Abstract

Denoising autoencoders have been previously shown to be competitive alternatives to restricted Boltzmann machines for unsupervised pretraining of each layer of a deep architecture. 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 model to that of a nonparametric 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 rank examples by their energy. It suggests a different way to apply score matching that is related to learning to denoise and does not require computing second derivatives. It justifies the use of tied weights between the encoder and decoder and suggests ways to extend the success 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, 2008) for learning the parameters of unnormalized density models over continuous-valued data, and the training of denoising autoencoders (Vincent, Larochelle, Bengio, & Manzagol, 2008; Vincent, Larochelle, Lajoie, Bengio, & Manzagol, 2010).

Score matching (SM) is an alternative to the maximum likelihood principle suitable for unnormalized probability density models whose partition function is intractable. Its relationship to maximum likelihood has been investigated by Lyu (2010), who formally relates the Fisher divergence that yields score matching and the Kullback-Leibler divergence that yields maximum likelihood. Interestingly, his formal analysis indicates that SM searches for parameters that are more robust to small-noise perturbations of the training data. Score matching has also been recast as a special case under the more general frameworks of generalized score matching (Lyu, 2010; Marlin, Swersky, Chen, & de Freitas, 2010) and minimum probability flow (Sohl-Dickstein et al., 2009), allowing generalizations of SM to discrete distributions (Hyvärinen, 2007b; Lyu, 2010; Sohl-Dickstein et al., 2009). The minimum probability flow paradigm is particularly interesting as it unifies several recent alternative parameter estimation methods, for both continuous and discrete data, under a single unified view.1 Recently, Kingma and LeCun (2010) investigated a regularized form of SM that adds a specific regularization term to the original SM objective. Its relationship to this work is 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, Osindero, and Teh (2006) for pretraining deep networks (Erhan et 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 infinitesimal variance (Hyvärinen, 2008), and have shown that training gaussian binary RBM with SM is equivalent to training a regular (nondenoising) autoencoder with an additional regularization term (Swersky, 2010). This note, however, is the first to recast the training of a DAE as a form of regularized SM. 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 with a brief presentation of the DAE architecture for continuous-valued inputs in section 2 and the SM technique in section 3. This allows us to introduce our formalism and precise terminology. In section 4, we connect the denoising autoencoder objective to SM. We conclude by a discussion on how our findings advance our understanding of both approaches.

1.1.  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 . The pdf for discrete distributions will be expressed with Dirac-deltas δ.

q(xUnknown true pdf.  
Dn = {x(1),  …,  x(n)Training set: i.i.d. sample from q 
 Empirical pdf associated with Dn 
 Smoothing kernel or noise model: isotropic gaussian of variance σ2 
 Joint pdf 
 Parzen density estimate based on Dn obtainable by marginalizing  
p(x; θ) Density model with parameters θ 
J1J2 Means J1(θ) and J2(θ) are equivalent optimization objectives2 
 Expectation with respect to distribution q 
u, v〉 = ∑iuivi Dot product between two vectors 
 Euclidean norm of vector u 
softplus(x) = log(1 + exWill be applied elementwise to vectors 
 Will be applied elementwise to vectors 
I Identity matrix 
WT Transpose of matrix W 
Wi Vector for ith row of W 
W·,j Vector for jth column of W 
q(xUnknown true pdf.  
Dn = {x(1),  …,  x(n)Training set: i.i.d. sample from q 
 Empirical pdf associated with Dn 
 Smoothing kernel or noise model: isotropic gaussian of variance σ2 
 Joint pdf 
 Parzen density estimate based on Dn obtainable by marginalizing  
p(x; θ) Density model with parameters θ 
J1J2 Means J1(θ) and J2(θ) are equivalent optimization objectives2 
 Expectation with respect to distribution q 
u, v〉 = ∑iuivi Dot product between two vectors 
 Euclidean norm of vector u 
softplus(x) = log(1 + exWill be applied elementwise to vectors 
 Will be applied elementwise to vectors 
I Identity matrix 
WT Transpose of matrix W 
Wi Vector for ith row of W 
W·,j Vector for jth 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 an overcomplete 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 pretraining deep networks (Vincent et al., 2008, 2010; Erhan et al., 2010). Denoising autoencoders have also been used in different contexts in the earlier works of LeCun (1987); Gallinari, LeCun, Thiria, and Fogelman-Soulie (1987); and Seung (1998).

In this study, we 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 loss.3 We will be using tied weights whereby encoder and decoder share the same linear transformation parameters. The considered corruption is additive isotropic gaussian noise. A detailed description of the architecture follows:

  • • 

    A training input xDn is first corrupted by additive gaussian noise of covariance σ2I, yielding corrupted input , . This corresponds to conditional density .

