Extracting and Composing Robust Features with Denoising
Autoencoders
Pascal Vincent vincentp@iro.umontreal.ca Hugo Larochelle larocheh@iro.umontreal.ca Yoshua Bengio bengioy@iro.umontreal.ca Pierre-Antoine Manzagol manzagop@iro.umontreal.ca Universit´e de Montr´e al,Dept.IRO,CP6128,Succ.Centre-Ville,Montral,Qubec,H3C3J7,Canada
Abstract
Previous work has shown that the difficul-
ties in learning deep generative or discrim-
inative models can be overcome by an ini-
tial unsupervised learning step that maps in-
puts to useful intermediate representations.
We introduce and motivate a new training
principle for unsupervised learning of a rep-
resentation based on the idea of making the
learned representations robust to partial cor-
ruption of the input pattern.This approach
can be used to train autoencoders,and these
denoising autoencoders can be stacked to ini-
tialize deep architectures.The algorithm can
be motivated from a manifold learning and
information theoretic perspective or from a
generative model perspective.Comparative
experiments clearly show the surprising ad-
vantage of corrupting the input of autoen-
coders on a pattern classification benchmark
suite.
1.Introduction
Recent theoretical studies indicate that deep architec-tures(Bengio&Le Cun,2007;Bengio,2007)may be needed to efficiently model complex distributions and achieve better generalization performance on challeng-ing recognition tasks.The belief that additional levels of functional composition will yield increased repre-sentational and modeling power is not new(McClel-land et al.,1986;Hinton,1989;Utgoff&Stracuzzi, 2002).However,in practice,learning in deep archi-tectures has proven to be difficult.One needs only Appearing in Proceedings of the25th International Confer-ence on Machine Learning,Helsinki,Finland,2008.Copy-right2008by the author(s)/owner(s).to ponder the difficult problem of inference in deep directed graphical models,due to“explaining away”. Also looking back at the history of multi-layer neural networks,their difficult optimization(Bengio et al., 2007;Bengio,2007)has long prevented reaping the ex-pected benefits of going beyond one or two hidden lay-ers.However this situation has recently changed with the successful approach of(Hinton e
t al.,2006;Hinton &Salakhutdinov,2006;Bengio et al.,2007;Ranzato et al.,2007;Lee et al.,2008)for training Deep Belief Networks and stacked autoencoders.
One key ingredient to this success appears to be the use of an unsupervised training criterion to perform a layer-by-layer initialization:each layer is atfirst trained to produce a higher level(hidden)represen-tation of the observed patterns,based on the repre-sentation it receives as input from the layer below, by optimizing a local unsupervised criterion.Each level produces a representation of the input pattern that is more abstract than the previous level’s,be-cause it is obtained by composing more operations. This initialization yields a starting point,from which a globalfine-tuning of the model’s parameters is then performed using another training criterion appropriate for the task at hand.This technique has been shown empirically to avoid getting stuck in the kind of poor solutions one typically reaches with random initializa-tions.While unsupervised learning of a mapping that produces“good”intermediate representations of the input pattern seems to be key,little is understood re-garding what constitutes“good”representations for initializing deep architectures,or what explicit crite-ria may guide learning such representations.We know of only a few algorithms that seem to work well for this purpose:Restricted Boltzmann Machines(RBMs) trained with contrastive divergence on one hand,and various types of autoencoders on the other.
The present research begins with the question of what
explicit criteria a good intermediate representation should satisfy.Obviously,it should at a minimum re-tain a certain amount of“information”about its input, while at the same time being constrained to a given a real-valued vector of a given size in the case of an autoencoder).A supplemental criterion that has been proposed for such models is sparsity of the representation(Ranzato et al.,2008;Lee et al.,2008). Here we hypothesize and investigate an additional spe-cific criterion:robustness to partial destruction of the ,partially destroyed inputs should yield almost the same representation.It is motivated by the following informal reasoning:a good represen-tation is expected to capture stable structures in the form of dependencies and regularities characteristic of the(unknown)distribution of its observed input.For high dimensional redundant input(such as images)at least,such structures are likely to depend on evidence gathered from a combination of many input dimen-sions.They should thus be recoverable from partial observation only.A hallmark of this is our human ability to recognize partially occluded or corrupted im-ages.Further evidence is our ability to form a high level concept associated to multiple modalities(such as image and sound)and recall it even when some of the modalities are missing.
