# Variational Autoencoders¶

Here we introduce variational encoders which, given data $(x_i)_{i=1}^m$, attempt to estimate the probability distribution $p(x)$ that generated the data. We assume that the data is defined by latent variables $z$ and that the marginal w.r.t $z$ of the joint distribution $p(x,z)$ is a standard Gaussian, i.e., $p(z)\sim N(0,I)$.

The approach of the variational encoder is to learn' the latent representation of data $x$ (i.e., $p(z|x)$ -- a probabilistic encoding), as well as the reconstruction $p(x|z)$, i.e., a probabilistic decoding.

Both, $p(x|z)$ as well as $p(z|x)$ are learned using neural networks: $p(x|z)\sim N(f(z,\theta),I)$ and $p(z|x)\sim N(f_\mu(x,\eta_\mu),f_{\sigma}(x,\eta_\sigma))$, and $f(z,\theta), f_\mu(x,\eta_\mu),f_{\sigma}(x,\eta_\sigma)$ are neural networks.

To train the variational autoencoder we want that $p(x|z)p(z|x)$ be maximized (so that data is actually well represented), and we also want to push the conditional distribution $p(z|x)$ to be as close to a standard Gaussian as possible.

The loss function therefore consists of a representation loss $|x-f(z,\theta)|^2$, where $z$ is drawn from $N(f_\mu(x,\eta_\mu),f_{\sigma}(x,\eta_\sigma))$, and a distribution loss $D_{KL}(N(f_\mu(x,\eta_\mu),f_{\sigma}(x,\eta_\sigma))||N(0,I))$, which can be evaluated explicitly (see the code below).

Since the node which samples from $N(f_\mu(x,\eta_\mu),f_{\sigma}(x,\eta_\sigma))$ cannot be differentiated, one resorts to the reparametrization trick': create an input node which draws $\varepsilon$ from $N(0,I)$ and a deterministic node which computes $f_\mu(x,\eta_\mu)+\varepsilon \cdot f_{\sigma}(x,\eta_\sigma)$. The resulting architecture can be trained using backprop.

Often, the logarithm of $f_{\sigma}(x,\eta_\sigma)$ is learned, instead of $f_{\sigma}(x,\eta_\sigma)$; this does not change the algorithm but results in better performance.

We may now "generate data" by drawing $z$ according to $N(0,I)$ and applying $f(z,\theta)$.

...and here comes the implementation...

Import libraries

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#import libraries
import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
import os #for creating output directory
from tensorflow.examples.tutorials.mnist import input_data


Set hyperparameters

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mnist = input_data.read_data_sets('../../MNIST_data', one_hot=True)
mb_size = 64 #minibatch size
z_dim = 100 #dimension of latent variable
X_dim = mnist.train.images.shape[1] #dimension of data
h_dim = 128 #dimension of hidden layer
lr = 1e-3 #learning rate


Define auxiliary routines

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#define plotting routine
def plot(samples):
fig = plt.figure(figsize=(4, 4))
gs = gridspec.GridSpec(4, 4)
gs.update(wspace=0.05, hspace=0.05)

for i, sample in enumerate(samples):
ax = plt.subplot(gs[i])
plt.axis('off')
ax.set_xticklabels([])
ax.set_yticklabels([])
ax.set_aspect('equal')
plt.imshow(sample.reshape(28, 28), cmap='Greys_r')

return fig


Initialization

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#define initialization
def xavier_init(size):
in_dim = size[0]
xavier_stddev = 1. / tf.sqrt(in_dim / 2.)
return tf.random_normal(shape=size, stddev=xavier_stddev)


Define Encoder

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# define the encoder
X = tf.placeholder(tf.float32, shape=[None, X_dim])

Q_W1 = tf.Variable(xavier_init([X_dim, h_dim]))
Q_b1 = tf.Variable(tf.zeros(shape=[h_dim]))

Q_W2_mu = tf.Variable(xavier_init([h_dim, z_dim]))
Q_b2_mu = tf.Variable(tf.zeros(shape=[z_dim]))

Q_W2_sigma = tf.Variable(xavier_init([h_dim, z_dim]))
Q_b2_sigma = tf.Variable(tf.zeros(shape=[z_dim]))

#define networks for mean and log-variance
def Q(X):
h = tf.nn.elu(tf.matmul(X, Q_W1) + Q_b1)
z_mu = tf.matmul(h, Q_W2_mu) + Q_b2_mu
z_logvar = tf.matmul(h, Q_W2_sigma) + Q_b2_sigma
return z_mu, z_logvar

#draw sample from encoder
def sample_z(mu, log_var):
eps = tf.random_normal(shape=tf.shape(mu))
return mu + tf.exp(log_var / 2) * eps


Define Decoder

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# define decoder
z = tf.placeholder(tf.float32, shape=[None, z_dim])

P_W1 = tf.Variable(xavier_init([z_dim, h_dim]))
P_b1 = tf.Variable(tf.zeros(shape=[h_dim]))

P_W2 = tf.Variable(xavier_init([h_dim, X_dim]))
P_b2 = tf.Variable(tf.zeros(shape=[X_dim]))

# output value of decoder, both with linear output unit and logit unit
def P(z):
h = tf.nn.elu(tf.matmul(z, P_W1) + P_b1)
logits = tf.matmul(h, P_W2) + P_b2
prob = tf.nn.sigmoid(logits)
return prob, logits


Define Training and run the model

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#train the VAE

z_mu, z_logvar = Q(X) #node for evaluating mean and logvar
z_sample = sample_z(z_mu, z_logvar) #node for sampling encoder
_, logits = P(z_sample) #node for decoder

# Sampling from random z
X_samples, _ = P(z)

# E[log P(X|z)]
recon_loss = tf.reduce_sum(tf.nn.sigmoid_cross_entropy_with_logits(logits=logits, labels=X), 1)
# D_KL(Q(z|X) || P(z|X)); calculate in closed form as both dist. are Gaussian
kl_loss = 0.5 * tf.reduce_sum(tf.exp(z_logvar) + z_mu**2 - 1. - z_logvar, 1)
# VAE loss
vae_loss = tf.reduce_mean(recon_loss + kl_loss)

sess = tf.Session()
sess.run(tf.global_variables_initializer())

#make directory for output
if not os.path.exists('out/'):
os.makedirs('out/')

i = 0

for it in range(1000000):
X_mb, _ = mnist.train.next_batch(mb_size) #get new minibatch

_, loss = sess.run([solver, vae_loss], feed_dict={X: X_mb})

if it % 1000 == 0:
print('Iter: {}'.format(it))
print('Loss: {:.4}'. format(loss))
print()

samples = sess.run(X_samples, feed_dict={z: np.random.randn(16, z_dim)}) #generte 16 random samples
#from trained VAE

fig = plot(samples)
#plt.savefig('out/{}.png'.format(str(i).zfill(3)), bbox_inches='tight')
plt.show()
i += 1
plt.close(fig)

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