Note

This section assumes the reader has already read through *Classifying MNIST digits using Logistic Regression*
and *Multilayer Perceptron*. Additionally it uses the following Theano functions
and concepts : T.tanh, shared variables, basic arithmetic ops, T.grad, Random numbers, floatX. If you intend to run the code on GPU also read GPU.

Note

The code for this section is available for download here.

The Denoising Autoencoder (dA) is an extension of a classical
autoencoder and it was introduced as a building block for deep networks
in [Vincent08]. We will start the tutorial with a short discussion on
*Autoencoders*.

See section 4.6 of [Bengio09] for an overview of auto-encoders.
An autoencoder takes an input and first
maps it (with an *encoder*) to a hidden representation
through a deterministic mapping, e.g.:

Where is a non-linearity such as the sigmoid.
The latent representation , or **code** is then mapped back (with a *decoder*) into a
**reconstruction** of same shape as
through a similar transformation, e.g.:

where ‘ does not indicate transpose, and
should be seen as a prediction of , given the code .
The weight matrix of the reverse mapping may be
optionally constrained by , which is
an instance of *tied weights*. The parameters of this model (namely
, ,
and, if one doesn’t use tied weights, also
) are optimized such that the average reconstruction
error is minimized. The reconstruction error can be measured in many ways, depending
on the appropriate distributional assumptions on the input given the code, e.g., using the
traditional *squared error* ,
or if the input is interpreted as either bit vectors or vectors of
bit probabilities by the reconstruction *cross-entropy* defined as :

The hope is that the code is a distributed representation that captures the coordinates along the main factors of variation in the data (similarly to how the projection on principal components captures the main factors of variation in the data). Because is viewed as a lossy compression of , it cannot be a good compression (with small loss) for all , so learning drives it to be one that is a good compression in particular for training examples, and hopefully for others as well, but not for arbitrary inputs. That is the sense in which an auto-encoder generalizes: it gives low reconstruction error to test examples from the same distribution as the training examples, but generally high reconstruction error to uniformly chosen configurations of the input vector.

If there is one linear hidden layer (the code) and
the mean squared error criterion is used to train the network, then the
hidden units learn to project the input in the span of the first
principal components of the data. If the hidden
layer is non-linear, the auto-encoder behaves differently from PCA,
with the ability to capture multi-modal aspects of the input
distribution. The departure from PCA becomes even more important when
we consider *stacking multiple encoders* (and their corresponding decoders)
when building a deep auto-encoder [Hinton06].

We want to implement an auto-encoder using Theano, in the form of a class, that could be afterwards used in constructing a stacked autoencoder. The first step is to create shared variables for the parameters of the autoencoder ( , and , since we are using tied weights in this tutorial ):

```
class AutoEncoder(object):
def __init__(self, numpy_rng, input=None, n_visible=784, n_hidden=500,
W=None, bhid=None, bvis=None):
"""
:type numpy_rng: numpy.random.RandomState
:param numpy_rng: number random generator used to generate weights
:type input: theano.tensor.TensorType
:paran input: a symbolic description of the input or None for standalone
dA
:type n_visible: int
:param n_visible: number of visible units
:type n_hidden: int
:param n_hidden: number of hidden units
:type W: theano.tensor.TensorType
:param W: Theano variable pointing to a set of weights that should be
shared belong the dA and another architecture; if dA should
be standalone set this to None
:type bhid: theano.tensor.TensorType
:param bhid: Theano variable pointing to a set of biases values (for
hidden units) that should be shared belong dA and another
architecture; if dA should be standalone set this to None
:type bvis: theano.tensor.TensorType
:param bvis: Theano variable pointing to a set of biases values (for
visible units) that should be shared belong dA and another
architecture; if dA should be standalone set this to None
"""
self.n_visible = n_visible
self.n_hidden = n_hidden
# note : W' was written as `W_prime` and b' as `b_prime`
if not W:
# W is initialized with `initial_W` which is uniformely sampled
# from -4*sqrt(6./(n_visible+n_hidden)) and 4*sqrt(6./(n_hidden+n_visible))
# the output of uniform if converted using asarray to dtype
# theano.config.floatX so that the code is runable on GPU
initial_W = numpy.asarray(numpy_rng.uniform(
low=-4 * numpy.sqrt(6. / (n_hidden + n_visible)),
high=4 * numpy.sqrt(6. / (n_hidden + n_visible)),
size=(n_visible, n_hidden)), dtype=theano.config.floatX)
W = theano.shared(value=initial_W, name='W')
if not bvis:
bvis = theano.shared(value=numpy.zeros(n_visible,
dtype=theano.config.floatX), name='bvis')
if not bhid:
bhid = theano.shared(value=numpy.zeros(n_hidden,
dtype=theano.config.floatX), name='bhid')
self.W = W
# b corresponds to the bias of the hidden
self.b = bhid
# b_prime corresponds to the bias of the visible
self.b_prime = bvis
# tied weights, therefore W_prime is W transpose
self.W_prime = self.W.T
# if no input is given, generate a variable representing the input
if input == None:
# we use a matrix because we expect a minibatch of several examples,
# each example being a row
self.x = T.dmatrix(name='input')
else:
self.x = input
self.params = [self.W, self.b, self.b_prime]
```

