Chapter 13

The Manifold Perspective on

Auto-Encoders

Manifold learning is an approach to machine learning that is capitalizing on the manifold

hypothesis (Cayton, 2005; Narayanan and Mitter, 2010): the data generating distribution

is assumed to concentrate near regions of low dimensionality. The notion of manifold in

mathematics refers to continuous spaces that locally resemble Euclidean space, and the

term we should be using is really submanifold, which corresponds to a subset which has

a manifold structure. The use of the term manifold in machine learning is much looser

than its use in mathematics, though:

• the data may not be strictly on the manifold, but only near it,

• the dimensionality many not be the same everywhere,

• the notion actually referred to in machine learning naturally extends to discrete

spaces.

252

⇓

Figure 13.1: Top: data sampled from a distribution that is actually concentrated near

a one-dimensional manifold. Bottom: the underlying manifold that the learner should

infer.

Indeed, although the very notions of a manifold or submanifold are deﬁned for con-

tinuous spaces, the more general notion of probability concentration applies equally well

to discrete data. It is a kind of informal prior assumption about the data generating dis-

tribution that seems particularly well-suited for AI tasks such as those involving images,

video, speech, music, text, etc. In all of these cases the natural data has the property

253

that randomly choosing conﬁgurations of the observed variables according to a factored

distribution (e.g. uniformly) are very unlikely to generate the kind of observations we

want to model. What is the probability of generating a natural looking image by choos-

ing pixel intensities independently of each other? What is the probability of generating

a meaningful natural language paragraph by independently choosing each character in

a string? Exponentially tiny. That is because the probability distribution of interest

concentrates in a tiny volume of the total space of conﬁgurations. That means that to

ﬁrst degree, the problem of characterizing the data generating distribution can be re-

duced to a binary classiﬁcation problem: is this conﬁguration probable or not?. Is this a

grammatically and semantically plausible sentence in English? Is this a natural-looking

image? Answering these questions tell us much more about the nature of natural lan-

guage or text than the additional information one would have by being able to assign

a precise probability to each possible sequence of characters or set of pixels. Hence,

simply characterizing where probability concentrates is a fundamental importance, and

this is what manifold learning algorithms attempt to do. Because it is a where question,

it is more about geometry than about probability distributions, although we ﬁnd both

views useful when designing learning algorithms for AI tasks.

tangent directions

tangent plane

Data on a curved manifold

Figure 13.2: A two-dimensional manifold near which training examples are concentrated,

along with a tangent plane and its associated tangent directions, forming a basis that

specify the directions of small moves one can make to stay on the manifold.

254

In addition to the property of probability concentration, there is another one that

characterizes the manifold hypothesis: when a conﬁguration is probable it is generally

surrounded (at least in some directions) by other probable conﬁgurations. If a conﬁgu-

ration of pixels looks like a natural image, then there are tiny changes one can make

to the image (like translating everything by 0.1 pixel to the left) which yield another

natural-looking image. The number of independent ways (each characterized by a num-

ber indicating how much or whether we do it) by which a probable conﬁguration can be

locally transformed into another probable conﬁguration indicates the local dimension of

the manifold. Whereas maximum likelihood procedures tend to concentrate probability

mass on the training examples (which can each become a local maximum of probability

when the model overﬁts), the manifold hypothesis suggests that good solutions instead

concentrate probability along ridges of high probability (or their high-dimensional gener-

alization) that connect nearby examples to each other. This is illustrated in Figure 13.1.

An important characterization of a manifold is the set of its tangent planes. At a point

x on a d-dimensional manifold, the tangent plane is given by d basis vectors that span

the local directions of variation allowed on the manifold. As illustrated in Figure 13.2,

these local directions specify how one can change x inﬁnitesimally while staying on the

manifold.

255

Figure 13.3: Non-parametric manifold learning procedures build a nearest neighbor

graph whose nodes are training examples and arcs connect nearest neighbors. Various

procedures can thus obtain the tangent plane associated with a neighborhood of the

graph, and a coordinate system that associates each training example with a real-valued

vector position. It is possible to generalize to new examples by a form of interpolation.

So long as the number of examples is large enough to cover the curvature and twists

of the manifold, these approaches work well. Images from the QMUL Multiview Face

Dataset (Gong et al., 2000).

