1

A Connection Between Score Matching
and Denoising Autoencoders
Pascal Vincent
vincentp@iro.umontreal.ca
Dept. IRO, Université de Montréal,
CP 6128, Succ. Centre-Ville, Montréal (QC) H3C 3J7, Canada.
Technical Report 1358
Département d’Informatique et de Recherche Opérationnelle
December 2010
THIS IS A PREPRINT VERSION OF A NOTE THAT HAS BEEN
ACCEPTED FOR PUBLICATION IN NEURAL COMPUTATION.
Keywords: autoencoder, energy based models, score matching, denoising, density
estimation.
Abstract
Denoising autoencoders have been previously shown to be competitive alternatives
to Restricted Boltzmann Machines for unsupervised pre-training of each layer of a deeparchitecture. We show that a simple denoising autoencoder training criterion is equivalent to matching the score (with respect to the data) of a specific energy based modelto that of a non-parametric Parzen density estimator of the data. This yields several
useful insights. It defines a proper probabilistic model for the denoising autoencoder
technique which makes it in principle possible to sample from them or to rank examples
by their energy. It suggests a different way to apply score matching that is related tolearning to denoise and does not require computing second derivatives. It justifies theuse of tied weights between the encoder and decoder, and suggests ways to extend thesuccess of denoising autoencoders to a larger family of energy-based models.

1

Introduction

This note uncovers an unsuspected link between the score matching technique (Hyvärinen, 2005; Hyvärinen, 2008) for learning the parameters of unnormalized density models over continuous-valued data, and the training of denoising autoencoders (Vincent
et al., 2008, 2010).
Score matching (SM) is an alternative to the maximum likelihood principle suitablefor unnormalized probability density models whose partition function is intractable. Itsrelationship to maximum likelihood has been investigated by Lyu (2009) who formally
relates the Fisher divergence that yields score matching and the Kullback-Leibler divergence that yields maximum likelihood. Interestingly, his formal analysis indicates thatscore matching searches for parameters that are more robust to small-noise perturbations of the training data (Lyu, 2009). Score matching has also been recast as a special
case under the more general frameworks of generalized score matching (Lyu, 2009;
Marlin et al., 2009) and minimum probability flow (Sohl-Dickstein et al., 2009), allowing generalizations of score matching to discrete distributions (Hyvärinen, 2007b; Lyu,
2009; Sohl-Dickstein et al., 2009). The minimum probability flow paradigm is particularly interesting as it unifies several recent alternative parameter-estimation methods,
both for continuous and discrete data, under a single unified view1 . Recently, Kingma
and LeCun (2010) investigated a regularized form of score matching which adds a specific regularization term to the original score matching objective. Its relationship to thepresent work will be discussed in detail in Section 5.
Denoising Autoencoders (DAE) were proposed by Vincent et al. (2008) as a simple
and competitive alternative to the Contrastive-Divergence-trained Restricted Boltzmann
Machines (RBM) used by Hinton et al. (2006) for pretraining deep networks (Erhanet al., 2010; Vincent et al., 2010). Previous studies have already pointed out connections between SM and Contrastive Divergence (Hyvärinen, 2007a; Sohl-Dickstein et al.,2009), have connected SM to optimal denoising for Gaussian noise with infinitesimalvariance (Hyvärinen, 2008) and have shown that training Gaussian Binary RBM with
SM is equivalent to training a regular (non-denoising) autoencoder with an additional
regularization term (Swersky, 2010). The present study however is the first to recast
the training of a DAE as a form of regularized score matching. This connection yields
insights relevant to both research directions and suggests a novel parameter estimation
technique that has its roots in both DAE and SM.
We begin by a very brief presentation of the DAE architecture for continuous-valued
inputs in Section 2 and of the SM technique in Section 3. This allows us to introduce
our formalism and precise terminology. Then, in section 4, we connect the denoising
autoencoder objective to score matching. We conclude by a discussion on how ourfindings advance our understanding of both approaches.