To validate our hypothesis and assess its usefulness as one of the guiding principles in learning dee
p architec-tures,we propose a modification to the autoencoder framework to explicitly integrate robustness to par-tially destroyed inputs.Section2describes the algo-rithm in details.Section3discusses links with other approaches in the literature.Section4is devoted to a closer inspection of the model from different theo-retical standpoints.In section5we verify empirically if the algorithm leads to a difference in performance. Section6concludes the study.
2.Description of the Algorithm
2.1.Notation and Setup
Let X and Y be two random variables with joint prob-ability density p(X,Y),with marginal distributions p(X)and p(Y).Throughout the text,we will use the following notation:Expectation:E E p(X)[f(X)]=
p(x)f(x)d x.Entropy:I H(X)=I H(p)= E E p(X)[−log p(X)].Conditional entropy:I H(X|Y)= E E p(X,Y)[−log p(X|Y)].Kullback-Leibler divergence:
I D KL(p q)=E E p(X)[log p(X)
q(X)
]
.Cross-entropy:I H(p q)= E E p(X)[−log q(X)]=I H(p)+I D KL(p q).Mutual infor-mation:I(X;Y)=I H(X)−I H(X|Y).Sigmoid:s(x)=
1
1+e−x and s(x)=(s(x1),...,s(x d))T.Bernoulli dis-
tribution with meanµ:Bµ(x).and by extension
Bµ(x)=(Bµ
1
(x1),...,Bµ
d
(x d)).
The setup we consider is the typical supervised learn-
ing setup with a training set of n(input,target)pairs
D n={(x(1),t(1))...,(x(n),t(n))},that we suppose
to be an i.i.d.sample from an unknown distribution
q(X,T)with corresponding marginals q(X)and q(T).
2.2.The Basic Autoencoder
We begin by recalling the traditional autoencoder
model such as the one used in(Bengio et al.,2007)
to build deep networks.An autoencoder takes an
input vector x∈[0,1]d,andfirst maps it to a hid-
den representation y∈[0,1]d through a deterministic
mapping y=fθ(x)=s(Wx+b),parameterized by
θ={W,b}.W is a d ×d weight matrix and b
is a bias vector.The resulting latent representation
y is then mapped back to a“reconstructed”vector
z∈[0,1]d in input space z=gθ (y)=s(W y+b )
withθ ={W ,b }.The weight matrix W of the
reverse mapping may optionally be constrained by
W =W T,in which case the autoencoder is said to
have tied weights.Each training x(i)is thus mapped
to a corresponding y(i)and a reconstruction z(i).The
parameters of this model are optimized to minimize
the average reconstruction error:
θ ,θ =arg min
θ,θ
1
n
n
i=1
L
x(i),z(i)
=arg min
θ,θ
1
n
n
i=1
数据
采集装置L
x(i),gθ (fθ(x(i)))
where L is a loss function such as the traditional
squared error L(x,z)= x−z 2.An alternative loss,
suggested by the interpretation of x and z as either
bit vectors or vectors of bit probabilities(Bernoullis)
is the reconstruction cross-entropy:
L I H(x,z)=I H(B x B z)
奥修书
=−
d
k=1
[x k log z k+(1−x k)log(1−z k)](2)
Note that if x is a binary vector,L I H(x,z)is a negative
log-likelihood for the example x,given the Bernoulli
parameters z.Equation1with L=L I H can be written
θ ,θ =arg min
θ,θ
E E q0(X)[L I H(X,gθ (fθ(X)))](3)
where q0(X)denotes the empirical distribution asso-
ciated to our n training inputs.This optimization will
typically be carried out by stochastic gradient descent.