Manifold learning has mostly focused on unsupervised learning procedures that at-

tempt to capture these manifolds. Most of the initial machine learning research on

learning non-linear manifolds has focused on non-parametric methods based on the

nearest-neighbor graph. This graph has one node per training example and edges con-

necting near neighbors. Basically, these methods (Sch¨olkopf et al., 1998; Roweis and

Saul, 2000; Tenenbaum et al., 2000; Brand, 2003; Belkin and Niyogi, 2003; Donoho and

Grimes, 2003; Weinberger and Saul, 2004; Hinton and Roweis, 2003; van der Maaten

and Hinton, 2008) associate each of these nodes with a tangent plane that spans the

directions of variations associated with the diﬀerence vectors between the example and

its neighbors, as illustrated in Figure 13.3.

256

Figure 13.4: If the tangent plane at each location is known, then they can be tiled to

form a global coordinate system or a density function. In the ﬁgure, each local patch

can be thought of as a local Euclidean coordinate system or as a local ﬂat Gaussian

(or “pancake”). The average of all these Gaussians would provide an estimated density

function, as in the Manifold Parzen algorithm (Vincent and Bengio, 2003) or its non-local

neural-net based variant (Bengio et al., 2006b).

A global coordinate system can then be obtained through an optimization or solving

a linear system. Figure 13.4 illustrates how a manifold can be tiled by a large number

of locally linear Gaussian-like patches (or “pancakes”, because the Gaussians are ﬂat

in the tangent directions). However, there is a fundamental diﬃculty with such non-

parametric neighborhood-based approaches to manifold learning, raised in Bengio and

Monperrus (2005): if the manifolds are not very smooth (they have many ups and

downs and twists), one may need a very large number of training examples to cover

each one of these variations, with no chance to generalize to unseen variations. Indeed,

these methods can only generalize the shape of the manifold by interpolating between

neighboring examples. Unfortunately, the manifolds of interest in AI have many ups and

downs and twists and strong curvature, as illustrated in Figure 13.5. This motivates the

use of distributed representations and deep learning for capturing manifold structure,

which is the subject of this chapter.

257

tangent directions

tangent image

tangent directions

tangent image

shifted

image

high−contrast image

Figure 13.5: When the data are images, the tangent vectors can also be visualized like

images. Here we show the tangent vector associated with translation: it corresponds

to the diﬀerence between an image and a slightly translated version. This basically

extracts edges, with the sign (shown as red or green color) indicating whether ink is

being added or subtracted when moving to the right. The ﬁgure illustrates the fact that

the tangent vectors along the manifolds associated with translation, rotation and such

apparently benign transformations of images, can be highly curved: the tangent vector

at a position x is very diﬀerent from the tangent vector at a nearby position (slightly

translated image). Note how the two tangent vectors shown have almost no overlap,

i.e., their dot product is nearly 0, and they are as far as two such images can be. If

the tangent vectors of a manifold change quickly, it means that the manifold is highly

curved. This is going to make it very diﬃcult to capture such manifolds (with a huge

numbers of twists, ups and downs) with local non-parametric methods such as graph-

based non-parametric manifold learning. This point was made in Bengio and Monperrus

(2005).

The hope of many manifold learning algorithms, including those based on deep

learning and auto-encoders, is that one learns an explicit or implicit coordinate system

for the leading factors of variation that explain most of the structure in the unknown

data generating distribution. An example of explicit coordinate system is one where the

dimensions of the representation (e.g., the outputs of the encoder, i.e., of the hidden

258

units that compute the “code” associated with the input) are directly the coordinates

that map the unknown manifold. Training examples of a face dataset in which the

images have been arranged visually on a 2-D manifold are shown in Figure 13.6, with

the images laid down so that each of the two axes corresponds to one of the two angles

of rotation of the face.

Figure 13.6: Training examples of a face dataset – the QMUL Multiview Face

Dataset (Gong et al., 2000) – for which the subjects were asked to move in such a

way as to cover the two-dimensional manifold corresponding to two angles of rotation.

We would like learning algorithms to be able to discover and disentangle such factors.

Figure 13.7 illustrates such a feat.

However, the objective is to discover such manifolds, and Figure 13.7 illustrates the

images generated by a variational auto-encoder (Kingma and Welling, 2014) when the

two-dimensional auto-encoder code (representation) is varied on the 2-D plane. Note

how the algorithm actually discovered two independent factors of variation: angle of

rotation and emotional expression.