Notation
We are interested in techniques that learn the parameters θ of a model by minimizing some objective function J(θ). For uniformity of notation, all distributions will be
represented by their probability density functions (pdf) on Rd . The pdf for discrete
distributions will be expressed with Dirac-deltas δ.
q(x) Unknown true pdf. x ∈ Rd .
Dn = {x , . . . , x(n) } Training set: i.i.d. sample from q.
(1)

1

Specifically score matching (Hyvärinen, 2005), minimum velocity learning (Movellan, 2008), andcertain forms of contrastive divergence (Hinton, 2002; Welling and Hinton, 2002) are all recast as minimizing the Kullback-Leibler divergence between the data distribution and the distribution obtained after
running, for infinitesimal time, a dynamic that would transform it into the model distribution (SohlDickstein et al., 2009).

2

P
q0 (x) = n1 ni=1 δ(kx − x(i) k) Empirical pdf associated with Dn .
12
1
qσ (x̃|x) = (2π)d/2
e− 2σ2 kx̃−xk Smoothing kernel or noise model: isotropicσdGaussian of variance σ 2 .
qσ (x̃, x) P
= qσ (x̃|x)q0 (x) Joint pdf.
1
qσ (x̃) = n nt=1 qσ (x̃|x(t) ) Parzen density estimate based on Dn
obtainable by marginalizing qσ (x̃, x).
p(x; θ) Density model with parameters θ.
J1 ^ J2 Means J1 (θ) and J2 (θ) are equivalent
optimization objectives 2 .
´Eq(x) [g(x)] = x q(x)g(x)dx
Expectation with respect to distribution q.
Phu, vi = p i ui vi Dot product between two vectors.
kuk = hu, ui Euclidean norm of vector u.
softplus(x) = log(1 + ex ) Will be applied elementwise to vectors.
sigmoid(x) = 1+e1−x = softplus0 (x) Will be applied elementwise to vectors.
I Identity matrix.
T
W
Transpose of matrix W.
Wi Vector for ith row of W.
W·,j Vector for j th column of W.

2

Denoising Autoencoders

Denoising Autoencoders (DAEs) are a simple modification of classical autoencoder
neural networks that are trained, not to reconstruct their input, but rather to denoise an
artificially corrupted version of their input (Vincent et al., 2008, 2010). Whereas anover-complete regular autoencoder can easily learn a useless identity mapping, a DAE
must extract more useful features in order to solve the much harder denoising problem.
DAEs have proven to be an empirically successful alternative to Restricted Boltzmann
Machines (RBM) for pre-training deep networks (Vincent et al., 2008, 2010; Erhanet al., 2010). Denoising autoencoders have also been used in different contexts in theearlier works of LeCun (1987); Gallinari et al. (1987); Seung (1998).
In this study, we will consider the denoising version of a simple classical autoencoder that uses a single sigmoidal hidden layer. Since data points originate from a continuous real valued distribution it is natural to use a linear decoder with a squared reconstruction loss3 . We will be using tied weights whereby encoder and decoder share thesame linear transformation parameters. The considered corruption is additive isotropic
Gaussian noise. A detailed description of the architecture follows:• A training input x ∈ Dn is first corrupted by additive Gaussian noise of covariance σ 2 I yielding corrupted input x̃ = x + ,  ∼ N (0, σ 2 I). This corresponds to2

Equivalence will be asserted when J2 = αJ1 + β with α > 0, β ∈ R. Indeed a gradient-based
optimization algorithm, when starting from some initial θ value, should land in the exact same minimum
whether optimizing J1 or J2 (this may however require learning rate adjustment to compensate for scaling
factor α).
3
as opposed to a linear+sigmoid decoder with a Bernoulli cross-entropy loss, which would be thepreferred choice for binary input.