Note that we pass the symbolic `input` to the autoencoder as a
parameter. This is such that later we can concatenate layers of
autoencoders to form a deep network: the symbolic output (the above) of
the k-th layer will be the symbolic input of the (k+1)-th.

Now we can express the computation of the latent representation and of the reconstructed signal:

```
def get_hidden_values(self, input):
""" Computes the values of the hidden layer """
return T.nnet.sigmoid(T.dot(input, self.W) + self.b)
def get_reconstructed_input(self, hidden):
""" Computes the reconstructed input given the values of the hidden layer """
return T.nnet.sigmoid(T.dot(hidden, self.W_prime) + self.b_prime)
```

And using these function we can compute the cost and the updates of one stochastic gradient descent step :

```
def get_cost_updates(self, learning_rate):
""" This function computes the cost and the updates for one trainng
step """
y = self.get_hidden_values(self.x)
z = self.get_reconstructed_input(y)
# note : we sum over the size of a datapoint; if we are using minibatches,
# L will be a vector, with one entry per example in minibatch
L = -T.sum(self.x * T.log(z) + (1 - self.x) * T.log(1 - z), axis=1)
# note : L is now a vector, where each element is the cross-entropy cost
# of the reconstruction of the corresponding example of the
# minibatch. We need to compute the average of all these to get
# the cost of the minibatch
cost = T.mean(L)
# compute the gradients of the cost of the `dA` with respect
# to its parameters
gparams = T.grad(cost, self.params)
# generate the list of updates
updates = []
for param, gparam in zip(self.params, gparams):
updates.append((param, param - learning_rate * gparam))
return (cost, updates)
```

We can now define a function that applied iteratively will update the
parameters `W`, `b` and `b_prime` such that the
reconstruction cost is approximately minimized.

```
autoencoder = AutoEncoder(numpy_rng=numpy.random.RandomState(1234), n_visible=784, n_hidden=500)
cost, updates = autoencoder.get_cost_updates(learning_rate=0.1)
train = theano.function([x], cost, updates=updates)
```