259

(a) Learned Frey Face manifold (b) Learned MNIST manifold

Figure 4: Visualisations of learned data manifold for generative models with two-dimensional latent

space, learned with AEVB. Since the prior of the latent space is Gaussian, linearly spaced coor-

dinates on the unit square were transformed through the inverse CDF of the Gaussian to produce

values of the latent variables z. For each of these values z, we plotted the corresponding generative

✓

(x|z) with the learned parameters ✓.

(a) 2-D latent space (b) 5-D latent space (c) 10-D latent space (d) 20-D latent space

Figure 5: Random samples from learned generative models of MNIST for different dimensionalities

of latent space.

B Solution of D



(z)||p

✓

(z)), Gaussian case

The variational lower bound (the objective to be maximized) contains a KL term that can often be

integrated analytically. Here we give the solution when both the prior p

✓

(z)=N(0, I) and the

posterior approximation q



(z|x

(i)

) are Gaussian. Let J be the dimensionality of z. Let µ and 

denote the variational mean and s.d. evaluated at datapoint i, and let µ

and 

simply denote the

j-th element of these vectors. Then:

✓

(z) log p(z) dz =

N(z; µ, 

) log N(z; 0, I) dz

= 

log(2⇡) 

j=1

(µ

+ 

)

(a) Learned Frey Face manifold (b) Learned MNIST manifold

Figure 4: Visualisations of learned data manifold for generative models with two-dimensional latent

space, learned with AEVB. Since the prior of the latent space is Gaussian, linearly spaced coor-

dinates on the unit square were transformed through the inverse CDF of the Gaussian to produce

values of the latent variables z. For each of these values z, we plotted the corresponding generative

✓

(x|z) with the learned parameters ✓.

(a) 2-D latent space (b) 5-D latent space (c) 10-D latent space (d) 20-D latent space

Figure 5: Random samples from learned generative models of MNIST for different dimensionalities

of latent space.

B Solution of D



(z)||p

✓

(z)), Gaussian case

The variational lower bound (the objective to be maximized) contains a KL term that can often be

integrated analytically. Here we give the solution when both the prior p

✓

(z)=N(0, I) and the

posterior approximation q



(z|x

(i)

) are Gaussian. Let J be the dimensionality of z. Let µ and 

denote the variational mean and s.d. evaluated at datapoint i, and let µ

and 

simply denote the

j-th element of these vectors. Then:

✓

(z) log p(z) dz =

N(z; µ, 

) log N(z; 0, I) dz

= 

log(2⇡) 

j=1

(µ

+ 

)

Figure 13.7: Two-dimensional representation space (for easier visualization), i.e., a Eu-

clidean coordinate system for Frey faces (left) and MNIST digits (right), learned by a

variational auto-encoder (Kingma and Welling, 2014). Figures reproduced with permis-

sion from the authors. The images shown are not examples from the training set but

actually generated by the model, simply by changing the 2-D “code”. On the left, one

dimension that has been discovered (horizontal) mostly corresponds to a rotation of the

face, while the other (vertical) corresponds to the emotional expression. The decoder

deterministically maps codes (here two numbers) to images. The encoder maps images

to codes (and adds noise, during training).

Another kind of interesting illustration of manifold learning involves the discovery

of distributed representations for words. Neural language models were initiated with the

work of Bengio et al. (2001b, 2003), in which a neural network is trained to predict

the next word in a sequence of natural language text, given the previous words, and

where each word is represented by a real-valued vector, called embedding or neural word

embedding.

Figure 13.8 shows such neural word embeddings reduced to two dimensions (orig-

inally 50 or 100) using the t-SNE non-linear dimensionality reduction algorithm (van

der Maaten and Hinton, 2008). The ﬁgures zooms into diﬀerent areas of the word-space

and illustrates that words that are semantically and syntactically close end up having

nearby embeddings.

260

Figure 13.8: Two-dimensional representation space (for easier visualization), of English

words, learned by a neural language model as in Bengio et al. (2001b, 2003), with t-SNE

for the non-linear dimensionality reduction from 100 to 2. Diﬀerent regions are zoomed

to better see the details. At the global level one can identify big clusters correspond-

ing to part-of-speech, while locally one sees mostly semantic similarity explaining the

neighborhood structure.

