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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Copyright 2018 The TensorFlow Authors.
Licensed under the Apache License, Version 2.0 (the "License").
Convolutional VAE
Step1:  Import TensorFlow and enable Eager execution
Step2:  Load the MNIST dataset
Each MNIST image is originally a vector of 784 integers, each of which is between 0-255 and represents the intensity of a pixel. We model each pixel with a Bernoulli distribution in our model, and we statically binarize the dataset.
Step3:  Use tf.data to create batches and shuffle the dataset
Step4:  Wire up the generative and inference network with tf.keras.Sequential
In our VAE example, we use two small ConvNets for the generative and inference network. Since these neural nets are small, we use tf.keras.Sequential to simplify our code. Let $x$ and $z$ denote the observation and latent variable respectively in the following descriptions. 
Generative Network
This defines the generative model which takes a latent encoding as input, and outputs the parameters for a conditional distribution of the observation, i.e. $p(x|z)$. Additionally, we use a unit Gaussian prior $p(z)$ for the latent variable.
Inference Network
This defines an approximate posterior distribution $q(z|x)$, which takes as input an observation and outputs a set of parameters for the conditional distribution of the latent representation. In this example, we simply model this distribution as a diagonal Gaussian. In this case, the inference network outputs the mean and log-variance parameters of a factorized Gaussian (log-variance instead of the variance directly is for numerical stability).
Reparameterization Trick
During optimization, we can sample from $q(z|x)$ by first sampling from a unit Gaussian, and then multiplying by the standard deviation and adding the mean. This ensures the gradients could pass through the sample to the inference network parameters.
Network architecture
For the inference network, we use two convolutional layers followed by a fully-connected layer. In the generative network, we mirror this architecture by using a fully-connected layer followed by three convolution transpose layers (a.k.a. deconvolutional layers in some contexts). Note, it's common practice to avoid using batch normalization when training VAEs, since the additional stochasticity due to using mini-batches may aggravate instability on top of the stochasticity from sampling.
Step5:  Define the loss function and the optimizer
VAEs train by maximizing the evidence lower bound (ELBO) on the marginal log-likelihood
Step6:  Training
We start by iterating over the dataset
During each iteration, we pass the image to the encoder to obtain a set of mean and log-variance parameters of the approximate posterior $q(z|x)$
We then apply the reparameterization trick to sample from $q(z|x)$
Finally, we pass the reparameterized samples to the decoder to obtain the logits of the generative distribution $p(x|z)$
Note
Step7:  Display an image using the epoch number
Step8:  Generate a GIF of all the saved images.
Step9:  To downlod the animation from Colab uncomment the code below
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Python Code:
# to generate gifs
!pip install imageio
Explanation: Copyright 2018 The TensorFlow Authors.
Licensed under the Apache License, Version 2.0 (the "License").
Convolutional VAE: An example with tf.keras and eager
<table class="tfo-notebook-buttons" align="left"><td>
<a target="_blank"  href="https://colab.research.google.com/github/tensorflow/tensorflow/blob/master/tensorflow/contrib/eager/python/examples/generative_examples/cvae.ipynb">
    <img src="https://www.tensorflow.org/images/colab_logo_32px.png" />Run in Google Colab</a>  
</td><td>
<a target="_blank"  href="https://github.com/tensorflow/tensorflow/tree/master/tensorflow/contrib/eager/python/examples/generative_examples/cvae.ipynb"><img width=32px src="https://www.tensorflow.org/images/GitHub-Mark-32px.png" />View source on GitHub</a></td></table>
This notebook demonstrates how to generate images of handwritten digits using tf.keras and eager execution by training a Variational Autoencoder. (VAE, [1], [2]).
End of explanation
from __future__ import absolute_import, division, print_function
# Import TensorFlow >= 1.9 and enable eager execution
import tensorflow as tf
tfe = tf.contrib.eager
tf.enable_eager_execution()
import os
import time
import numpy as np
import glob
import matplotlib.pyplot as plt
import PIL
import imageio
from IPython import display
Explanation: Import TensorFlow and enable Eager execution
End of explanation
(train_images, _), (test_images, _) = tf.keras.datasets.mnist.load_data()
train_images = train_images.reshape(train_images.shape[0], 28, 28, 1).astype('float32')
test_images = test_images.reshape(test_images.shape[0], 28, 28, 1).astype('float32')
# Normalizing the images to the range of [0., 1.]
train_images /= 255.
test_images /= 255.
# Binarization
train_images[train_images >= .5] = 1.
train_images[train_images < .5] = 0.
test_images[test_images >= .5] = 1.
test_images[test_images < .5] = 0.
