# Chapter 6: Project: Handwritten Digit Generation with VAEs

## 6.5 Enhancing Digit Generation with Beta-VAE

Beta-VAE is an extension of the standard VAE that introduces a hyperparameter \( \beta \) to control the balance between the reconstruction loss and the KL divergence in the loss function. This modification can encourage the model to learn more disentangled representations in the latent space, which can be particularly useful for generating high-quality and diverse images. In this section, we will enhance our digit generation project by implementing a Beta-VAE and exploring its benefits.

### 6.5.1 Understanding Beta-VAE

The key idea behind Beta-VAE is to introduce a hyperparameter ( \beta ) in the loss function to weight the KL divergence term. By adjusting ( \beta ), we can control the trade-off between the fidelity of the reconstructions and the regularity of the latent space.

**Beta-VAE Loss Function:**

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

When ( \beta > 1 ), the model places more emphasis on the KL divergence, promoting disentangled representations. Conversely, when ( \beta < 1 ), the model focuses more on accurate reconstructions.

### 6.5.2 Implementing Beta-VAE

We will modify our existing VAE implementation to incorporate the ( \beta ) hyperparameter. This involves updating the loss function and recompiling the model.

**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

# Define the 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

# Build the 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')

# Build the 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=4.0):

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))

This code contains the following main steps:

- Import necessary libraries and modules from TensorFlow.
- Define a custom
`Sampling`

layer that uses the reparameterization trick to sample from the latent space. - Define functions to build the encoder and decoder parts of the VAE. The encoder takes in input data and outputs parameters of the latent space distribution (mean and log variance), as well as a sampled latent vector. The decoder takes in a latent vector and outputs a reconstructed data point.
- Define the input shape and latent dimension, and build the encoder and decoder.
- Define the Beta-VAE model, which takes in input data, passes it through the encoder to get a sampled latent vector, and then passes the latent vector through the decoder to get the reconstructed data.
- Define a custom loss function for the Beta-VAE, which is a combination of reconstruction loss (how well the VAE can reconstruct the input data) and KL divergence loss (how well the latent space distribution matches a target distribution).
- Compile the Beta-VAE model with the Adam optimizer and the custom loss function.
- Train the Beta-VAE model on some training data 'x_train' and validate it on some testing data 'x_test'.

### 6.5.3 Evaluating Beta-VAE

After training the Beta-VAE, we will evaluate its performance using the same metrics and techniques we used for the standard VAE. This will help us understand the impact of the ( \beta ) parameter on the model's ability to generate high-quality and diverse images.

**Example: Evaluating Beta-VAE**

`# Calculate reconstruction loss for the Beta-VAE`

reconstructed_images = beta_vae.predict(x_test)

beta_reconstruction_loss = np.mean(binary_crossentropy(x_test, reconstructed_images))

print(f"Beta-VAE Reconstruction Loss: {beta_reconstruction_loss}")

# Calculate KL Divergence for the Beta-VAE

beta_kl_divergence = calculate_kl_divergence(encoder, x_test)

print(f"Beta-VAE KL Divergence: {beta_kl_divergence}")

# Generate images for evaluation

n_samples = 1000

random_latent_vectors = np.random.normal(size=(n_samples, latent_dim))

beta_generated_images = decoder.predict(random_latent_vectors)

beta_generated_images = beta_generated_images.reshape((n_samples, 28, 28, 1))

# Calculate Inception Score for Beta-VAE

beta_is_mean, beta_is_std = calculate_inception_score(beta_generated_images)

print(f"Beta-VAE Inception Score: {beta_is_mean} ± {beta_is_std}")

# Calculate FID for Beta-VAE

beta_fid_score = calculate_fid(real_images, beta_generated_images)

print(f"Beta-VAE FID Score: {beta_fid_score}")

This Python code snippet is for evaluating the performance of a Beta-VAE

- It first calculates the reconstruction loss by comparing the original test images with the images reconstructed by the Beta-VAE.
- It then calculates the Kullback-Leibler (KL) divergence as a measure of how one probability distribution is different from a second, reference probability distribution.
- It generates images from the Beta-VAE using random latent vectors.
- It calculates the Inception Score (IS), a metric used to evaluate the quality of generated images in generative models.
- Lastly, it calculates the Fréchet Inception Distance (FID), another metric for assessing the quality of the images generated by the model, by comparing them with real images.