2.3.The Denoising Autoencoder
To test our hypothesis and enforce robustness to par-tially destroyed inputs we modify the basic autoen-coder we just described.We will now train it to recon-struct a clean “repaired”input from a corrupted ,par-tially destroyed one.This is done by first corrupting the initial input x to get a partially destroyed version ˜x by means of a stochastic mapping ˜x ∼q D (˜x |x ).In our experiments,we considered the following corrupt-ing process,parameterized by the desired proportion νof “destruction”:for each input x ,a fixed number νd of components are chosen at random,and their value is forced to 0,while the others are left untouched.All information about the chosen components is thus re-moved from that particuler input pattern,and the au-toencoder will be trained to “fill-in”these artificially introduced “blanks”.Note that alternative corrupting
noises could be considered 1.The corrupted input ˜x
is then mapped,as with the basic autoencoder,to a hid-den representation y =f θ(˜x )=s (W ˜x +b )from which we reconstruct a z =g θ (y )=s (W
y +b )(see figure 1for a schematic representation of the process).As before the parameters are trained to minimize the av-erage reconstruction error L I H (x ,z )=I H (B x B z )over a training have z as close as possible to the uncorrupted input x .But the key difference is that z
is now a deterministic function of ˜x
rather than x and thus the result of a stochastic mapping of x
.
x
x
y
z
Figure 1.An example x is corrupted to ˜x
.The autoen-coder then maps it to y and attempts to reconstruct x .
Let us define the joint distribution
q 0(X, X,Y )=q 0(X )q D ( X |X )δf θ(e X )
(Y )(4)
where δu (v )puts mass 0when u =v .Thus Y is a
deterministic function of X
.q 0(X, X,Y )is param-eterized by θ.The objective function minimized by stochastic gradient descent becomes:
arg min θ,θ
E E q 0(X,e X ) L I H X,g θ (f θ( X ))
.
(5)So from the point of view of the stochastic gradient de-scent algorithm,in addition to picking an input sam-ple from the training set,we will also produce a ran-dom corrupted version of it,and take a gradient step
1
The approach we describe and our analysis is not spe-cific to a particular kind of corrupting noise.
公众安全感
towards reconstructing the uncorrupted version from the corrupted version.Note that in this way,the au-toencoder cannot learn the identity,unlike the basic autoencoder,thus removing the constraint that d <d or the need to regularize specifically to avoid such a trivial solution.
2.4.Layer-wise Initialization and Fine Tuning The basic autoencoder has been used as a building block to train deep networks (Bengio et al.,2007),with the representation of the k -th layer used as input for the (k +1)-th,and the (k +1)-th layer trained after the k -th has been trained.After a few layers have been trained,the parameters are used as initialization for a network optimized with respect to a supervised train-ing criterion.This greedy layer-wise procedure has been shown to yield significantly better local minima than random initialization of deep networks ,achieving better generalization on a number of tasks (Larochelle et al.,2007).
The procedure to train a deep network using the de-noising autoencoder is similar.The only difference is how each layer is ,to minimize the crite-rion in eq.5instead of eq.3.Note that the corrup-tion process q D is only used during training,but not for propagating representations from the raw input to higher-level representations.Note also that when layer k is trained,it receives as input the uncorrupted out-put of the previous layers.
3.Relationship to Other Approaches
Our training procedure for the denoising autoencoder involves learning to recover a clean input from a cor-rupted version,a task known as denoising .The prob-lem of image denoising,in particular,has been exten-sively studied in the image processing community and many recent developments rely on machine learning approaches (Roth and Black (2005);Elad and Aharon (2006);Hammond and Simoncelli (2007)).A particular form of gated autoencoders has also been used for denoising in Memisevic (2007).Denoising us-ing autoencoders was actually introduced much ear-lier (LeCun,1987;Gallinari et al.,1987),as an alter-native to Hopfield models (Hopfield,1982).Our ob-jective however is fundamentally different from that of developing a competitive image denoising algorithm.We investigate explicit robustness to corrupting noise as a novel criterion guiding the learning of suitable in-termediate representations to initialize a deep network.Thus our corruption+de
noising procedure is applied not only on the input,but also recursively to interme-diate representations.
The approach also bears some resemblance to the well known technique of augmenting the training data with stochastically“transformed”augment-ing a training set by transforming original bitmaps through small rotations,translations,and scalings is known to improvefinal classification performance.In contrast to this technique our approach does not use any prior knowledge of image topology,nor does it pro-duce extra labeled examples for supervised training. We use corrupted patterns in a spe-cific to images)unsupervised initialization step,while the supervised training phase uses the unmodified orig-inal data.
