[Gluon] Add VAE demo (apache#18758)

* add VAE demo

* minor changes

* change format to md

* minor changes

* add liscence

* Update VAE.md

* update vae demo

* remove unnecessary files

example/probability/VAE/VAE.md

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+
+
+# VAE with Gluon.probability 
+
+In this example, we will demonstrate how you can implement a Variational Auto-encoder(VAE) with Gluon.probability and MXNet's latest NumPy API.
+
+
+```{.python .input}
+import numpy as np
+import mxnet as mx
+from mxnet import autograd, gluon, np, npx
+from mxnet.gluon import nn
+import mxnet.gluon.probability as mgp
+import matplotlib.pyplot as plt
+
+# Switch numpy-compatible semantics on.
+npx.set_np()
+
+# Set context for model context, here we choose to use GPU. 
+model_ctx = mx.gpu(0)
+```
+
+## Dataset
+
+We will use MNIST here for simplicity purpose.
+
+
+```{.python .input}
+def load_data(batch_size):
+    mnist_train = gluon.data.vision.MNIST(train=True)
+    mnist_test = gluon.data.vision.MNIST(train=False)
+    num_worker = 4
+    transformer = gluon.data.vision.transforms.ToTensor()
+    return (gluon.data.DataLoader(mnist_train.transform_first(transformer),
+                                batch_size, shuffle=True,
+                                num_workers=num_worker),
+          gluon.data.DataLoader(mnist_test.transform_first(transformer),
+                                batch_size, shuffle=False,
+                                num_workers=num_worker))
+                                 
+```
+
+## Model definition
+
+
+```{.python .input}
+class VAE(gluon.HybridBlock):
+    def __init__(self, n_hidden=256, n_latent=2, n_layers=1, n_output=784, act_type='relu', **kwargs):
+        r"""
+        n_hidden : number of hidden units in each layer
+        n_latent : dimension of the latent space
+        n_layers : number of layers in the encoder and decoder network
+        n_output : dimension of the observed data
+        """
+        self.soft_zero = 1e-10
+        self.n_latent = n_latent
+        self.output = None
+        self.mu = None
+        super(VAE, self).__init__(**kwargs)
+        self.encoder = nn.HybridSequential()
+        for _ in range(n_layers):
+            self.encoder.add(nn.Dense(n_hidden, activation=act_type))
+        self.encoder.add(nn.Dense(n_latent*2, activation=None))
+        self.decoder = nn.HybridSequential()
+        for _ in range(n_layers):
+            self.decoder.add(nn.Dense(n_hidden, activation=act_type))
+        self.decoder.add(nn.Dense(n_output, activation='sigmoid'))
+        
+    def encode(self, x):
+        r"""
+        Given a batch of x,
+        return the encoder's output
+        """
+        # [loc_1, ..., loc_n, log(scale_1), ..., log(scale_n)]
+        h = self.encoder(x)
+
+        # Extract loc and log_scale from the encoder output.
+        loc_scale = np.split(h, 2, 1)
+        loc = loc_scale[0]
+        log_scale = loc_scale[1]
+
+        # Convert log_scale back to scale.
+        scale = np.exp(log_scale)
+
+        # Return a Normal object.
+        return mgp.Normal(loc, scale)
+    
+    def decode(self, z):
+        r"""
+        Given a batch of samples from z,
+        return the decoder's output
+        """
+        return self.decoder(z)
+
+    def forward(self, x):
+        r"""
+        Given a batch of data x,
+        return the negative of Evidence Lower-bound,
+        i.e. an objective to minimize.
+        """
+        # prior p(z)
+        pz = mgp.Normal(0, 1)
+        
+        # posterior q(z|x)
+        qz_x = self.encode(x) 
+        
+        # Sampling operation qz_x.sample() is automatically reparameterized.
+        z = qz_x.sample() 
+
+        # Reconstruction result
+        y = self.decode(z) 
+        
+        # Gluon.probability can help you calculate the analytical kl-divergence
+        # between two distribution objects.