13.1 Manifold Learning via Regularized Auto-Encoders

Auto-encoders have been introduced in Section 10.1. What is their connection to man-

ifold learning? This is what we discuss here.

We denote f the encoder function, with h = f(x) the representation of x, and g the

decoding function, with ˆx = g(h) the reconstruction of x, although in some cases the

encoder is a conditional distribution q(h|x) and the decoder is a conditional distribution

P (x|h).

What all auto-encoders have in common, when they are prevented from simply

learning the identity function for all possible input x, is that training them involves a

compromise between two “forces”:

1. Learning a representation h of training examples x such that x can be approxi-

mately recovered from h through a decoder. Note that this needs not be true for

261

any x, only for those that are probable under the data generating distribution.

2. Some constraint or regularization is imposed, either on the code h or on the com-

position of the encoder/decoder, so as to prevent the auto-encoder from achieving

perfect reconstruction everywhere. We can think of these constraints or regulariza-

tion as a preference for solutions in which the representation is as constant as pos-

sible in as many directions as possible. In the case of the bottleneck auto-encoder

a ﬁxed number of representation dimensions is allowed, that is smaller than the

dimension of x. In the case of sparse auto-encoders (Section 13.3) the represen-

tation elements h

are pushed towards 0. In the case of denoising auto-encoders

(Section 13.4), the encoder/decoder function is encouraged to be contractive (have

small derivatives). In the case of the contractive auto-encoder (Section 13.5), the

encoder function alone is encouraged to be contractive, while the decoder function

is tied (by symmetric weights) to the encoder function. In the case of the vari-

ational auto-encoder (Section 17.7.1), a prior log P (h) is imposed on h to make

its distribution concentrate as much as possible. Note how in the limit, for all of

these cases, the regularization prefers constant representations.

How can these two forces (reconstruction error on one hand, and lack of representa-

tional power on the other hand) be reconciled? The solution of the optimization problem

is that only the variations that are needed to distinguish training examples need to be

represented. If the data generating distribution concentrates near a low-dimensional

manifold, this yields representations that implicitly capture a local coordinate for this

manifold: only the variations tangent to the manifold around x need to correspond to

changes in h = f(x). Hence the encoder learns a mapping from the embedding space x

to a representation space, a mapping that is only sensitive to changes along the manifold

directions, but that is insensitive to changes orthogonal to the manifold. This idea is

illustrated in Figure 13.9.

262

Figure 13.9: A regularized auto-encoder or a bottleneck auto-encoder has to reconcile

two forces: reconstruction error (which forces it to keep enough information to distin-

guish training examples from each other), and a regularizer or constraint that aims at

reducing its representational ability, to make it as insensitive as possible to the input

in as many directions as possible. The solution is for the learned representation to

be sensitive to changes along the manifold (green arrow going to the right, tangent to

the manifold) but invariant to changes orthogonal to the manifold (blue arrow going

down). This yields to contraction of the representation in the directions orthogonal to

the manifold.

13.2 Probabilistic Interpretation of Reconstruction Error

as Log-Likelihood

Although traditional auto-encoders (like traditional neural networks) were introduced

with an associated training loss, just like for neural networks, that training loss can

generally be given a probabilistic interpretation as a log-likelihood.

We have already covered that in general for feedforward neural networks in Sec-

tion 6.2.2. Like prediction error for regular feedforward neural networks, reconstruction

error for auto-encoders does not have to be squared error. When we view the loss as

negative log-likelihood, the interpretation of the reconstruction error as

L = −log P(x|h)

where h is the representation, which may generally be obtained through an encoder

taking x as input.

263

h = f(x)

L = log P (x|g(f(x)))

Figure 13.10: The computational graph of an auto-encoder, which is trained to maximize

the probability assigned by the decoder g to the data point x, given the output of the

encoder h = f(x). The training objective is thus L = −log P (x|g(f(x))), which ends

being squared reconstruction error if we choose a Gaussian reconstruction distribution

with mean g(f (x)), and cross-entropy if we choose a factorized Bernoulli reconstruction

distribution with means g(f(x)).