3

1

2

1
conditional density qσ (x̃|x) = (2π)d/2
e− 2σ2 kx̃−xk .
σd• The corrupted version x̃ is encoded into a hidden representation h ∈ Rdh throughan affine mapping followed by a nonlinearity:h = encode(x̃) = sigmoid(Wx̃+b), where x̃ ∈ Rd , h ∈ (0, 1)dh , W is a dh ×dmatrix and b ∈ Rdh .
• The hidden representation h is decoded into reconstruction xr through affine
mapping: xr = decode(h) = WT h + c, where c ∈ Rd .
• The parameters θ = {W, b, c} are optimized so that the expected squared reconstruction error kxr − xk2 is minimized, i.e. the objective function being minimized by such a denoising autoencoder (DAE) is:
JDAEσ (θ) = Eqσ (x̃,x) kdecode(encode(x̃)) − xk2hi2T= Eqσ (x̃,x) W sigmoid(Wx̃ + b)) + c − x
.
(1)

3

Score Matching

3.1

Explicit Score Matching (ESM)

Score Matching was introduced by Hyvärinen (2005) as a technique to learn the parameters θ of probability density models p(x; θ) with intractable partition function Z(θ),
where p can be written asp(x; θ) =1
exp(−E(x; θ)).
Z(θ)

E is called the energy function. Following Hyvärinen (2005), we will call score thep(x;θ). Bewaregradient of the log density with respect to the data vector: ψ(x; θ) = ∂ log∂xthat this usage differs slightly from traditional statistics terminology where score usually refers to the derivative of the log likelihood with respect to parameters, whereas
here we are talking about a score with respect to the data. The core principle of scorep(x;θ)
best matches thematching (Hyvärinen, 2005) is to learn θ so that ψ(x; θ) = ∂ log∂x∂ log q(x)corresponding score of the true distribution, i.e.
. The corresponding objective∂xfunction to be minimized is the expected squared error between these two vectors, i.e.
"#2
1
∂ log q(x)JESM q (θ) = Eq(x)ψ(x; θ) −.
2∂x
We refer to this formulation as explicit score matching (ESM).
Note that the score ψ(x; θ) does not depend on troublesome Z(θ). But since q is unknown, we do not have explicit regression targets ∂ log∂xq(x) . Hyvärinen (2005) mentions
in passing that non-parametric methods might be used to estimate those, and we shalllater pay closer attention to this possibility.

4

3.2

Implicit Score Matching (ISM)

Hyvärinen (2005) instead proceeds by proving the following remarkable property:
"
#
"
#
d
2
X
1
1
∂ψi (x; θ)∂ log q(x)2Eq(x)= Eq(x)kψ(x; θ)k +ψ(x; θ) −
+C1
2
∂x
2
∂xi
i=1
|
{z
} |
{z
}
JESM q (θ)

JISM q (θ)(2)∂ log p(x;θ), and C1 is a constant that does not depend∂xi

where ψi (x; θ) = ψ(x; θ)i =on θ. This yields an implicit score matching objective JISM q that no longer requires
having an explicit score target for q but is nevertheless equivalent to JESM q . Hyvärinen(2005) formally shows that, provided q(x) and ψ(x; θ) satisfy some weak regularity
conditions4 , we have
JESM q ^ JISM q

3.3

(3)

Finite Sample Version of Implicit Score Matching

Since we only have samples Dn from q, Hyvärinen proposes to optimize the finite
sample version of JISM q which, following our notation, we shall write JISM q0 :
#
"
d
X
∂ψ
(x;
θ)
1
i
kψ(x; θ)k2 +JISM q0 (θ) = Eq0 (x)
2
∂x
i
i=1
!
n
d
(t)
X
1X 1
∂ψ
(x
;
θ)
2
i
ψ(x(t) ; θ) +=.
(4)
n t=1 2∂xii=1JISM q0 is asymptotically equivalent to JISM q when n → ∞ and hence asymptotically equivalent to objective JESM q . This can be summarized as
JESM q ^ JISM q ^ lim JISM q0 .

