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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:
 Convolutional Autoencoder
Sticking with the MNIST dataset, let's improve our autoencoder's performance using convolutional layers. We'll build a convolutional autoencoder to compress the MNIST dataset. 
The encoder portion will be made of convolutional and pooling layers and the decoder will be made of transpose convolutional layers that learn to "upsample" a compressed representation.
<img src='notebook_ims/autoencoder_1.png' />
Compressed Representation
A compressed representation can be great for saving and sharing any kind of data in a way that is more efficient than storing raw data. In practice, the compressed representation often holds key information about an input image and we can use it for denoising images or oher kinds of reconstruction and transformation!
<img src='notebook_ims/denoising.png' width=60%/>
Let's get started by importing our libraries and getting the dataset.
Step1:  Visualize the Data
Step2:  Convolutional  Autoencoder
Encoder
The encoder part of the network will be a typical convolutional pyramid. Each convolutional layer will be followed by a max-pooling layer to reduce the dimensions of the layers. 
Decoder
The decoder though might be something new to you. The decoder needs to convert from a narrow representation to a wide, reconstructed image. For example, the representation could be a 7x7x4 max-pool layer. This is the output of the encoder, but also the input to the decoder. We want to get a 28x28x1 image out from the decoder so we need to work our way back up from the compressed representation. A schematic of the network is shown below.
<img src='notebook_ims/conv_enc_1.png' width=640px>
Here our final encoder layer has size 7x7x4 = 196. The original images have size 28x28 = 784, so the encoded vector is 25% the size of the original image. These are just suggested sizes for each of the layers. Feel free to change the depths and sizes, in fact, you're encouraged to add additional layers to make this representation even smaller! Remember our goal here is to find a small representation of the input data.
Transpose Convolutions, Decoder
This decoder uses transposed convolutional layers to increase the width and height of the input layers. They work almost exactly the same as convolutional layers, but in reverse. A stride in the input layer results in a larger stride in the transposed convolution layer. For example, if you have a 3x3 kernel, a 3x3 patch in the input layer will be reduced to one unit in a convolutional layer. Comparatively, one unit in the input layer will be expanded to a 3x3 path in a transposed convolution layer. PyTorch provides us with an easy way to create the layers, nn.ConvTranspose2d. 
It is important to note that transpose convolution layers can lead to artifacts in the final images, such as checkerboard patterns. This is due to overlap in the kernels which can be avoided by setting the stride and kernel size equal. In this Distill article from Augustus Odena, et al, the authors show that these checkerboard artifacts can be avoided by resizing the layers using nearest neighbor or bilinear interpolation (upsampling) followed by a convolutional layer. 
We'll show this approach in another notebook, so you can experiment with it and see the difference.
TODO
Step3:  Training
Here I'll write a bit of code to train the network. I'm not too interested in validation here, so I'll just monitor the training loss and the test loss afterwards. 
We are not concerned with labels in this case, just images, which we can get from the train_loader. Because we're comparing pixel values in input and output images, it will be best to use a loss that is meant for a regression task. Regression is all about comparing quantities rather than probabilistic values. So, in this case, I'll use MSELoss. And compare output images and input images as follows
Step4:  Checking out the results
Below I've plotted some of the test images along with their reconstructions. These look a little rough around the edges, likely due to the checkerboard effect we mentioned above that tends to happen with transpose layers.
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Python Code:
import torch
import numpy as np
from torchvision import datasets
import torchvision.transforms as transforms
# convert data to torch.FloatTensor
transform = transforms.ToTensor()
# load the training and test datasets
train_data = datasets.MNIST(root='data', train=True,
                                   download=True, transform=transform)
test_data = datasets.MNIST(root='data', train=False,
                                  download=True, transform=transform)
# Create training and test dataloaders
num_workers = 0
# how many samples per batch to load
batch_size = 20
# prepare data loaders
train_loader = torch.utils.data.DataLoader(train_data, batch_size=batch_size, num_workers=num_workers)
test_loader = torch.utils.data.DataLoader(test_data, batch_size=batch_size, num_workers=num_workers)
Explanation: Convolutional Autoencoder
Sticking with the MNIST dataset, let's improve our autoencoder's performance using convolutional layers. We'll build a convolutional autoencoder to compress the MNIST dataset. 
The encoder portion will be made of convolutional and pooling layers and the decoder will be made of transpose convolutional layers that learn to "upsample" a compressed representation.
<img src='notebook_ims/autoencoder_1.png' />
Compressed Representation
A compressed representation can be great for saving and sharing any kind of data in a way that is more efficient than storing raw data. In practice, the compressed representation often holds key information about an input image and we can use it for denoising images or oher kinds of reconstruction and transformation!
<img src='notebook_ims/denoising.png' width=60%/>
Let's get started by importing our libraries and getting the dataset.
End of explanation
import matplotlib.pyplot as plt
%matplotlib inline
    