### 6.5.4 Visualizing Beta-VAE Results

We will visually inspect the images generated by the Beta-VAE to assess their quality and diversity. This qualitative evaluation will help us understand how the Beta-VAE improves over the standard VAE.

**Example: Visualizing Beta-VAE Generated Images**

`# Visualize generated images from Beta-VAE`

visualize_generated_images(decoder, latent_dim)

# Perform latent space traversal for Beta-VAE

latent_space_traversal(decoder, latent_dim)

# Explore specific latent features for Beta-VAE

explore_latent_features(decoder, latent_dim, example_feature_vector)

`visualize_generated_images(decoder, latent_dim)`

: This function is used to visualize images generated by the Beta-VAE model. The "decoder" and "latent_dim" parameters are likely the decoder part of the model and the dimensionality of the latent space, respectively.`latent_space_traversal(decoder, latent_dim)`

: This function probably performs a traversal of the latent space of the Beta-VAE. This is a technique used to explore and understand the learned representations in the latent space.`explore_latent_features(decoder, latent_dim, example_feature_vector)`

: This function is likely used to explore specific features in the latent space of the Beta-VAE. The "example_feature_vector" parameter is probably a specific vector in the latent space for which the function will generate and display an image.

**Summary**

In this section, we enhanced our digit generation project by implementing a Beta-VAE. We introduced the ( \beta ) hyperparameter to control the balance between the reconstruction loss and the KL divergence, promoting more disentangled representations in the latent space. We updated our loss function, trained the Beta-VAE, and evaluated its performance using both quantitative and qualitative methods.

The Beta-VAE demonstrated improved performance in generating high-quality and diverse images, highlighting the benefits of using the ( \beta ) parameter to control the trade-off between reconstruction accuracy and latent space regularity. By understanding and leveraging these techniques, you can enhance your generative modeling projects and achieve better results in various applications.

## 6.5 Enhancing Digit Generation with Beta-VAE

Beta-VAE is an extension of the standard VAE that introduces a hyperparameter \( \beta \) to control the balance between the reconstruction loss and the KL divergence in the loss function. This modification can encourage the model to learn more disentangled representations in the latent space, which can be particularly useful for generating high-quality and diverse images. In this section, we will enhance our digit generation project by implementing a Beta-VAE and exploring its benefits.

### 6.5.1 Understanding Beta-VAE

The key idea behind Beta-VAE is to introduce a hyperparameter ( \beta ) in the loss function to weight the KL divergence term. By adjusting ( \beta ), we can control the trade-off between the fidelity of the reconstructions and the regularity of the latent space.

**Beta-VAE Loss Function:**

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

When ( \beta > 1 ), the model places more emphasis on the KL divergence, promoting disentangled representations. Conversely, when ( \beta < 1 ), the model focuses more on accurate reconstructions.

### 6.5.2 Implementing Beta-VAE

We will modify our existing VAE implementation to incorporate the ( \beta ) hyperparameter. This involves updating the loss function and recompiling the model.

**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

# Define the 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

# Build the 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')

# Build the 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=4.0):

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))

This code contains the following main steps:

- Import necessary libraries and modules from TensorFlow.
- Define a custom
`Sampling`

layer that uses the reparameterization trick to sample from the latent space. - Define functions to build the encoder and decoder parts of the VAE. The encoder takes in input data and outputs parameters of the latent space distribution (mean and log variance), as well as a sampled latent vector. The decoder takes in a latent vector and outputs a reconstructed data point.
- Define the input shape and latent dimension, and build the encoder and decoder.
- Define the Beta-VAE model, which takes in input data, passes it through the encoder to get a sampled latent vector, and then passes the latent vector through the decoder to get the reconstructed data.
- Define a custom loss function for the Beta-VAE, which is a combination of reconstruction loss (how well the VAE can reconstruct the input data) and KL divergence loss (how well the latent space distribution matches a target distribution).
- Compile the Beta-VAE model with the Adam optimizer and the custom loss function.
- Train the Beta-VAE model on some training data 'x_train' and validate it on some testing data 'x_test'.

### 6.5.3 Evaluating Beta-VAE

After training the Beta-VAE, we will evaluate its performance using the same metrics and techniques we used for the standard VAE. This will help us understand the impact of the ( \beta ) parameter on the model's ability to generate high-quality and diverse images.