(5)

n→∞

What happens in the transition from JISM q to finite sample version JISM q0 is however not entirely clear. Concerns regarding the stability of the resulting criterion wereraised by Kingma and LeCun (2010), who propose instead to optimize a regularizedversion of JISM q0 :JISM reg (θ) = JISM q0 (θ) + λ

2
d 
X
∂ψi (x(t) ; θ)
∂xi

i=1

,

(6)

where the strength of the additional regularization term is controlled by hyperparameterλ. The relationship between this criterion and the criteria we propose below will befurther discussed in Section 5.
4Namely: q(x) and ψ(x; θ) are differentiable, Eq(x)hi2
Eq(x) kψ(x; θ)k is finite and limkxk→∞ q(x)ψ(x; θ) = 0.

5

∂ log q(x)∂x2is finite, and for any θ,4Linking Score Matching to the Denoising Autoencoder
Objective

4.1

Matching the Score of a Non-Parametric Estimator

As previously stated, the possibility of matching the score ψ(x; θ) with an explicit target score for q obtained through non-parametric estimation was mentioned but not pursued in Hyvärinen (2005). We now examine this possibility more closely. Explicitly
matching ψ(x; θ) with the score of Parzen windows density estimator qσ (x̃) yields thefollowing objective:#"
2
1∂ log qσ (x̃).
(7)
JESM qσ (θ) = Eqσ (x̃)
ψ(x̃; θ) −
2
∂ x̃


2
∂ log qσ (x̃)
For σ > 0, qσ is differentiable, decreases to 0 at infinity, and Eqσ (x̃)∂ x̃is finite. All regularity conditions are satisfied, so the same equivalence with ISM as in
Eq. 3 holds, i.e.
JESM qσ ^ JISM qσ .
(8)Note that this equivalence however breaks in the limit σ → 0, because qσ no longer
satisfies these regularity conditions, and JESM qσ can no longer be computed (whereas
JISM qσ remains well-behaved).

4.2

Denoising Score Matching (DSM)

Let us now consider a slightly different objective, that is inspired by both the Score
Matching principle and by the Denoising Autoencoder approach of using pairs of clean
and corrupted examples (x, x̃). For joint density qσ (x̃, x) = qσ (x̃|x)q0 (x), we define
the following denoising score matching (DSM) objective:#"
2∂ log qσ (x̃|x)1
ψ(x̃; θ) −.
(9)
JDSM qσ (θ) = Eqσ (x,x̃)2∂ x̃
The underlying intuition is that following the gradient ψ of the log density at some
corrupted point x̃ should ideally move us towards the clean sample x. Note that withthe considered Gaussian kernel we have
∂ log qσ (x̃|x)1= 2 (x − x̃).
(10)
∂ x̃
σ
Direction σ12 (x − x̃) clearly corresponds to moving from x̃ back towards clean sample
x, and we want ψ to match that as best it can.
Now this alternate objective, inspired by denoising autoencoders, is equivalent to
explicit score matching. Formally,
JESM qσ ^ JDSM qσ(11)

The proof is in the appendix, and does not depend on the particular form of qσ (x̃|x)or q(x) as long as log qσ (x̃|x) is differentiable with respect to x̃.
64.3

An Energy Function that Yields the Denoising Autoencoder Objective

Let us now choose for model p the form
1
exp(−E(x; θ))Z(θ)
Phhc, xi − 12 kxk2 + dj=1
softplus (hWj , xi + bj )E(x; W, b, c) = −.
| {z }
σ2p(x; θ) =(12)

θWe then have
∂ log p(x; θ)
∂xi
∂E
= −
∂xi

ψi (x; θ) =

1
=
σ2
=

1
σ2

dh
X

∂ (hWj , xi + bj )ci − xi +softplus0 (hWj , xi + bj )
∂xi
j=1
!
dh
X
ci − xi +sigmoid (hWj , xi + bj ) Wji