# obtain one batch of training images
dataiter = iter(train_loader)
images, labels = dataiter.next()
images = images.numpy()
# get one image from the batch
img = np.squeeze(images[0])
fig = plt.figure(figsize = (5,5)) 
ax = fig.add_subplot(111)
ax.imshow(img, cmap='gray')
Explanation: Visualize the Data
End of explanation
import torch.nn as nn
import torch.nn.functional as F
# define the NN architecture
class ConvAutoencoder(nn.Module):
    def __init__(self):
        super(ConvAutoencoder, self).__init__()
        ## encoder layers ##
        # conv layer (depth from 1 --> 16), 3x3 kernels
        self.conv1 = nn.Conv2d(1, 16, 3, padding=1)  
        # conv layer (depth from 16 --> 4), 3x3 kernels
        self.conv2 = nn.Conv2d(16, 4, 3, padding=1)
        # pooling layer to reduce x-y dims by two; kernel and stride of 2
        self.pool = nn.MaxPool2d(2, 2)
        
        ## decoder layers ##
        ## a kernel of 2 and a stride of 2 will increase the spatial dims by 2
        self.t_conv1 = nn.ConvTranspose2d(4, 16, 2, stride=2)
        self.t_conv2 = nn.ConvTranspose2d(16, 1, 2, stride=2)
    def forward(self, x):
        ## encode ##
        # add hidden layers with relu activation function
        # and maxpooling after
        x = F.relu(self.conv1(x))
        x = self.pool(x)
        # add second hidden layer
        x = F.relu(self.conv2(x))
        x = self.pool(x)  # compressed representation
        
        ## decode ##
        # add transpose conv layers, with relu activation function
        x = F.relu(self.t_conv1(x))
        # output layer (with sigmoid for scaling from 0 to 1)
        x = F.sigmoid(self.t_conv2(x))
                