TRAIN_BUF = 60000
BATCH_SIZE = 100
TEST_BUF = 10000
Explanation: Load the MNIST dataset
Each MNIST image is originally a vector of 784 integers, each of which is between 0-255 and represents the intensity of a pixel. We model each pixel with a Bernoulli distribution in our model, and we statically binarize the dataset.
End of explanation
train_dataset = tf.data.Dataset.from_tensor_slices(train_images).shuffle(TRAIN_BUF).batch(BATCH_SIZE)
test_dataset = tf.data.Dataset.from_tensor_slices(test_images).shuffle(TEST_BUF).batch(BATCH_SIZE)
Explanation: Use tf.data to create batches and shuffle the dataset
End of explanation
class CVAE(tf.keras.Model):
  def __init__(self, latent_dim):
    super(CVAE, self).__init__()
    self.latent_dim = latent_dim
    self.inference_net = tf.keras.Sequential(
      [
          tf.keras.layers.InputLayer(input_shape=(28, 28, 1)),
          tf.keras.layers.Conv2D(
              filters=32, kernel_size=3, strides=(2, 2), activation=tf.nn.relu),
          tf.keras.layers.Conv2D(
              filters=64, kernel_size=3, strides=(2, 2), activation=tf.nn.relu),
          tf.keras.layers.Flatten(),
          # No activation
          tf.keras.layers.Dense(latent_dim + latent_dim),
      ]
    )
    self.generative_net = tf.keras.Sequential(
        [
          tf.keras.layers.InputLayer(input_shape=(latent_dim,)),
          tf.keras.layers.Dense(units=7*7*32, activation=tf.nn.relu),
          tf.keras.layers.Reshape(target_shape=(7, 7, 32)),
          tf.keras.layers.Conv2DTranspose(
              filters=64,
              kernel_size=3,
              strides=(2, 2),
              padding="SAME",
              activation=tf.nn.relu),
          tf.keras.layers.Conv2DTranspose(
              filters=32,
              kernel_size=3,
              strides=(2, 2),
              padding="SAME",
              activation=tf.nn.relu),
          # No activation
          tf.keras.layers.Conv2DTranspose(
              filters=1, kernel_size=3, strides=(1, 1), padding="SAME"),
        ]
    )
  def sample(self, eps=None):
    if eps is None:
      eps = tf.random_normal(shape=(100, self.latent_dim))
    return self.decode(eps, apply_sigmoid=True)
  def encode(self, x):
    mean, logvar = tf.split(self.inference_net(x), num_or_size_splits=2, axis=1)
    return mean, logvar
  def reparameterize(self, mean, logvar):
    eps = tf.random_normal(shape=mean.shape)
    return eps * tf.exp(logvar * .5) + mean
  def decode(self, z, apply_sigmoid=False):
    logits = self.generative_net(z)
    if apply_sigmoid:
      probs = tf.sigmoid(logits)
      return probs
    return logits
Explanation: Wire up the generative and inference network with tf.keras.Sequential
In our VAE example, we use two small ConvNets for the generative and inference network. Since these neural nets are small, we use tf.keras.Sequential to simplify our code. Let $x$ and $z$ denote the observation and latent variable respectively in the following descriptions. 
Generative Network
This defines the generative model which takes a latent encoding as input, and outputs the parameters for a conditional distribution of the observation, i.e. $p(x|z)$. Additionally, we use a unit Gaussian prior $p(z)$ for the latent variable.
Inference Network
This defines an approximate posterior distribution $q(z|x)$, which takes as input an observation and outputs a set of parameters for the conditional distribution of the latent representation. In this example, we simply model this distribution as a diagonal Gaussian. In this case, the inference network outputs the mean and log-variance parameters of a factorized Gaussian (log-variance instead of the variance directly is for numerical stability).
Reparameterization Trick
During optimization, we can sample from $q(z|x)$ by first sampling from a unit Gaussian, and then multiplying by the standard deviation and adding the mean. This ensures the gradients could pass through the sample to the inference network parameters.
Network architecture
For the inference network, we use two convolutional layers followed by a fully-connected layer. In the generative network, we mirror this architecture by using a fully-connected layer followed by three convolution transpose layers (a.k.a. deconvolutional layers in some contexts). Note, it's common practice to avoid using batch normalization when training VAEs, since the additional stochasticity due to using mini-batches may aggravate instability on top of the stochasticity from sampling.