  • • 

    The corrupted version is encoded into a hidden representation through an affine mapping followed by a nonlinearity: , where , W is a dh × d matrix and .

  • • 

    The hidden representation h is decoded into reconstruction xr through affine mapping: xr = decode(h) = WTh + c, where .

  • • 
    The parameters θ = {W, b, c} are optimized so that the expected squared reconstruction error ‖xrx2 is minimized, that is, the objective function being minimized by such a DAE is
    formula
    2.1

3.  Score Matching

3.1.  Explicit Score Matching.

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 as
formula
E is called the energy function. Following Hyvärinen (2005), we will call score the gradient of the log density with respect to the data vector: . Beware that this use 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 SM (Hyvärinen, 2005) is to learn θ so that best matches the corresponding score of the true distribution: . The corresponding objective function to be minimized is the expected squared error between these two vectors:
formula
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 . Hyvärinen (2005) mentions in passing that nonparametric methods might be used to estimate those, and we shall later pay closer attention to this possibility.

3.2.  Implicit Score Matching.

Hyvärinen (2005) instead proceeds by proving the following remarkable property:
formula
3.1
where , and C1 is a constant that does not depend on θ. This yields an implicit SM objective JISMq that no longer requires having an explicit score target for q but is nevertheless equivalent to JESMq. Hyvärinen (2005) formally shows that provided q(x) and ψ(x; θ) satisfy some weak regularity conditions,4 we have
formula
3.2

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 JISMq which, following our notation, we shall write as :
formula
3.3
is asymptotically equivalent to JISMq when n → ∞ and hence asymptotically equivalent to objective JESMq. This can be summarized as
formula
3.4
What happens in the transition from JISMq to finite-sample version is, however, not entirely clear. Concerns regarding the stability of the resulting criterion were raised by Kingma and LeCun (2010), who instead propose optimizing a regularized version of ,
formula
3.5
where the strength of the additional regularization term is controlled by hyperparameter λ. The relationship between this criterion and the criteria we propose below are further discussed in section 5.

4.  Linking Score Matching to the Denoising Autoencoder Objective

4.1.  Matching the Score of a Nonparametric Estimator.

The possibility of matching the score ψ(x; θ) with an explicit target score for q obtained through nonparametric 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 yields the following objective:
formula
4.1
For σ>0, qσ is differentiable, decreases to 0 at infinity, and is finite. All regularity conditions are satisfied, so the same equivalence with ISM as in equation 3.2 holds:
formula
4.2

Note that this equivalence breaks in the limit σ → 0 because qσ no longer satisfies these regularity conditions and JESMqσ can no longer be computed (whereas remains well behaved).

4.2.  Denoising Score Matching.

Let us now consider a slightly different objective, which is inspired by both the SM principle and the DAE approach of using pairs of clean and corrupted examples . For joint density , we define the following denoising score matching (DSM) objective:
formula
4.3
The underlying intuition is that following the gradient ψ of the log density at some corrupted point, should ideally move us toward the clean sample x. Note that with the considered gaussian kernel, we have
formula
4.4
Direction clearly corresponds to moving from back toward clean sample x, and we want ψ to match that as best it can.
This objective, inspired by denoising autoencoders, is equivalent to explicit SM. Formally,
formula
4.5

The proof is in the appendix and does not depend on the particular form of as long as is differentiable with respect to .

4.3.  An Energy Function That Yields the Denoising Autoencoder Objective.

Let us now choose for model p the form
formula
4.6
We then have
formula
which we can write as the single equation
formula
4.7
Substituting equations 4.4 and 4.7 in the expression for , equation 4.3, we get, for this choice of Parzen kernel and density model,
formula
We have thus shown that
formula
4.8

5.  Discussion

Putting together equations 4.2, 4.5, and 4.8, we can write, for σ>0,
formula
5.1

In summary, training the denoising autoencoder defined in section 2 is equivalent to performing SM (explicit or implicit) with the energy function of equation 4.6 on Parzen density estimate qσ. Such training would typically use stochastic gradient descent, whereby samples from qσ are obtained by corrupting samples from Dn. And it may be carried out with any of these four optimization objective formulations.5

We introduced the kernel-smoothed empirical distribution qσ to show a connection between SM and a simple DAE. Interestingly, the regularized SM criterion JISMreg (see equation 3.5) that Kingma and LeCun (2010) recently introduced with the very different motivation of curing possible instabilities, was derived by approximating what amounts to 6 From this perspective, our four qσ-based criteria in equation 5.1, including the DAE, may be seen as alternative approximation-free forms of regularized score matching. A key difference is that, as is done with DAE 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 approximation of , centered on the training samples only (i.e., sampling from q0). Also, whereas JISMreg, like the other ISM criteria, requires computing second derivatives, the stochastic version of our novel criterion does not, and thus appears much simpler to implement.