!

j=1

1
=
(ci − xi + hW·i , sigmoid(Wx + b)i) ,σ2
which we can write as the single equation
1
TW
sigmoid(Wx
+
b)
+
c
−
x
.
(13)
σ2
Substituting Eq. 10 and Eq. 13 in the expression for JDSM qσ (Eq. 9) we get, for thischoice of Parzen kernel and density model,ψ(x; θ) ="JDSM qσ (θ) = Eqσ (x,x̃)

1∂ log qσ (x̃|x)ψ(x̃; θ) −
2
∂ x̃

"

2

#


1
1 1
= Eqσ (x,x̃)
WT sigmoid(Wx̃ + b) + c − x̃ − 2 (x − x̃)
2
2 σ
σ
h
i
1
2
T
=
E
W
sigmoid(Wx̃
+
b)
+
c
−
x
q
(x,x̃)
2σ 4 σ1=
JDAEσ (θ).
2σ 4

2

#

We have thus shown that
JDSM qσ ^ JDAEσ

7

(14)

5

Discussion

Putting together equations 8, 11 and 14, we can write, for σ > 0,
JISM qσ ^ JESM qσ ^ JDSM qσ ^ JDAEσ .

(15)

In summary, training the denoising autoencoder defined in section 2 is equivalent to performing score matching (explicit or implicit) with the energy function of
Eq. 12 on Parzen density estimate qσ . Such a training would typically use stochastic
gradient descent, whereby samples from qσ are obtained by corrupting samples fromDn . And it may be carried out with any of these four optimization objective formulations5 .
We introduced the kernel-smoothed empirical distribution qσ to show a connection
between score matching and a simple denoising autoencoder. Interestingly, the regularized score matching criterion JISM reg (Eq. 6) that Kingma and LeCun (2010) recently introduced with the very different motivation of curing possible instabilities, wasderived by approximating6 what amounts to JISM qσ . From this perspective our fourqσ -based criteria in Eq. 15, including the denoising autoencoder, may be seen as alternative approximation-free forms of regularized score matching. A key difference isthat, as is done with denoising autoencoder training, we would optimize stochastic versions of these approximation-free regularized criteria by corrupting training examples(i.e. sampling form qσ ), whereas Kingma and LeCun (2010) optimize an approximationof JISM qσ , centered on the training samples only (i.e. sampling from q0 ). Also, whereas
JISM reg , like the other ISM criteria, requires computing second derivatives, the stochastic version of our novel JDSM qσ criterion does not, and thus appears much simpler toimplement.
Note that the energy function in Eq. 12 is particular in that its scaling, which we
may call its temperature, is chosen to match the corrupting noise level σ 2 . This isrequired only to establish the last equivalence with the specific DAE we considered.
But regarding the generic objectives JISM qσ ^ JESM qσ ^ JDSM qσ , their σ may inprinciple be chosen irrespective of the form or temperature of whatever energy function
is to be learnt. Interestingly, the energy function in Eq. 12, that we designed to yield
the equivalence with our denoising autoencoder objective, happens to be very similar
to the free energy of a Restricted Boltzmann Machine with binary hidden units andGaussian visible units (Welling et al., 2005; Bengio et al., 2007; Swersky, 2010). The
major difference is that this latter free energy does not have a global temperature scalingof the whole expression7 . We designed Eq. 12 to exactly yield the denoising versionof the classic autoencoder described in Section 2. But with tied weights, it may be
preferable to allow for an extra positive scaling parameter α for the reconstruction, sothat there at least exists an equivalent reparametrization of the model for scaled input
5Note however that while these qσ -based objectives are formally equivalent, their stochastic gradientdescent optimization, based on sampling a limited number of corrupted examples, is likely to behave
differently for each objective.
6
A first order Taylor expansion and a diagonal Hessian approximation are used.
7
Specifically, in the free energy of a Gaussian-binary RBM, the softplus terms are not divided by σ 2
nor scaled in any way.