An advantage of this view is that it immediately tells us what kind of loss function

one should use depending on the nature of the input. If the input is real-valued and

unbounded, then squared error is a reasonable choice of reconstruction error, and cor-

responds to P (x|h) being Normal. If the input is a vector of bits, then cross-entropy

is a more reasonable choice, and corresponds to P (x|h) =



P (x

|h) with x

|h being

Bernoulli-distributed. We then view the decoder g(h) as computing the parameters of

the reconstruction distribution, i.e., P(x|h) = P(x|g(h)).

Another advantage of this view is that we can think about the training of the decoder

as estimating the conditional distribution P (x|h), which comes handy in the probabilistic

interpretation of denoising auto-encoders, allowing us to talk about the distribution P (x)

implicitly represented by the auto-encoder (see Section 13.4 below and Section 17.8

for more details). In the same spirit, we can rethink the notion of encoder from a

simple function to a conditional distribution Q(h|x), with a special case being when

Q(h|x) is a Dirac at some particular value. Equivalently, thinking about the encoder

as a distribution corresponds to injecting noise inside the auto-encoder. This view is

developed further in Sections 17.7.1 and 17.9.

264

13.3 Sparse Representations

Sparse auto-encoders are auto-encoders which learn a sparse representation, i.e., one

whose elements are often either zero or close to zero. Sparse coding was introduced

in Section 10.2.4 as a linear factor model in which the prior on the representation

encourages values at or near 0. In Section 13.3.1, we see how ordinary auto-encoders

can be prevented from learning a useless identity transformation by using a sparsity

penalty rather than a bottleneck. The main diﬀerence between a sparse auto-encoder

and sparse coding is that sparse coding has no explicit parametric encoder, whereas

sparse auto-encoders have one. The “encoder” of sparse coding is the algorithm that

performs the approximate inference, i.e., looks for

∗

(x) = arg max

log P (h|x) = arg min

||x −(b + W h)||

− log P (h) (13.1)

where P(h) is a “sparse” prior that puts more probability mass around h = 0, such as

the Laplacian prior, with factorized marginals

P (h

) =

λ|h

(13.2)

or the Student-t prior, with factorized marginals

P (h

) ∝

(1 +

)

ν+1

. (13.3)

The advantages of such a non-parametric encoder and the sparse coding approach over

sparse auto-encoders are that

1. it can in principle minimize the combination of reconstruction error and log-prior

better than any parametric encoder,

2. performs what is called explaining away (see Figure 9.8), i.e., it allows to “choose”

some “explanations” (hidden factors) and inhibits the others.

The disadvantages are that

1. computing time for encoding the given input x, i.e., performing inference (comput-

ing the representation h that goes with the given x) can substantially larger than

with a parametric encoder (because an optimization must be performed), and

2. the resulting encoder function can be highly non-smooth and non-linear (with two

nearby x’s being associated with very diﬀerent h’s), potentially making it more

diﬃcult for the downstream layers to properly generalize.

In Section 13.3.2, we describe PSD (Predictive Sparse Decomposition), which com-

bines a non-parametric encoder (as in sparse coding, with the representation obtained

via an optimization) and a parametric encoder (like in the sparse auto-encoder). Sec-

tion 13.4 introduces the Denoising Auto-Encoder (DAE), which puts pressure on the

265

representation by requiring it to extract information about the underlying distribution

and where it concentrates, so as to be able to denoise a corrupted input. Section13.5

describes the Contractive Auto-Encoder (CAE), which optimizes an explicit regulariza-

tion penalty that aims at making the representation as insensitive as possible to the

input, while keeping the information suﬃcient to reconstruct the training examples.

13.3.1 Sparse Auto-Encoders

A sparse auto-encoder is simply an auto-encoder whose training criterion involves a

sparsity penalty Ω(h) in addition to the reconstruction error:

L = −log P (x|g(h)) + Ω(h) (13.4)

where g(h) is the decoder output and typically we have h = f(x), the encoder output.

We can think of that penalty Ω(h) simply as a regularizer or as a log-prior on the

representations h. For example, the sparsity penalty corresponding to the Laplace prior

is the absolute value sparsity penalty (Glorot et al., 2011a):

Ω(h) = λ



−log P(h) =



log

+ λ|h

| = mconst + Ω(h) (13.5)

where the constant term depends only of λ and not h (which we typically ignore in the

training criterion because we consider λ as a hyper-parameter rather than a parameter).