One serious potential issue with auto-encoders is that if there is no other constraint besides minimizing the reconstruction error, then an auto-encoder with inputs and an encoding of dimension at least could potentially just learn the identity function, for which many encodings would be useless (e.g., just copying the input), i.e., the autoencoder would not differentiate test examples (from the training distribution) from other input configurations. Surprisingly, experiments reported in [Bengio07] nonetheless suggest that in practice, when trained with stochastic gradient descent, non-linear auto-encoders with more hidden units than inputs (called overcomplete) yield useful representations (in the sense of classification error measured on a network taking this representation in input). A simple explanation is based on the observation that stochastic gradient descent with early stopping is similar to an L2 regularization of the parameters. To achieve perfect reconstruction of continuous inputs, a one-hidden layer auto-encoder with non-linear hidden units (exactly like in the above code) needs very small weights in the first (encoding) layer (to bring the non-linearity of the hidden units in their linear regime) and very large weights in the second (decoding) layer. With binary inputs, very large weights are also needed to completely minimize the reconstruction error. Since the implicit or explicit regularization makes it difficult to reach large-weight solutions, the optimization algorithm finds encodings which only work well for examples similar to those in the training set, which is what we want. It means that the representation is exploiting statistical regularities present in the training set, rather than learning to replicate the identity function.

There are different ways that an auto-encoder with more hidden units
than inputs could be prevented from learning the identity, and still
capture something useful about the input in its hidden representation.
One is the addition of sparsity (forcing many of the hidden units to
be zero or near-zero), and it has been exploited very successfully
by many [Ranzato07] [Lee08]. Another is to add randomness in the transformation from
input to reconstruction. This is exploited in Restricted Boltzmann
Machines (discussed later in *Restricted Boltzmann Machines (RBM)*), as well as in
Denoising Auto-Encoders, discussed below.

The idea behind denoising autoencoders is simple. In order to force
the hidden layer to discover more robust features and prevent it
from simply learning the identity, we train the
autoencoder to *reconstruct the input from a corrupted version of it*.

The denoising auto-encoder is a stochastic version of the auto-encoder. Intuitively, a denoising auto-encoder does two things: try to encode the input (preserve the information about the input), and try to undo the effect of a corruption process stochastically applied to the input of the auto-encoder. The latter can only be done by capturing the statistical dependencies between the inputs. The denoising auto-encoder can be understood from different perspectives ( the manifold learning perspective, stochastic operator perspective, bottom-up – information theoretic perspective, top-down – generative model perspective ), all of which are explained in [Vincent08]. See also section 7.2 of [Bengio09] for an overview of auto-encoders.

In [Vincent08], the stochastic corruption process
consists in randomly setting some of the inputs (as many as half of them)
to zero. Hence the denoising auto-encoder is trying to *predict the corrupted (i.e. missing)
values from the uncorrupted (i.e., non-missing) values*, for randomly selected subsets of
missing patterns. Note how being able to predict any subset of variables
from the rest is a sufficient condition for completely capturing the
joint distribution between a set of variables (this is how Gibbs
sampling works).

To convert the autoencoder class into a denoising autoencoder class, all we need to do is to add a stochastic corruption step operating on the input. The input can be corrupted in many ways, but in this tutorial we will stick to the original corruption mechanism of randomly masking entries of the input by making them zero. The code below does just that :

```
from theano.tensor.shared_randomstreams import RandomStreams
def get_corrupted_input(self, input, corruption_level):
""" This function keeps ``1-corruption_level`` entries of the inputs the same
and zero-out randomly selected subset of size ``coruption_level``
Note : first argument of theano.rng.binomial is the shape(size) of
random numbers that it should produce
second argument is the number of trials
third argument is the probability of success of any trial
this will produce an array of 0s and 1s where 1 has a probability of
1 - ``corruption_level`` and 0 with ``corruption_level``
"""
return self.theano_rng.binomial(size=input.shape, n=1, p=1 - corruption_level) * input
```

In the stacked autoencoder class (*Stacked Autoencoders*) the
weights of the `dA` class have to be shared with those of an
corresponding sigmoid layer. For this reason, the constructor of the `dA` also gets Theano
variables pointing to the shared parameters. If those parameters are left
to `None`, new ones will be constructed.

