z_log_var, z = cvae_encoder([inputs, labels])

outputs = cvae_decoder([z, labels])

cvae = Model([inputs, labels], outputs, name='cvae')

# Define the CVAE loss function

def cvae_loss(inputs, outputs, z_mean, z_log_var):

reconstruction_loss = tf.keras.losses.binary_crossentropy(inputs, outputs)

reconstruction_loss *= input_shape[0]

kl_loss = 1 + z_log_var - K.square(z_mean) - K.exp(z_log_var)

kl_loss = K.sum(kl_loss, axis=-1)

kl_loss *= -0.5

return K.mean(reconstruction_loss + kl_loss)

# Compile the CVAE model

cvae.compile(optimizer='adam', loss=lambda x, y: cvae_loss(x, y, z_mean, z_log_var))

# Prepare the labels for training

y_train = tf.keras.utils.to_categorical(y_train, num_classes)

y_test = tf.keras.utils.to_categorical(y_test, num_classes)

# Train the CVAE model

cvae.fit([x_train, y_train], x_train, epochs=50, batch_size=128, validation_data=([x_test, y_test], x_test))

In this example:

**Summary**