Similarly, the sparsity penalty corresponding to the Student-t prior (Olshausen and

Field, 1997) is

Ω(h) =



ν + 1

log(1 +

) (13.6)

where ν is considered to be a hyper-parameter.

TODO: reference to early LeCun work with sparse coding and sparse auto-encoders

However, the sparsity penalty does not need to have a probabilistic interpretation.

For example, Goodfellow et al. (2009) successfully used the following sparsity penalty,

which does not try to bring h

all the way down to 0, but only towards some low target

value such as ρ = 0.05.

Ω(h) =



ρ log h

+ (1 −ρ) log(1 −h

) (13.7)

where 0 < h

< 1, usually with h

= sigmoid(a

). This is just the cross-entropy between

the Bernoulli distributions with probability p = h

and the target Bernoulli distribution

with probability p = ρ.

One way to achieve actual zeros in h for sparse (and denoising) auto-encoders was

introduced in Glorot et al. (2011c). The idea is to use a half-rectiﬁer (a.k.a. sim-

ply as “rectiﬁer”) or ReLU (Rectiﬁed Linear Unit) as the output non-linearity of the

266

encoder. With a prior that actually pushes the representations to zero (like the ab-

solute value penalty), one can thus indirectly control the average number of zeros in

the representation. The same strategy was ﬁrst successfully used for deep feedforward

networks in Glorot et al. (2011a), achieving for the ﬁrst time the ability to train fairly

deep supervised networks without the need for unsupervised pre-training.

Interestingly, the “regularizer” used in sparse auto-encoders does not conform to the

classical interpretation of regularizers as priors on the parameters. That interpretation

of the regularizer comes from the MAP (Maximum A Posteriori) point estimation (see

Section 5.5.1) of parameters associated with the Bayesian view of parameters as random

variables and the joint distribution of data x and parameters θ (see Section 5.7):

arg max

P (θ|x) = arg max

log P (x|θ) + log P(θ)

where the ﬁrst term is the usual data log-likelihood term and the second term, the

log-prior over parameters, incorporates the preference over particular values of θ.

Here, instead, the regularizer corresponds to a log-prior over latent variables, or

equivalently in the case of sparse auto-encoders, predictive sparse decomposition and

contractive auto-encoders, as a preference over a learned function of the data. This

makes such a regularizer data-dependent, unlike the classical parameter log-prior. Specif-

ically, in the case of the sparse auto-encoder, it says that we prefer an encoder whose

output produces values closer to 0. Indirectly (when we marginalize over the training

distribution), this is also indicating a preference over parameters, of course.

13.3.2 Predictive Sparse Decomposition

Predictive Sparse Decomposition (PSD)...

13.4 Denoising Auto-Encoders

The Denoising Auto-Encoder (DAE) was ﬁrst proposed as a means of forcing an auto-

encoder to learn to capture the data distribution without an explicit constraint on either

the dimension or the sparsity of the learned representation. It was motivated by the

idea that in order to fully capture a complex distribution, an auto-encoder needs to have

at least as many hidden units as needed by the complexity of that distribution. Hence

its dimensionality should not be restricted to the input dimension.

The principle of the denoising auto-encoder is deceptively simple and illustrated in

Figure 13.11: the encoder sees as input a corrupted version of the input, but the decoder

tries to reconstruct the clean uncorrupted input.

267

˜x

C(˜x|x)

L = log P (x|g(f (˜x)))

h = f(˜x)

Figure 13.11: The computational graph of a denoising auto-encoder, which is trained to

reconstruct the clean data point x from it corrupted version ˜x, i.e., to minimize the loss

L = −log P (x|g(f(˜x))), where ˜x is a corrupted version of the data example x, obtained

through a given corruption process C(˜x|x).

Mathematically, and following the notations used in this chapter, this can be for-

malized as follows. We introduce a corruption process C(

X|X) which represents a

conditional distribution over corrupted samples

X, given a data sample X. The auto-

encoder then learns a reconstruction distribution P(X|

X) estimated from training pairs

(x, ˜x), as follows:

1. Sample a training example X = x from the data generating distribution (the

training set).

2. Sample a corrupted version

X = ˜x from the conditional distribution C(

X|X = x).