8

data8 . This is easily obtained in the energy function by multiplying the sum of softplus
terms in Eq. 12 by α. We may even allow an arbitrary rescaling factor αj for eachhidden layer dimension independently by multiplying each softplus term by its own
rescaling parameter αj , which yields the following more flexible energy function:
!
dh
X
1
1
E(x; W, b, c, α, σm ) = − 2 hc, xi − kxk2 +αj softplus (hWj , xi + bj ) .
|
{z
}
σm
2
j=1
θ

Here we have also included, as model parameter, a σm (where m stands for model)
distinct from the noise σ of the training objective9 .
Our qσ -based objectives JISM qσ , JESM qσ , or JDSM qσ can be used as alternatives tothe finite sample objective JISM q0 (Eq. 4) advocated in Hyvärinen (2005) for learning
unnormalized densities. Note that JISM q0 is a special case of JISM qσ obtained in thelimit of σ → 0. Also, since Kingma and LeCun (2010) showed that it may be preferable
to use a regularized criterion (that they derived form smoothed empirical distributionqσ ), it is likely that our qσ -based criteria may, for σ > 0, yield better generalization
performance than the JISM q0 advocated in Hyvärinen (2005)10 . It seems that σ could
allow one to choose an optimal bias-variance tradeoff for the finite-sample estimationof the true score matching gradient of interest ∇θ JESM q = ∇θ JISM q . While ∇θ JISM q0
is an unbiased estimator of it, ∇θ JISM qσ = ∇θ JESM qσ = ∇θ JDSM qσ will generally bebiased when σ > 0 but are also likely to have a lower variance.
Among the three equivalent score matching objectives based on qσ , objective JDSM qσ
appears particularly interesting as a novel alternative formulation. It was motivated byboth the SM principle and the DAE principle. From DAE it borrows the idea of learning
to denoise artificially corrupted samples, and from SM it borrows the idea of learninga score function derived from an unnormalized density. JDSM qσ may prove simpler andmore efficient in practice than the mathematically equivalent JISM qσ , as it does not
require computing second derivatives.
Our result is also a significant advance for DAEs. First, we have defined a proper
energy function for the considered DAE through Eq. 12. This will enable many previously impossible or ill-defined operations on a trained DAE, for example deciding
which is the more likely among several inputs, or sampling from a trained DAE using Hybrid Monte-Carlo (Duane et al., 1987). Second, whereas using the same weightmatrix (“tied weights”) for the encoder and decoder is justified for RBMs, the encoderdecoder framework does not constrain that choice. Previous work on DAEs (Vincent
et al., 2008; Erhan et al., 2010; Vincent et al., 2010) explored both options, often finding tied weights to yield better empirical results. Within the SM framework presented
here, using tied weights between encoder and decoder now has a proper justification,
8

If for example one multiplies the input values by 100, one can obtain the same hidden representation
as before by dividing W by 100. But because of the tied weights this means that the reconstruction would
also be divided by 100 (i.e. there is no equivalent reparametrization), unless it can be compensated by anadditional scaling of the reconstruction by a parameter α.
9We would however have to set σm = σ to recover a recognizable denoising autoencoder objective.
10
It is also noteworthy that the experimental results of Vincent et al. (2008, 2010) on DAE showed thatthe best models, judged by their ability to extract useful features, were obtained for non negligible valuesof the noise parameters. Moreover this way of controlling the model’s capacity worked much better thaneither reducing the hidden layer size or than traditional weight decay.

9since it follows naturally from differentiating the energy. Third, this framework opens
the door to new variants that would naturally fall out from other choices of the energyfunction.

Acknowledgements
We would like to thank Yoshua Bengio, Olivier Delalleau and the other members of
the Lisa Lab who provided timely feedback, as well as two anonymous referees whosethoughtful comments and suggestions helped improve the manuscript. This research
was supported by NSERC, MITACS and FQRNT.