The final denoising autoencoder class becomes :

```
class dA(object):
"""Denoising Auto-Encoder class (dA)
A denoising autoencoders tries to reconstruct the input from a corrupted
version of it by projecting it first in a latent space and reprojecting
it afterwards back in the input space. Please refer to Vincent et al.,2008
for more details. If x is the input then equation (1) computes a partially
destroyed version of x by means of a stochastic mapping q_D. Equation (2)
computes the projection of the input into the latent space. Equation (3)
computes the reconstruction of the input, while equation (4) computes the
reconstruction error.
.. math::
\tilde{x} ~ q_D(\tilde{x}|x) (1)
y = s(W \tilde{x} + b) (2)
x = s(W' y + b') (3)
L(x,z) = -sum_{k=1}^d [x_k \log z_k + (1-x_k) \log( 1-z_k)] (4)
"""
def __init__(self, numpy_rng, theano_rng=None, input=None, n_visible=784, n_hidden=500,
W=None, bhid=None, bvis=None):
"""
Initialize the dA class by specifying the number of visible units (the
dimension d of the input ), the number of hidden units ( the dimension
d' of the latent or hidden space ) and the corruption level. The
constructor also receives symbolic variables for the input, weights and
bias. Such a symbolic variables are useful when, for example the input is
the result of some computations, or when weights are shared between the
dA and an MLP layer. When dealing with SdAs this always happens,
the dA on layer 2 gets as input the output of the dA on layer 1,
and the weights of the dA are used in the second stage of training
to construct an MLP.
:type numpy_rng: numpy.random.RandomState
:param numpy_rng: number random generator used to generate weights
:type theano_rng: theano.tensor.shared_randomstreams.RandomStreams
:param theano_rng: Theano random generator; if None is given one is generated
based on a seed drawn from `rng`
:type input: theano.tensor.TensorType
:paran input: a symbolic description of the input or None for standalone
dA
:type n_visible: int
:param n_visible: number of visible units
:type n_hidden: int
:param n_hidden: number of hidden units
:type W: theano.tensor.TensorType
:param W: Theano variable pointing to a set of weights that should be
shared belong the dA and another architecture; if dA should
be standalone set this to None
:type bhid: theano.tensor.TensorType
:param bhid: Theano variable pointing to a set of biases values (for
hidden units) that should be shared belong dA and another
architecture; if dA should be standalone set this to None
:type bvis: theano.tensor.TensorType
:param bvis: Theano variable pointing to a set of biases values (for
visible units) that should be shared belong dA and another
architecture; if dA should be standalone set this to None
"""
self.n_visible = n_visible
self.n_hidden = n_hidden
# create a Theano random generator that gives symbolic random values
if not theano_rng :
theano_rng = RandomStreams(rng.randint(2 ** 30))
# note : W' was written as `W_prime` and b' as `b_prime`
if not W:
# W is initialized with `initial_W` which is uniformely sampled
# from -4.*sqrt(6./(n_visible+n_hidden)) and 4.*sqrt(6./(n_hidden+n_visible))
# the output of uniform if converted using asarray to dtype
# theano.config.floatX so that the code is runable on GPU
initial_W = numpy.asarray(numpy_rng.uniform(
low=-4 * numpy.sqrt(6. / (n_hidden + n_visible)),
high=4 * numpy.sqrt(6. / (n_hidden + n_visible)),
size=(n_visible, n_hidden)), dtype=theano.config.floatX)
W = theano.shared(value=initial_W, name='W')
if not bvis:
bvis = theano.shared(value = numpy.zeros(n_visible,
dtype=theano.config.floatX), name='bvis')
if not bhid:
bhid = theano.shared(value=numpy.zeros(n_hidden,
dtype=theano.config.floatX), name='bhid')
self.W = W
# b corresponds to the bias of the hidden
self.b = bhid
# b_prime corresponds to the bias of the visible
self.b_prime = bvis
# tied weights, therefore W_prime is W transpose
self.W_prime = self.W.T
self.theano_rng = theano_rng
# if no input is given, generate a variable representing the input
if input == None:
# we use a matrix because we expect a minibatch of several examples,
# each example being a row
self.x = T.dmatrix(name='input')
else:
self.x = input
self.params = [self.W, self.b, self.b_prime]
def get_corrupted_input(self, input, corruption_level):
""" This function keeps ``1-corruption_level`` entries of the inputs the same
and zero-out randomly selected subset of size ``coruption_level``
Note : first argument of theano.rng.binomial is the shape(size) of
random numbers that it should produce
second argument is the number of trials
third argument is the probability of success of any trial
this will produce an array of 0s and 1s where 1 has a probability of
1 - ``corruption_level`` and 0 with ``corruption_level``
"""
return self.theano_rng.binomial(size=input.shape, n=1, p=1 - corruption_level) * input
def get_hidden_values(self, input):
""" Computes the values of the hidden layer """
return T.nnet.sigmoid(T.dot(input, self.W) + self.b)
def get_reconstructed_input(self, hidden ):
""" Computes the reconstructed input given the values of the hidden layer """
return T.nnet.sigmoid(T.dot(hidden, self.W_prime) + self.b_prime)
def get_cost_updates(self, corruption_level, learning_rate):
""" This function computes the cost and the updates for one trainng
step of the dA """
tilde_x = self.get_corrupted_input(self.x, corruption_level)
y = self.get_hidden_values( tilde_x)
z = self.get_reconstructed_input(y)
# note : we sum over the size of a datapoint; if we are using minibatches,
# L will be a vector, with one entry per example in minibatch
L = -T.sum(self.x * T.log(z) + (1 - self.x) * T.log(1 - z), axis=1 )
# note : L is now a vector, where each element is the cross-entropy cost
# of the reconstruction of the corresponding example of the
# minibatch. We need to compute the average of all these to get
# the cost of the minibatch
cost = T.mean(L)
# compute the gradients of the cost of the `dA` with respect
# to its parameters
gparams = T.grad(cost, self.params)
# generate the list of updates
updates = []
for param, gparam in zip(self.params, gparams):
updates.append((param, param - learning_rate * gparam))
return (cost, updates)
```