3. Use (x, ˜x) as a training example for estimating the auto-encoder reconstruction

distribution P(x|˜x) = P (x|g(h)) with h the output of encoder f(˜x) and g(h) the

output of the decoder.

Typically we can simply perform gradient-based approximate minimization (such as

minibatch gradient descent) on the negative log-likelihood −log P (x|h), i.e., the de-

noising reconstruction error, using back-propagation to compute gradients, just like for

regular feedforward neural networks (the only diﬀerence being the corruption of the

input and the choice of target output).

We can view this training objective as performing stochastic gradient descent on the

denoising reconstruction error, but where the “noise” now has two sources:

1. the choice of training sample x from the data set, and

2. the random corruption applied to x to obtain ˜x.

268

We can therefore consider that the DAE is performing stochastic gradient descent

on the following expectation:

−E

x∼Q(X)

˜x∼C(˜x|x)

log P (x|g(f(˜x)))

where Q(X) is the training distribution.

C(˜x|x)

˜x

g(f(˜x)) ⇡ E[x|˜x]

Figure 13.12: A denoising auto-encoder is trained to reconstruct the clean data point x

from it corrupted version ˜x. In the ﬁgure, we illustrate the corruption process C(˜x|x)

(grey circle of equiprobable corruptions) acting on examples x (red crosses) lying near

a low-dimensional manifold near which probability concentrates. When the denoising

auto-encoder is trained to minimize the average of squared error ||g(f(˜x) − x||

, the

reconstruction estimates E[x|˜x], which points approximately orthogonally towards the

manifold. It thus learns a vector ﬁeld g(f(x)) − x (the green arrows) and it turns out

that this vector ﬁeld estimates the gradient ﬁeld

∂ log Q(x)

∂x

, where Q is the unknown data

generating distribution.

13.4.1 Learning a Vector Field that Estimates a Gradient Field

As illustrated in Figure 13.12, a very important property of DAEs is that their training

criterion makes the auto-encoder learn a vector ﬁeld g(f(x)) that estimates the gradient

ﬁeld (or score)

∂ log Q(x)

∂x

. This has been proven (Alain and Bengio, 2012, 2013) in the

case where the corruption and the reconstruction distributions are Gaussian (and of

course x is continuous-valued), i.e., with the squared error denoising error

||g(f(˜x)) −x||

269

and corruption

X = ˜x|x) = N(˜x; µ = x, Σ = σ

with noise variance σ

Figure 13.13: Vector ﬁeld learned by a denoising auto-encoder around a 1-D curved

manifold near which the data (orange circles) concentrates in a 2-D space. Each arrow

is proportional to the reconstruction minus input vector of the auto-encoder and points

towards higher probability according to the implicitly estimated probability distribution.

Note that the vector ﬁelds has zeros at both peaks of the estimated density function (on

the data manifolds) and at troughs (local minima) of that density function, e.g., on the

curve that separates diﬀerent arms of the spiral or in the middle of it.

More precisely, the main theorem states that

g(f(x))−x

is a consistent estimator of

∂ log Q(x)

∂x

g(f(x)) − x

→

∂ log Q(x)

∂x

so long as f and g have suﬃcient capacity to represent the true score (and assuming

that the expected training criterion can be minimized, as usual when proving consistency

associated with a training objective).

270

Although it was intuitively appealing that in order to denoise correctly one must

capture the training distribution, this result makes it mathematically very clear. Fig-

ure 13.13 (see details of experiment in Alain and Bengio (2013)) illustrates this. Note

how the norm of reconstruction error (i.e. the norm of the vectors shown in the ﬁgure)

is related to but diﬀerent from the energy (unnormalized log-density) associated with

the estimated model. The energy should be low only where the probability is high. The

reconstruction error (norm of the estimated score vector) is also low where probability

is high, but it can also be low at maxima of energy (minima of probability).

13.4.2 Turning the Gradient Field into a Generative Model

Since each application of the encoder-decoder pair brings us towards a more probable

conﬁguration, we can use this to actually sample from the estimated distribution. If you

consider most Monte-Carlo Markov Chain (MCMC) algorithms, they have two elements:

1. move from lower probability conﬁgurations towards higher probability conﬁgura-

tions, and

2. inject randomness so that the chain moves around (and does not stay stuck at

some peak of probability, or mode) and has a chance to visit the whole space, with

probability equal to the underlying model.