        return x
# initialize the NN
model = ConvAutoencoder()
print(model)
Explanation: Convolutional  Autoencoder
Encoder
The encoder part of the network will be a typical convolutional pyramid. Each convolutional layer will be followed by a max-pooling layer to reduce the dimensions of the layers. 
Decoder
The decoder though might be something new to you. The decoder needs to convert from a narrow representation to a wide, reconstructed image. For example, the representation could be a 7x7x4 max-pool layer. This is the output of the encoder, but also the input to the decoder. We want to get a 28x28x1 image out from the decoder so we need to work our way back up from the compressed representation. A schematic of the network is shown below.
<img src='notebook_ims/conv_enc_1.png' width=640px>
Here our final encoder layer has size 7x7x4 = 196. The original images have size 28x28 = 784, so the encoded vector is 25% the size of the original image. These are just suggested sizes for each of the layers. Feel free to change the depths and sizes, in fact, you're encouraged to add additional layers to make this representation even smaller! Remember our goal here is to find a small representation of the input data.
Transpose Convolutions, Decoder
This decoder uses transposed convolutional layers to increase the width and height of the input layers. They work almost exactly the same as convolutional layers, but in reverse. A stride in the input layer results in a larger stride in the transposed convolution layer. For example, if you have a 3x3 kernel, a 3x3 patch in the input layer will be reduced to one unit in a convolutional layer. Comparatively, one unit in the input layer will be expanded to a 3x3 path in a transposed convolution layer. PyTorch provides us with an easy way to create the layers, nn.ConvTranspose2d. 
It is important to note that transpose convolution layers can lead to artifacts in the final images, such as checkerboard patterns. This is due to overlap in the kernels which can be avoided by setting the stride and kernel size equal. In this Distill article from Augustus Odena, et al, the authors show that these checkerboard artifacts can be avoided by resizing the layers using nearest neighbor or bilinear interpolation (upsampling) followed by a convolutional layer. 
We'll show this approach in another notebook, so you can experiment with it and see the difference.
TODO: Build the network shown above.
Build the encoder out of a series of convolutional and pooling layers. 
When building the decoder, recall that transpose convolutional layers can upsample an input by a factor of 2 using a stride and kernel_size of 2.
End of explanation
# specify loss function
criterion = nn.MSELoss()
# specify loss function
optimizer = torch.optim.Adam(model.parameters(), lr=0.001)
# number of epochs to train the model
n_epochs = 30
for epoch in range(1, n_epochs+1):
    # monitor training loss
    train_loss = 0.0
    
    ###################
    # train the model #
    ###################
    for data in train_loader:
        # _ stands in for labels, here
        # no need to flatten images
        images, _ = data
        # clear the gradients of all optimized variables
        optimizer.zero_grad()
        # forward pass: compute predicted outputs by passing inputs to the model
        outputs = model(images)
        # calculate the loss
        loss = criterion(outputs, images)
        # backward pass: compute gradient of the loss with respect to model parameters
        loss.backward()
        # perform a single optimization step (parameter update)
        optimizer.step()
        # update running training loss
        train_loss += loss.item()*images.size(0)
            
    # print avg training statistics 
    train_loss = train_loss/len(train_loader)
    print('Epoch: {} \tTraining Loss: {:.6f}'.format(
        epoch, 
        train_loss
        ))
Explanation: Training
Here I'll write a bit of code to train the network. I'm not too interested in validation here, so I'll just monitor the training loss and the test loss afterwards. 
We are not concerned with labels in this case, just images, which we can get from the train_loader. Because we're comparing pixel values in input and output images, it will be best to use a loss that is meant for a regression task. Regression is all about comparing quantities rather than probabilistic values. So, in this case, I'll use MSELoss. And compare output images and input images as follows:
loss = criterion(outputs, images)
Otherwise, this is pretty straightfoward training with PyTorch. Since this is a convlutional autoencoder, our images do not need to be flattened before being passed in an input to our model.
End of explanation
# obtain one batch of test images
dataiter = iter(test_loader)
images, labels = dataiter.next()
# get sample outputs
output = model(images)
# prep images for display
images = images.numpy()
# output is resized into a batch of iages
output = output.view(batch_size, 1, 28, 28)
# use detach when it's an output that requires_grad
output = output.detach().numpy()
# plot the first ten input images and then reconstructed images
fig, axes = plt.subplots(nrows=2, ncols=10, sharex=True, sharey=True, figsize=(25,4))
# input images on top row, reconstructions on bottom
for images, row in zip([images, output], axes):
    for img, ax in zip(images, row):
        ax.imshow(np.squeeze(img), cmap='gray')
        ax.get_xaxis().set_visible(False)
        ax.get_yaxis().set_visible(False)
Explanation: Checking out the results
Below I've plotted some of the test images along with their reconstructions. These look a little rough around the edges, likely due to the checkerboard effect we mentioned above that tends to happen with transpose layers.
End of explanation
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