There is a well known link between“training with noise”and regularization:they are equivalent for small additive noise(Bishop,1995).By contrast,our cor-ruption process is a large,non-additive,destruction of information.We train autoencoders to”fill in the blanks”,not merely be smooth functions(regulariza-tion).Also in our experience,regularized autoencoders (i.e.with weight decay)do not yield the quantitative jump in performance and the striking qualitative dif-ference observed in thefilters that we get with denois-ing autoencoders.
There are also similarities with the work of(Doi et al., 2006)on robust coding over noisy channels.In their framework,a linear encoder is to encode a clean input for optimal transmission over a noisy channel to a de-coder that reconstructs the input.This work was later extended to robustness to noise in the input,in a pro-posal for a model of retinal coding(Doi&Lewicki, 2007).Though some of the inspiration behind our work comes from neural coding and computation,our goal is not to account for experimental data of neu-ronal activity as in(Doi&Lewicki,2007).Also,the non-linearity of our denoising autoencoder is crucial for its use in initializing a deep neural network.
It may be objected that,if our goal is to handle missing values correctly,we could have more naturally defined a proper latent variable generative model,and infer the posterior over the latent(hidden)representation in the presence of missing inputs.But this usually requires a costly marginalization2which has to be carried out for each new example.By contrast,our approach tries to learn a fast and robust deterministic mapping fθfrom examples of already corrupted inputs.The bur-den is on learning such a constrained mapping during training,rather than on unconstrained inference at use time.We expect this may force the model to capture implicit invariances in the data,and result in interest-2as for RBMs,where it is exponential in the number of missing values ing features.Also note that in section4.2we will see how our learning algorithm for the denoising autoen-coder can be viewed as a form of variational inference in a particular generative model.
4.Analysis of Denoising Autoencoders The above intuitive motivation for the denoising au-toencoder was given with the perspective of discover-ing robust representations.In the following,which can be skipped without hurting the remainder of the paper, we propose alternative perspectives on the algorithm.
4.1.Manifold Learning Perspective
The process of mapping a corrupted example to an uncorrupted one can be visualized in Figure2,with a low-dimensional manifold near which the data con-centrate.We learn a stochastic operator p(X| X)that maps an X to an X,p(X| X)=B g
θ
(fθ(e X))
(X).The corrupted examples will be much more likely to be outside and farther from the manifold than the uncor-rupted ones.Hence the stochastic operator p(X| X) learns a map that tends to go from lower probability points X to high probability points X,generally on or near the manifold.Note that when X is farther from the manifold,p(X| X)should learn to make big-ger steps,to reach the manifold.
At the limit we see that the operator should map even far away points to a small volume near the manifold.
The denoising autoencoder can thus be seen as a way to define and learn a manifold.The intermediate rep-resentation Y=f(X)can be interpreted as a coordi-nate system for points on the manifold(this is most clear if we force the dimension of Y to be smaller than the dimension of X).More generally,one can think of Y=f(X)as a representation of X which is well suited to capture the main variations in the ,on the manifold.When additional criteria(such as sparsity) are introduced in the learning model,one can no longer directly view Y=f(X)as an explicit low-dimensional coordinate system for points on the manifold,but it retains the property of capturing the main factors of variation in the data.
4.2.Top-down,Generative Model Perspective In this section we recover the training criterion for our denoising autoencoder(eq.5)from a generative model perspective.Specifically we show that training the denoising autoencoder as described in section2.3 is equivalent to maximizing a variational bound on a particular generative model.