End of explanation
def log_normal_pdf(sample, mean, logvar, raxis=1):
  log2pi = tf.log(2. * np.pi)
  return tf.reduce_sum(
      -.5 * ((sample - mean) ** 2. * tf.exp(-logvar) + logvar + log2pi),
      axis=raxis)
def compute_loss(model, x):
  mean, logvar = model.encode(x)
  z = model.reparameterize(mean, logvar)
  x_logit = model.decode(z)
  cross_ent = tf.nn.sigmoid_cross_entropy_with_logits(logits=x_logit, labels=x)
  logpx_z = -tf.reduce_sum(cross_ent, axis=[1, 2, 3])
  logpz = log_normal_pdf(z, 0., 0.)
  logqz_x = log_normal_pdf(z, mean, logvar)
  return -tf.reduce_mean(logpx_z + logpz - logqz_x)
def compute_gradients(model, x):
  with tf.GradientTape() as tape:
    loss = compute_loss(model, x)
  return tape.gradient(loss, model.trainable_variables), loss
optimizer = tf.train.AdamOptimizer(1e-4)
def apply_gradients(optimizer, gradients, variables, global_step=None):
  optimizer.apply_gradients(zip(gradients, variables), global_step=global_step)
Explanation: Define the loss function and the optimizer
VAEs train by maximizing the evidence lower bound (ELBO) on the marginal log-likelihood:
$$\log p(x) \ge \text{ELBO} = \mathbb{E}_{q(z|x)}\left[\log \frac{p(x, z)}{q(z|x)}\right].$$
In practice, we optimize the single sample Monte Carlo estimate of this expectation:
$$\log p(x| z) + \log p(z) - \log q(z|x),$$
where $z$ is sampled from $q(z|x)$.
Note: we could also analytically compute the KL term, but here we incorporate all three terms in the Monte Carlo estimator for simplicity.
End of explanation
epochs = 100
latent_dim = 50
num_examples_to_generate = 16
# keeping the random vector constant for generation (prediction) so
# it will be easier to see the improvement.
random_vector_for_generation = tf.random_normal(
    shape=[num_examples_to_generate, latent_dim])
model = CVAE(latent_dim)
def generate_and_save_images(model, epoch, test_input):
  predictions = model.sample(test_input)
  fig = plt.figure(figsize=(4,4))
  for i in range(predictions.shape[0]):
      plt.subplot(4, 4, i+1)
      plt.imshow(predictions[i, :, :, 0], cmap='gray')
      plt.axis('off')
  # tight_layout minimizes the overlap between 2 sub-plots
  plt.savefig('image_at_epoch_{:04d}.png'.format(epoch))
  plt.show()
generate_and_save_images(model, 0, random_vector_for_generation)
for epoch in range(1, epochs + 1):
  start_time = time.time()
  for train_x in train_dataset:
    gradients, loss = compute_gradients(model, train_x)
    apply_gradients(optimizer, gradients, model.trainable_variables)
  end_time = time.time()
  if epoch % 1 == 0:
    loss = tfe.metrics.Mean()
    for test_x in test_dataset:
      loss(compute_loss(model, test_x))
    elbo = -loss.result()
    display.clear_output(wait=False)
    print('Epoch: {}, Test set ELBO: {}, '
          'time elapse for current epoch {}'.format(epoch,
                                                    elbo,
                                                    end_time - start_time))
    generate_and_save_images(
        model, epoch, random_vector_for_generation)
Explanation: Training
We start by iterating over the dataset
During each iteration, we pass the image to the encoder to obtain a set of mean and log-variance parameters of the approximate posterior $q(z|x)$
We then apply the reparameterization trick to sample from $q(z|x)$
Finally, we pass the reparameterized samples to the decoder to obtain the logits of the generative distribution $p(x|z)$
Note: Since we use the dataset loaded by keras with 60k datapoints in the training set and 10k datapoints in the test set, our resulting ELBO on the test set is slightly higher than reported results in the literature which uses dynamic binarization of Larochelle's MNIST.
Generate Images
After training, it is time to generate some images
We start by sampling a set of latent vectors from the unit Gaussian prior distribution $p(z)$
The generator will then convert the latent sample $z$ to logits of the observation, giving a distribution $p(x|z)$
Here we plot the probabilities of Bernoulli distributions
End of explanation
def display_image(epoch_no):
  return PIL.Image.open('image_at_epoch_{:04d}.png'.format(epoch_no))
display_image(epochs)  # Display images
Explanation: Display an image using the epoch number
End of explanation
with imageio.get_writer('cvae.gif', mode='I') as writer:
  filenames = glob.glob('image*.png')
  filenames = sorted(filenames)
  last = -1
  for i,filename in enumerate(filenames):
    frame = 2*(i**0.5)
    if round(frame) > round(last):
      last = frame
    else:
      continue
    image = imageio.imread(filename)
    writer.append_data(image)
  image = imageio.imread(filename)
  writer.append_data(image)
    
# this is a hack to display the gif inside the notebook
os.system('cp cvae.gif cvae.gif.png')
display.Image(filename="cvae.gif.png")
Explanation: Generate a GIF of all the saved images.
End of explanation
#from google.colab import files
#files.download('cvae.gif')
Explanation: To downlod the animation from Colab uncomment the code below:
End of explanation
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