**Example: Evaluating Beta-VAE**

`# Calculate reconstruction loss for the Beta-VAE`

reconstructed_images = beta_vae.predict(x_test)

beta_reconstruction_loss = np.mean(binary_crossentropy(x_test, reconstructed_images))

print(f"Beta-VAE Reconstruction Loss: {beta_reconstruction_loss}")

# Calculate KL Divergence for the Beta-VAE

beta_kl_divergence = calculate_kl_divergence(encoder, x_test)

print(f"Beta-VAE KL Divergence: {beta_kl_divergence}")

# Generate images for evaluation

n_samples = 1000

random_latent_vectors = np.random.normal(size=(n_samples, latent_dim))

beta_generated_images = decoder.predict(random_latent_vectors)

beta_generated_images = beta_generated_images.reshape((n_samples, 28, 28, 1))

# Calculate Inception Score for Beta-VAE

beta_is_mean, beta_is_std = calculate_inception_score(beta_generated_images)

print(f"Beta-VAE Inception Score: {beta_is_mean} ± {beta_is_std}")

# Calculate FID for Beta-VAE

beta_fid_score = calculate_fid(real_images, beta_generated_images)

print(f"Beta-VAE FID Score: {beta_fid_score}")

This Python code snippet is for evaluating the performance of a Beta-VAE

- It first calculates the reconstruction loss by comparing the original test images with the images reconstructed by the Beta-VAE.
- It then calculates the Kullback-Leibler (KL) divergence as a measure of how one probability distribution is different from a second, reference probability distribution.
- It generates images from the Beta-VAE using random latent vectors.
- It calculates the Inception Score (IS), a metric used to evaluate the quality of generated images in generative models.
- Lastly, it calculates the Fréchet Inception Distance (FID), another metric for assessing the quality of the images generated by the model, by comparing them with real images.

### 6.5.4 Visualizing Beta-VAE Results

We will visually inspect the images generated by the Beta-VAE to assess their quality and diversity. This qualitative evaluation will help us understand how the Beta-VAE improves over the standard VAE.

**Example: Visualizing Beta-VAE Generated Images**

`# Visualize generated images from Beta-VAE`

visualize_generated_images(decoder, latent_dim)

# Perform latent space traversal for Beta-VAE

latent_space_traversal(decoder, latent_dim)

# Explore specific latent features for Beta-VAE

explore_latent_features(decoder, latent_dim, example_feature_vector)

`visualize_generated_images(decoder, latent_dim)`

: This function is used to visualize images generated by the Beta-VAE model. The "decoder" and "latent_dim" parameters are likely the decoder part of the model and the dimensionality of the latent space, respectively.`latent_space_traversal(decoder, latent_dim)`

: This function probably performs a traversal of the latent space of the Beta-VAE. This is a technique used to explore and understand the learned representations in the latent space.`explore_latent_features(decoder, latent_dim, example_feature_vector)`

: This function is likely used to explore specific features in the latent space of the Beta-VAE. The "example_feature_vector" parameter is probably a specific vector in the latent space for which the function will generate and display an image.

**Summary**

In this section, we enhanced our digit generation project by implementing a Beta-VAE. We introduced the ( \beta ) hyperparameter to control the balance between the reconstruction loss and the KL divergence, promoting more disentangled representations in the latent space. We updated our loss function, trained the Beta-VAE, and evaluated its performance using both quantitative and qualitative methods.

The Beta-VAE demonstrated improved performance in generating high-quality and diverse images, highlighting the benefits of using the ( \beta ) parameter to control the trade-off between reconstruction accuracy and latent space regularity. By understanding and leveraging these techniques, you can enhance your generative modeling projects and achieve better results in various applications.

## 6.5 Enhancing Digit Generation with Beta-VAE

Beta-VAE is an extension of the standard VAE that introduces a hyperparameter \( \beta \) to control the balance between the reconstruction loss and the KL divergence in the loss function. This modification can encourage the model to learn more disentangled representations in the latent space, which can be particularly useful for generating high-quality and diverse images. In this section, we will enhance our digit generation project by implementing a Beta-VAE and exploring its benefits.

### 6.5.1 Understanding Beta-VAE

The key idea behind Beta-VAE is to introduce a hyperparameter ( \beta ) in the loss function to weight the KL divergence term. By adjusting ( \beta ), we can control the trade-off between the fidelity of the reconstructions and the regularity of the latent space.