+        KL = mgp.kl_divergence(qz_x, pz).sum(1)
+        
+        # We assume p(x|z) ~ Bernoulli, therefore we compute the reconstruction
+        # loss with binary cross entropy.
+        logloss = np.sum(x * np.log(y + self.soft_zero) + (1 - x)
+                         * np.log(1 - y + self.soft_zero), axis=1)
+        loss = -logloss + KL
+        return loss
+```
+
+## Training
+
+
+```{.python .input}
+def train(net, n_epoch, print_period, train_iter, test_iter):
+    net.initialize(mx.init.Xavier(), ctx=model_ctx)
+    net.hybridize()
+    trainer = gluon.Trainer(net.collect_params(), 'adam',
+                          {'learning_rate': .001})
+    training_loss = []
+    validation_loss = []
+    for epoch in range(n_epoch):
+        epoch_loss = 0
+        epoch_val_loss = 0
+
+        n_batch_train = 0
+        for batch in train_iter:
+            n_batch_train += 1
+            data = batch[0].as_in_context(model_ctx).reshape(-1, 28 * 28)
+            with autograd.record():
+                loss = net(data)
+            loss.backward()
+            trainer.step(data.shape[0])
+            epoch_loss += np.mean(loss)
+
+        n_batch_val = 0
+        for batch in test_iter:
+            n_batch_val += 1
+            data = batch[0].as_in_context(model_ctx).reshape(-1, 28 * 28)
+            loss = net(data)
+            epoch_val_loss += np.mean(loss)
+
+        epoch_loss /= n_batch_train
+        epoch_val_loss /= n_batch_val
+
+        training_loss.append(epoch_loss)
+        validation_loss.append(epoch_val_loss)
+
+        if epoch % max(print_period, 1) == 0:
+            print('Epoch{}, Training loss {:.2f}, Validation loss {:.2f}'.format(
+              epoch, float(epoch_loss), float(epoch_val_loss)))
+```
+
+
+```{.python .input}
+n_hidden = 128
+n_latent = 40
+n_layers = 3
+n_output = 784
+batch_size = 128
+model_prefix = 'vae_gluon_{}d{}l{}h.params'.format(
+  n_latent, n_layers, n_hidden)
+net = VAE(n_hidden=n_hidden, n_latent=n_latent, n_layers=n_layers,
+        n_output=n_output)
+net.hybridize()
+n_epoch = 50
+print_period = n_epoch // 10
+train_set, test_set = load_data(batch_size)
+train(net, n_epoch, print_period, train_set, test_set)
+```
+
+
+## Reconstruction visualiztion
+
+To verify the effictiveness of our model, we first take a look at how well our model can reconstruct the data.
+
+
+```{.python .input}
+# Grab a batch from the test set
+qz_x = None
+for batch in test_set:
+    data = batch[0].as_in_context(model_ctx).reshape(-1, 28 * 28)
+    qz_x = net.encode(data)
+    break
+```
+
+
+```{.python .input}
+num_samples = 4
+fig, axes = plt.subplots(nrows=num_samples, ncols=2, figsize=(4, 6), subplot_kw={'xticks': [], 'yticks': []})
+axes[0, 0].set_title('Original image')
+axes[0, 1].set_title('reconstruction')
+for i in range(num_samples):
+    axes[i, 0].imshow(data[i].squeeze().reshape(28, 28).asnumpy(), cmap='gray')
+    axes[i, 1].imshow(net.decode(qz_x.sample())[i].reshape(28, 28).asnumpy(), cmap='gray')
+```
+
+
+![png](./VAE_11_0.png)
+
+
+## Sample generation
+
+One of the most important difference between Variational Auto-encoder and Auto-encoder is VAE's capabilities of generating new samples.
+
+To achieve that, one simply needs to feed a random sample from $p(z) \sim \mathcal{N}(0,1)$ to the decoder network.
+
+
+```{.python .input}
+def plot_samples(samples, h=5, w=10):
+    fig, axes = plt.subplots(nrows=h,
+                             ncols=w,
+                             figsize=(int(1.4 * w), int(1.4 * h)),
+                             subplot_kw={'xticks': [], 'yticks': []})
+    for i, ax in enumerate(axes.flatten()):
+        ax.imshow(samples[i], cmap='gray')
+```
+
+
+```{.python .input}
+n_samples = 20
+noise = np.random.randn(n_samples, n_latent).as_in_context(model_ctx)
+dec_output = net.decode(noise).reshape(-1, 28, 28).asnumpy()
+plot_samples(dec_output, 4, 5)
+```
+
+
+![png](./VAE_14_0.png)
+