Consider the generative model p(X, X,Y)= p(Y)p(X|Y)p( X|X)where p(X|Y)=B s(W Y+b )and
p ( X
|X )=q D ( X |X ).p (Y )is a uniform prior over Y ∈[0,1]d
.This defines a generative model with pa-rameter set θ ={W ,b }.We will use the previ-ously defined q 0(X, X,Y )=q 0(X )q D ( X |X )δf θ(e X )
(Y )(equation 4)as an auxiliary model in the context of a variational approximation of the log-likelihood of p ( X
).Note that we abuse notation to make it lighter,and use the same letters X , X
and Y for different sets of random variables representing the same quan-tity under different distributions:p or q 0.Keep in mind that whereas we had the dependency structure
X → X
→Y for q or q 0,we have Y →X → X for p .Since p contains a corruption operation at the last
generative stage,we propose to fit p ( X
)
to corrupted training samples.Performing maximum likelihood fit-ting for samples drawn from q 0( X
)corresponds to min-imizing the cross-entropy,or maximizing
H =max θ
{−I H (q 0( X ) p ( X ))}=max θ
{E E q 0(e X )[log p ( X )]}.(6)
Let q
(X,Y | X )be a conditional density,the quan-tity L (q , X )=E E q (X,Y |e X ) log p (X,e X,Y )q (X,Y |e X ) is a lower
bound on log p ( X
)since the following can be shown to be true for any q :
log p ( X
)
读者杂志
=L (q , X )+I D KL (q (X,Y | X ) p (X,Y | X ))Also it is easy to verify that the bound is tight when q (X,Y | X )=p (X,Y | X ),where the I D KL becomes 0.We can thus write log p ( X
)=max q L (q , X ),and consequently rewrite equation 6as
H =max θ
骗纸{E E q 0(e X )[max q
L (q
, X )]}=max θ ,q
{E E q 0(e X )[L (q , X )]}
(7)
Figure 2.Manifold learning perspective.Suppose
training data (×)concentrate near a low-dimensional man-ifold.Corrupted examples (.
)obtained by applying cor-ruption process q D (e X
|X )will lie farther from the manifold.The model learns with p (X |e X
)to “project them back”onto the manifold.Intermediate representation Y can be inter-preted as a coordinate system for points on the manifold.
where we moved the maximization outside of the ex-pectation because an unconstrained q (X,Y | X
)can in principle perfectly model the conditional distribu-tion needed to maximize L (q , X
)for any X .Now if we replace the maximization over an unconstrained q by the maximization over the parameters θof our q 0(appearing in f θthat maps an x to a y ),we get
a lower bound on H :H ≥max θ ,θ{E E q 0(e X )[L (q 0, X )]}
Maximizing this lower bound,we find
arg max θ,θ
{E E q 0(e X )[L (q 0, X )]}=arg max θ,θ E E q 0(X,e X,Y )
log p (X, X,
Y )q 0(X,Y | X
)
=arg max θ,θ
E E q 0(X,e X,Y ) log p (X, X,Y )
+E E q 0(e X )
I
H [q 0
(X,Y | X )] =arg max θ,θ
E E q 0(X,e X,Y ) log p (X, X,Y )
.
Note that θonly occurs in Y =f θ( X
),and θ only occurs in p (X |Y ).The last line is therefore obtained
because q 0(X | X
)∝q D ( X |X )q 0(X )(none of which de-pends on (θ,θ )),and q 0(Y | X
)is ,its entropy is constant,irrespective of (θ,θ ).Hence the
entropy of q 0(X,Y | X
)
=q 0(Y | X )q 0(X | X ),does not vary with (θ,θ ).Finally,following from above,we obtain our training criterion (eq.5):
arg max θ,θ E E q 0(e X )[L (q 0, X )]=arg max θ,θ
E E q 0(X,e X,Y )[log[p (Y )p (X |Y )p ( X |X )]]=arg max θ,θ
E E q 0(X,e X,Y )[log p (X |Y )]
=arg max θ,θ
E E q 0(X,e X )[log p (X |Y =f θ( X ))]
=arg min θ,θ
E E q 0(X,e X ) L I H
X,g θ (f θ( X ))
where the third line is obtained because (θ,θ )have no influence on E E q 0(X,e X,Y )[log p (Y )]because we chose p (Y )stant,nor on
E E q 0(X,e X )[log p ( X |X )],and the last line is obtained
by inspection of the definition of L I H in eq.2,when
p (X |Y =f θ( X ))is a B g θ
(f θ(e X
)).4.3.Other Theoretical Perspectives
Information Theoretic Perspective:Consider
X ∼q (X ),q unknown,Y =f θ( X
).It can easily be shown (Vincent et al.,2008)that minimizing the expected reconstruction error amounts to maximizing
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