**Beta-VAE Loss Function:**

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

When ( \beta > 1 ), the model places more emphasis on the KL divergence, promoting disentangled representations. Conversely, when ( \beta < 1 ), the model focuses more on accurate reconstructions.

### 6.5.2 Implementing Beta-VAE

We will modify our existing VAE implementation to incorporate the ( \beta ) hyperparameter. This involves updating the loss function and recompiling the model.

**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

# Define the 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

# Build the 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')

# Build the 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=4.0):

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))

This code contains the following main steps:

- Import necessary libraries and modules from TensorFlow.
- Define a custom
`Sampling`

layer that uses the reparameterization trick to sample from the latent space. - Define functions to build the encoder and decoder parts of the VAE. The encoder takes in input data and outputs parameters of the latent space distribution (mean and log variance), as well as a sampled latent vector. The decoder takes in a latent vector and outputs a reconstructed data point.
- Define the input shape and latent dimension, and build the encoder and decoder.
- Define the Beta-VAE model, which takes in input data, passes it through the encoder to get a sampled latent vector, and then passes the latent vector through the decoder to get the reconstructed data.
- Define a custom loss function for the Beta-VAE, which is a combination of reconstruction loss (how well the VAE can reconstruct the input data) and KL divergence loss (how well the latent space distribution matches a target distribution).
- Compile the Beta-VAE model with the Adam optimizer and the custom loss function.
- Train the Beta-VAE model on some training data 'x_train' and validate it on some testing data 'x_test'.

### 6.5.3 Evaluating Beta-VAE

After training the Beta-VAE, we will evaluate its performance using the same metrics and techniques we used for the standard VAE. This will help us understand the impact of the ( \beta ) parameter on the model's ability to generate high-quality and diverse images.

**Example: Evaluating Beta-VAE**

`# Calculate reconstruction loss for the Beta-VAE`

reconstructed_images = beta_vae.predict(x_test)

beta_reconstruction_loss = np.mean(binary_crossentropy(x_test, reconstructed_images))

print(f"Beta-VAE Reconstruction Loss: {beta_reconstruction_loss}")

# Calculate KL Divergence for the Beta-VAE

beta_kl_divergence = calculate_kl_divergence(encoder, x_test)

print(f"Beta-VAE KL Divergence: {beta_kl_divergence}")

# Generate images for evaluation

n_samples = 1000

random_latent_vectors = np.random.normal(size=(n_samples, latent_dim))

beta_generated_images = decoder.predict(random_latent_vectors)

beta_generated_images = beta_generated_images.reshape((n_samples, 28, 28, 1))

# Calculate Inception Score for Beta-VAE

beta_is_mean, beta_is_std = calculate_inception_score(beta_generated_images)

print(f"Beta-VAE Inception Score: {beta_is_mean} ± {beta_is_std}")

# Calculate FID for Beta-VAE

beta_fid_score = calculate_fid(real_images, beta_generated_images)

print(f"Beta-VAE FID Score: {beta_fid_score}")

This Python code snippet is for evaluating the performance of a Beta-VAE

- It first calculates the reconstruction loss by comparing the original test images with the images reconstructed by the Beta-VAE.
- It then calculates the Kullback-Leibler (KL) divergence as a measure of how one probability distribution is different from a second, reference probability distribution.
- It generates images from the Beta-VAE using random latent vectors.
- It calculates the Inception Score (IS), a metric used to evaluate the quality of generated images in generative models.
- Lastly, it calculates the Fréchet Inception Distance (FID), another metric for assessing the quality of the images generated by the model, by comparing them with real images.

### 6.5.4 Visualizing Beta-VAE Results

We will visually inspect the images generated by the Beta-VAE to assess their quality and diversity. This qualitative evaluation will help us understand how the Beta-VAE improves over the standard VAE.