Note that the energy function in equation 4.6 is particular in that its scaling, which we may call its temperature, is chosen to match the corrupting noise level σ2. This is required only to establish the last equivalence with the specific DAE we considered. But regarding the generic objectives , their σ may in principle be chosen regardless of the form or temperature of whatever energy function is to be learned. Interestingly, the energy function in equation 4.6, which 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 and gaussian visible units (Welling, Rosen-Zvi, & Hinton, 2005; Bengio, Lamblin, Popovici, & Larochelle, 2007; Swersky, 2010). The major difference is that this latter free energy does not have a global temperature scaling of the whole expression.7 We designed equation 4.6 to exactly yield the denoising version of the classic autoencoder described in section 2. But with tied weights, it may be preferable to allow an extra positive scaling parameter α for the reconstruction, so that there at least exists an equivalent reparameterization of the model for scaled input data.8 This is easily obtained in the energy function by multiplying the sum of softplus terms in equation 4.6 by α. We may even allow an arbitrary rescaling factor αj for each hidden-layer dimension independently by multiplying each softplus term by its own rescaling parameter αj, which yields the following more flexible energy function:
formula
Here we have also included, as model parameter, a σm (where m stands for model) distinct from the noise σ of the training objective.9

Our qσ-based objectives , , or can be used as alternatives to the finite sample objective (see equation 3.3) advocated in Hyvärinen (2005) for learning unnormalized densities. Note that is a special case of obtained in the limit of σ → 0. Also, since Kingma and LeCun (2010) showed that it may be preferable to use a regularized criterion (that they derived from smoothed empirical distribution qσ), it is likely that our qσ-based criteria may, for σ>0, yield better generalization performance than the advocated in Hyvärinen (2005).10 It seems that σ could allow one to choose an optimal bias-variance trade-off for the finite-sample estimation of the true SM gradient of interest ∇θJESMq = ∇θJISMq. While is an unbiased estimator of it, will generally be biased when σ>0 but are also likely to have a lower variance.

Among the three equivalent SM objectives based on qσ, objective appears particularly interesting as a novel alternative formulation. It was motivated by both the SM and the DAE principles. From DAE, it borrows the idea of learning to denoise artificially corrupted samples, and from SM, it borrows the idea of learning a score function derived from an unnormalized density. may prove simpler and more efficient in practice than the mathematically equivalent because 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 equation 4.6. 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, Kennedy, Pendleton, & Roweth, 1987). Second, whereas using the same weight matrix (“tied weights”) for the encoder and decoder is justified for RBMs, the encoder-decoder framework does not constrain that choice. Previous work on DAEs (Vincent et al., 2008, 2010; Erhan 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, 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 energy function.

Appendix:  Proof That Equation 4.5

The explicit score matching criterion using the Parzen density estimator is defined in equation 4.1 as
formula
which we can develop as
formula
A.1
where is a constant that does not depend on θ, and
formula
Substituting this expression for S(θ) in equation A.1 yields
formula
A.2
We also have defined in equation 4.3,
formula
which we can develop as
formula
A.3
where is a constant that does not depend on θ.

Looking at equations A.2 and A.3, we see that . We have thus shown that the two optimization objectives are equivalent.

Acknowledgments

I thank Yoshua Bengio, Olivier Delalleau, and the other members of the Lisa Lab who provided timely feedback, as well as two anonymous referees whose thoughtful comments and suggestions helped improve this note. This research was supported by NSERC, MITACS, and FQRNT.

Notes

1

Specifically SM (Hyvärinen, 2005), minimum velocity learning (Movellan, 2008), and certain forms of contrastive divergence (Hinton, 2002; Welling & 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 (Sohl-Dickstein et al., 2009).

2

Equivalence will be asserted when J2 = αJ1 + β with α>0, . 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 the preferred choice for binary input.

4

Namely, q(x) and ψ(x; θ) are differentiable, is finite, and for any θ, is finite and limx‖→∞q(x)ψ(x; θ) = 0.

5

Note, however, that while these qσ-based objectives are formally equivalent, their stochastic gradient descent 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 σ2 or scaled in any way.

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 reparameterization), unless it can be compensated by an additional scaling of the reconstruction by a parameter α.

9

We 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 that the best models, judged by their ability to extract useful features, were obtained for nonnegligible values of the noise parameters. Moreover, this way of controlling the model’s capacity worked much better than either reducing the hidden layer size or than traditional weight decay.

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