It is easy now to construct an instance of our `dA` class and train
it.

```
# allocate symbolic variables for the data
index = T.lscalar() # index to a [mini]batch
x = T.matrix('x') # the data is presented as rasterized images
######################
# BUILDING THE MODEL #
######################
rng = numpy.random.RandomState(123)
theano_rng = RandomStreams(rng.randint(2 ** 30))
da = dA(numpy_rng=rng, theano_rng=theano_rng, input=x,
n_visible=28 * 28, n_hidden=500)
cost, updates = da.get_cost_updates(corruption_level=0.2,
learning_rate=learning_rate)
train_da = theano.function([index], cost, updates=updates,
givens = {x: train_set_x[index * batch_size: (index + 1) * batch_size]})
start_time = time.clock()
############
# TRAINING #
############
# go through training epochs
for epoch in xrange(training_epochs):
# go through trainng set
c = []
for batch_index in xrange(n_train_batches):
c.append(train_da(batch_index))
print 'Training epoch %d, cost ' % epoch, numpy.mean(c)
end_time = time.clock
training_time = (end_time - start_time)
print ('Training took %f minutes' % (pretraining_time / 60.))
```

In order to get a feeling of what the network learned we are going to plot the filters (defined by the weight matrix). Bare in mind however, that this does not provide the entire story, since we neglect the biases and plot the weights up to a multiplicative constant (weights are converted to values between 0 and 1).

To plot our filters we will need the help of `tile_raster_images` (see
*Plotting Samples and Filters*) so we urge the reader to familiarize himself with
it. Also using the help of PIL library, the following lines of code will
save the filters as an image :

```
image = PIL.Image.fromarray(tile_raster_images(X=da.W.get_value(borrow=True).T,
img_shape=(28, 28), tile_shape=(10, 10),
tile_spacing=(1, 1)))
image.save('filters_corruption_30.png')
```

To run the code :

```
python dA.py
```

The resulted filters when we do not use any noise are :

The filters for 30 percent noise :