**Example: Visualizing Beta-VAE Generated Images**

`# Visualize generated images from Beta-VAE`

visualize_generated_images(decoder, latent_dim)

# Perform latent space traversal for Beta-VAE

latent_space_traversal(decoder, latent_dim)

# Explore specific latent features for Beta-VAE

explore_latent_features(decoder, latent_dim, example_feature_vector)

`visualize_generated_images(decoder, latent_dim)`

: This function is used to visualize images generated by the Beta-VAE model. The "decoder" and "latent_dim" parameters are likely the decoder part of the model and the dimensionality of the latent space, respectively.`latent_space_traversal(decoder, latent_dim)`

: This function probably performs a traversal of the latent space of the Beta-VAE. This is a technique used to explore and understand the learned representations in the latent space.`explore_latent_features(decoder, latent_dim, example_feature_vector)`

: This function is likely used to explore specific features in the latent space of the Beta-VAE. The "example_feature_vector" parameter is probably a specific vector in the latent space for which the function will generate and display an image.

**Summary**

In this section, we enhanced our digit generation project by implementing a Beta-VAE. We introduced the ( \beta ) hyperparameter to control the balance between the reconstruction loss and the KL divergence, promoting more disentangled representations in the latent space. We updated our loss function, trained the Beta-VAE, and evaluated its performance using both quantitative and qualitative methods.

The Beta-VAE demonstrated improved performance in generating high-quality and diverse images, highlighting the benefits of using the ( \beta ) parameter to control the trade-off between reconstruction accuracy and latent space regularity. By understanding and leveraging these techniques, you can enhance your generative modeling projects and achieve better results in various applications.

## 6.5 Enhancing Digit Generation with Beta-VAE

### 6.5.1 Understanding Beta-VAE

**Beta-VAE Loss Function:**

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

### 6.5.2 Implementing Beta-VAE

**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

# Define the 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

# Build the 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')

# Build the 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=4.0):

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))

This code contains the following main steps:

- Import necessary libraries and modules from TensorFlow.
- Define a custom
`Sampling`

layer that uses the reparameterization trick to sample from the latent space. - Define the input shape and latent dimension, and build the encoder and decoder.
- Compile the Beta-VAE model with the Adam optimizer and the custom loss function.

### 6.5.3 Evaluating Beta-VAE

**Example: Evaluating Beta-VAE**

`# Calculate reconstruction loss for the Beta-VAE`

reconstructed_images = beta_vae.predict(x_test)

beta_reconstruction_loss = np.mean(binary_crossentropy(x_test, reconstructed_images))

print(f"Beta-VAE Reconstruction Loss: {beta_reconstruction_loss}")

# Calculate KL Divergence for the Beta-VAE

beta_kl_divergence = calculate_kl_divergence(encoder, x_test)

print(f"Beta-VAE KL Divergence: {beta_kl_divergence}")

# Generate images for evaluation

n_samples = 1000

random_latent_vectors = np.random.normal(size=(n_samples, latent_dim))

beta_generated_images = decoder.predict(random_latent_vectors)

beta_generated_images = beta_generated_images.reshape((n_samples, 28, 28, 1))

# Calculate Inception Score for Beta-VAE

beta_is_mean, beta_is_std = calculate_inception_score(beta_generated_images)

print(f"Beta-VAE Inception Score: {beta_is_mean} ± {beta_is_std}")

# Calculate FID for Beta-VAE

beta_fid_score = calculate_fid(real_images, beta_generated_images)

print(f"Beta-VAE FID Score: {beta_fid_score}")

This Python code snippet is for evaluating the performance of a Beta-VAE

- It generates images from the Beta-VAE using random latent vectors.

### 6.5.4 Visualizing Beta-VAE Results

**Example: Visualizing Beta-VAE Generated Images**

`# Visualize generated images from Beta-VAE`

visualize_generated_images(decoder, latent_dim)

# Perform latent space traversal for Beta-VAE

latent_space_traversal(decoder, latent_dim)

# Explore specific latent features for Beta-VAE

explore_latent_features(decoder, latent_dim, example_feature_vector)

`visualize_generated_images(decoder, latent_dim)`

: This function is used to visualize images generated by the Beta-VAE model. The "decoder" and "latent_dim" parameters are likely the decoder part of the model and the dimensionality of the latent space, respectively.`latent_space_traversal(decoder, latent_dim)`

: This function probably performs a traversal of the latent space of the Beta-VAE. This is a technique used to explore and understand the learned representations in the latent space.`explore_latent_features(decoder, latent_dim, example_feature_vector)`

: This function is likely used to explore specific features in the latent space of the Beta-VAE. The "example_feature_vector" parameter is probably a specific vector in the latent space for which the function will generate and display an image.

**Summary**