References
Bengio, Y., Lamblin, P., Popovici, D., and Larochelle, H. (2007). Greedy layer-wisetraining of deep networks. In B. Schölkopf, J. Platt, and T. Hoffman, editors, Advances in Neural Information Processing Systems 19 (NIPS’06), pages 153–160. MIT
Press.
Duane, S., Kennedy, A., Pendleton, B., and Roweth, D. (1987). Hybrid Monte Carlo.
Phys. Lett. B, 195, 216–222.
Erhan, D., Bengio, Y., Courville, A., Manzagol, P.-A., Vincent, P., and Bengio, S.
(2010). Why does unsupervised pre-training help deep learning? Journal of Machine
Learning Research, 11, 625–660.
Gallinari, P., LeCun, Y., Thiria, S., and Fogelman-Soulie, F. (1987). Memoires associatives distribuees. In Proceedings of COGNITIVA 87, Paris, La Villette.
Hinton, G. E. (2002). Training products of experts by minimizing contrastive divergence. Neural Computation, 14, 1771–1800.
Hinton, G. E., Osindero, S., and Teh, Y.-W. (2006). A fast learning algorithm for deep
belief nets. Neural Computation, 18, 1527–1554.
Hyvärinen, A. (2005). Estimation of non-normalized statistical models using scorematching. Journal of Machine Learning Research, 6, 695–709.
Hyvärinen, A. (2007a). Connections between score matching, contrastive divergence,and pseudolikelihood for continuous-valued variables. IEEE Transactions on Neural
Networks, 18, 1529–1531.
Hyvärinen, A. (2007b). Some extensions of score matching. Computational Statistics
and Data Analysis, 51, 2499–2512.
Hyvärinen, A. (2008). Optimal approximation of signal priors. Neural Computation,
20(12), 3087–3110.

10

Kingma, D. and LeCun, Y. (2010). Regularized estimation of image statistics by scorematching. In J. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. Zemel, and A. Culotta,
editors, Advances in Neural Information Processing Systems 23 (NIPS’10), pages
1126–1134. MIT Press.
LeCun, Y. (1987). Modèles connexionistes de l’apprentissage. Ph.D. thesis, Universitéde Paris VI.
Lyu, S. (2009). Interpretation and generalization of score matching. In Proceedings ofthe 25th Conference in Uncertainty in Artificial Intelligence (UAI’09).
Marlin, B., Swersky, K., Chen, B., and de Freitas, N. (2009). Inductive principles for
restricted boltzmann machine learning. In Proceedings of The Thirteenth International Conference on Artificial Intelligence and Statistics (AISTATS’10), volume 9,
pages 509–516.
Movellan, J. R. (2008). Contrastive divergence in gaussian diffusions. Neural Computation, 20(9), 2238–2252.
Seung, S. H. (1998). Learning continuous attractors in recurrent networks. In M. Jordan,
M. Kearns, and S. Solla, editors, Advances in Neural Information Processing Systems
10 (NIPS’97), pages 654–660. MIT Press.
Sohl-Dickstein, J., Battaglino, P., and DeWeese, M. R. (2009). Minimum probability
flow learning. Technical report, arXiv:0906.4779.
Swersky, K. (2010). Inductive Principles for Learning Restricted Boltzmann Machines.
Master’s thesis, University of British Columbia.
Vincent, P., Larochelle, H., Bengio, Y., and Manzagol, P.-A. (2008). Extracting andcomposing robust features with denoising autoencoders. In W. W. Cohen, A. McCallum, and S. T. Roweis, editors, ICML’08, pages 1096–1103. ACM.
Vincent, P., Larochelle, H., Lajoie, I., Bengio, Y., and Manzagol, P.-A. (2010). Stacked
denoising autoencoders: learning useful representations in a deep network with alocal denoising criterion. Journal of Machine Learning Research, 11, 3371–3408.
Welling, M. and Hinton, G. E. (2002). A new learning algorithm for mean field Boltzmann machines. In ICANN ’02: Proceedings of the International Conference on
Artificial Neural Networks, pages 351–357, London, UK. Springer-Verlag.
Welling, M., Rosen-Zvi, M., and Hinton, G. E. (2005). Exponential family harmoniums
with an application to information retrieval. In L. Saul, Y. Weiss, and L. Bottou,
editors, Advances in Neural Information Processing Systems 17 (NIPS’04), pages
1481–1488, Cambridge, MA. MIT Press.