So conceptually all one needs to do is perform encode-decode operations as well as inject

noise, as hinted at in (Mesnil et al., 2012; Rifai et al., 2012). But how much and how?

271

˜x

= g(h

)

= f(˜x

)

P (X|!)

X|X)

t+1

Figure 13.14: Each step of the Markov chain associated with a trained denoising auto-

encoder, that generates the samples from the probabilistic model implicitly trained by

the denoising reconstruction criterion. Each step consists in (a) injecting corruption C

in state x, yielding ˜x, (b) encoding it with f, (c) decoding the result with g, and (d) from

this, sampling a new state from the reconstruction distribution P(X|ω = g(f (˜x))). In

the typical squared reconstruction error case, g(h) = ˆx and estimates E[x|˜x], corruption

consists in adding Gaussian noise and sampling from P(X|ω) consists in adding another

Gaussian noise to the reconstruction ˆx. The latter noise level should correspond to the

mean squared error of reconstructions, whereas the injected noise is a hyper-parameter

that controls the mixing speed as well as the extent to which the estimator smoothes

the empirical distribution (Vincent, 2011). In the ﬁgure, only the C and P conditionals

are stochastic steps (f and g are deterministic computations), although noise can also

be injected inside the auto-encoder, as in generative stochastic networks (Bengio et al.,

2014)

The answer has been given by Bengio et al. (2013) (Theorem 1), which state that

each step of the Markov chain that generates from the estimated distribution consists

of the following sub-steps, illustrated in Figure 13.14:

1. starting from the previous state x, inject corruption noise, sampling ˜x from C(˜x|x).

2. Encode ˜x into h = f(˜x).

3. Decode h to obtain the parameters ω = g(h) of P (X|ω = g(h)).

4. Sample the next state x from P (X|ω = g(h)).

The theorem states that if the auto-encoder P (X|

X) forms a consistent estimator of

corresponding true conditional distribution, then the stationary distribution of the above

272

Markov chain forms a consistent estimator (albeit an implicit one) of the data generating

distribution of X.

P (

x|x)

P (

x|x)

P (

x|x)

P (x|

Figure 13.15: Illustration of one step of the sampling Markov chain associated with a

denoising auto-encoder (see also Figure 13.14). In the ﬁgure, the data (black circles) are

sitting near a low-dimensional manifold (a spiral, here), and the two stochastic steps of

the Markov chain are ﬁrst to corrupt x (clean car image, green circle) into ˜x (noisy car

image, blue circle) via C(˜x|x) (here an isotropic Gaussian noise in green), and then to

sample a new x via the estimated denoising P(x|˜x). Note how there are many possible

x which could have given rise to ˜x, and these all lie on the manifold in the neighborhood

of ˜x, hence the ﬂattened shape of P (x|˜x) (in blue). Modiﬁed from a ﬁgure ﬁrst created

and graciously authorized by Jason Yosinski.

Figure 13.15 illustrates the sampling process of the DAE in a way that complements

Figure 13.14, with a speciﬁc imagined example. For a more elaborate discussion of the

probabilistic nature of denoising auto-encoders, and their generalization (Bengio et al.,

2014), Generative Stochastic Networks (GSNs), see Section 17.9. In particular, the noise

does not have to be injected only in the input, and it could be injected anywhere along

the chain. GSNs also generalize DAEs by allowing the state of the Markov chain to be

extended beyond the visible variable x, to include also some latent variable h. Finally,

Section 17.9 discusses training strategies for DAEs that are aimed at making it a better

generative model and not just a good feature learner.

273

See also Section 15.5 for the connection to score matching, an induction principle

that is an alternative to the maximum likelihood principle.

13.5 Contractive Auto-Encoders

The Contractive Auto-Encoder (CAE) introduces an explicit regularizer on the code

h = f (x), encouraging the derivatives of f to be as small as possible. If it weren’t for

the opposing force of reconstruction error which attempts to make the code h keep all

the information necessary to reconstruct training examples, the CAE penalty would yield

a code h that is constant and does not depend on the input x. The compromise between

these two forces yields an auto-encoder whose derivatives

∂h

∂x

are tiny in most directions,

except those that are needed to reconstruct training examples, i.e., the directions that

are tangent to the manifold near which data concentrate.

274