11Appendix
Proof that JESM qσ ^ JDSM qσ (11)The explicit score matching criterion using the Parzen density estimator is defined in
Eq. 7 as"#2
1
∂ log qσ (x̃)
JESM qσ (θ) = Eqσ (x̃)
ψ(x̃; θ) −2
∂ x̃
which we can develop as
JESM qσ (θ) = Eqσ (x̃)
where C2 = Eqσ (x̃)S(θ) =
=
=
=
=
=
=
=
=
=

1
2

∂ log qσ (x̃)
∂ x̃

2


1
2
kψ(x̃; θ)k − S(θ) + C2
2

(16)


is a constant that does not depend on θ, and∂ log qσ (x̃)
Eqσ (x̃) ψ(x̃; θ),
∂ x̃


ˆ
∂ log qσ (x̃)
qσ (x̃) ψ(x̃; θ),
dx̃
∂ x̃
x̃
*
+
ˆ
∂
q
(x̃)
σ
qσ (x̃) ψ(x̃; θ), ∂ x̃dx̃qσ (x̃)x̃ˆ ∂qσ (x̃) dx̃ψ(x̃; θ),
∂ x̃
x̃

ˆ 
ˆ
∂
ψ(x̃; θ),q0 (x)qσ (x̃|x)dx dx̃
∂ x̃ xx̃ˆ ˆ∂qσ (x̃|x)ψ(x̃; θ), q0 (x)dx dx̃
∂ x̃
x̃
x

ˆ 
ˆ
∂ log qσ (x̃|x)dx dx̃
ψ(x̃; θ), q0 (x)qσ (x̃|x)
∂ x̃
x̃
x


ˆ ˆ
∂ log qσ (x̃|x)q0 (x)qσ (x̃|x) ψ(x̃; θ),
dxdx̃
∂ x̃
x̃ x


ˆ ˆ
∂ log qσ (x̃|x)qσ (x̃, x) ψ(x̃; θ),
dxdx̃
∂ x̃
x̃ x


∂ log qσ (x̃|x)Eqσ (x̃,x) ψ(x̃; θ),.
∂ x̃

Substituting this expression for S(θ) in Eq. 16 yields

1
2
JESM qσ (θ) = Eqσ (x̃)
kψ(x̃; θ)k2∂ log qσ (x̃|x)+ C2 .
−Eqσ (x,x̃) ψ(x̃; θ),
∂ x̃
12

(17)

We also have defined in Eq. 9,"
JDSM qσ (θ) = Eqσ (x,x̃)

1∂ log qσ (x̃|x)ψ(x̃; θ) −
2
∂ x̃

2

#
,

which we can develop as
12
kψ(x̃; θ)k
JDSM qσ (θ) = Eqσ (x̃)
2∂ log qσ (x̃|x)−Eqσ (x,x̃) ψ(x̃; θ),
+ C3
(18)
∂ x̃


2
∂
log
q
(x̃|x)
σwhere C3 = Eqσ (x,x̃) 21
is a constant that does not depend on θ.
∂ x̃Looking at equations 17 and 18 we see that JESM qσ (θ) = JDSM qσ (θ) + C2 − C3 .
We have thus shown that the two optimization objectives are equivalent.

13