Sparse Penalty in Deep Belief Networks: Using the Mixed Norm Constraint
Authors: Xanadu Halkias, Sébastien PARIS, Herve Glotin
ICLR 2013

Abstract: Deep Belief Networks (DBN) have been successfully applied on popular machine learning tasks. Specifically, when applied on hand-written digit recognition, DBNs have achieved approximate accuracy rates of 98.8%. In an effort to optimize the data representation achieved by the DBN and maximize their descriptive power, recent advances have focused on inducing sparse constraints at each layer of the DBN. In this paper we present a theoretical approach for sparse constraints in the DBN using the mixed norm for both non-overlapping and overlapping groups. We explore how these constraints affect the classification accuracy for digit recognition in three different datasets (MNIST, USPS, RIMES) and provide initial estimations of their usefulness by altering different parameters such as the group size and overlap percentage.

Sparse Penalty in Deep Belief Networks: Using the
Mixed Norm Constraint
Xanadu C. Halkias
DYNI, LSIS, Universit`e du Sud,
Avenue de l’Universit´e - BP20132, 83957 LA GARDE CEDEX - FRANCE
xanadu.halkias@univ-tln.fr
S´ebastien Paris
DYNI, LSIS CNRS UMR 7296, Aix-Marseille University Domaine universitaire de Saint J´erˆome
Avenue Escadrille Normandie Niemen,13397 MARSEILLE Cedex 20, FRANCE
sebastien.paris@lsis.org
Herv´e Glotin
DYNI, LSIS CNRS UMR 7296, Universit´e Sud Toulon-Var, Institut Universitaire de France
Avenue de l’Universit´e - BP20132, 83957 LA GARDE CEDEX - FRANCE
glotin@univ-tln.fr
Abstract
Deep Belief Networks (DBN) have been successfully applied on popular ma-
chine learning tasks. Speciﬁcally, when applied on hand-written digit recogni-
tion, DBNs have achieved approximate accuracy rates of 98.8%. In an effort to
optimize the data representation achieved by the DBN and maximize their descrip-
tive power, recent advances have focused on inducing sparse constraints at each
layer of the DBN. In this paper we present a theoretical approach for sparse con-
straints in the DBN using the mixed norm for both non-overlapping and overlap-
ping groups. We explore how these constraints affect the classiﬁcation accuracy
for digit recognition in three different datasets (MNIST, USPS, RIMES) and pro-
vide initial estimations of their usefulness by altering different parameters such as
the group size and overlap percentage.
1 Introduction
Restricted Boltzmann Machines (RBMs) are Energy Based Models (EBMs) that have been exten-
sively used for a diverse set of machine learning applications mainly due to their generative and
unsupervised learning framework. These applications range from image scene recognition and gen-
eration [? ], video-sequence recognition [? ] and dimensionality reduction [? ].
An equally important aspect of RBMs is that they serve as the building blocks of DBNs [ ? ]. Their
use as such has been favored in the machine learning community due to the conditional independence
between the hidden units in the RBM that allows for the efﬁcient and computationally tractable
implementation of deep architectures.
In recent years, sparsity has become an important requirement in both shallow [add cite] and deep
architectures. Although primarily used in statistics for optimization tasks in order to overcome
the curse of dimensionality in various applications, it also serves as a way to emulate biologically
plausible models of the human visual cortex, where it has been shown that sparsity is an integral
process in the hierarchical processing of visual information [? ? ?].
1
arXiv:1301.3533v2  [cs.NE]  22 Feb 2013
Moreover, an added beneﬁt of using sparse constraints in the form of mixed norm regularizers in
deep architectures is that they can alleviate their restrictive nature by allowing implicit interactions
between the hidden units in the RBMs. Mixed norm regularizers such as l1,2 have been extensively
used in statistics and machine learning [ ? ]. In this paper we provide initial results when inducing
sparse constraints by using a mixed norm regularizer on the activation probabilities of the RBMs.
The mixed norm is applied on both non-overlapping and overlapping groups. We also show that this
regularizer can be used to train DBNs, and offer results for the task of digit recognition using several
datasets.
2 Restricted Boltzmann Machines
An RBM is a type of two layer neural network comprised of a visible layer that represents the
observed data x and a hidden layer that represents the hidden variables h. The addition of these
hidden units allows the model an increased capacity in expressing the underlying distribution of the
observed data.
RBMs are energy based models and as such they deﬁne a probability distribution through an energy
function as seen in Eq. 1
p(x,h) = e−E(x,h)
Z (1)
Where Z, provided in Eq. 2, is called the partition function and is a normalizing factor ensuring that
Eq. 1 is a probability.
Z =
∑
x,h
e−E(x,h) (2)
In the case of an RBM the energy function E(x,h) is deﬁned in Eq. 3:
Eθ(x,h) = −
I∑
i=1
J∑
j=1
xihjwij −
I∑
i=1
bixi −
J∑
j=1
ajhj, (3)
b are the visible unit biases and aare the hidden unit biases.
In the common case where we are using stochastic binary units for both visible and hidden units,
then the conditional probabilities of activation are obtained by:
p(xi = 1|h) = σ(bi + ∑
jhjwij)
p(hj = 1|x) = σ(aj + ∑
ixiwij), (4)
where σis the sigmoid function and
σ(f(x)) ≜ 1
1 + e−f(x) . (5)
Since an RBM does not allow for connections amongst hidden units or amongst visible units we can
easily obtain Eq. 6.
p(x|h) = ∏
i
p(xi|h)
p(h|x) = ∏
j
p(hj|x) (6)
Intuitively, the observed data, x will be modeled by those hidden units, h that are expressed with
a high conditional probability p(hj|x). The goal of adding sparse constraints to the network is to
allow for the salient activation of the hidden units based on the differences of the observed data. As
a result, we can achieve an initial clustering of the observed data that will increase the discriminative
power of the model.
2.1 Training an RBM
RBMs are energy based, generative models that are trained to model the marginal probability p(x)
of the observed data where:
p(x) =
∑
h∈{0,1}J
p(x,h). (7)
2
In general, energy based models can be learnt by performing gradient descent on the negative log-
likelihood of the observed data. Speciﬁcally, to learn the parameters of the network we need to
compute the gradient provided in Eq. 8 given the observed (training) data xl.
−∂logp(x)
∂θ = ⟨∂Eθ(xl,h)
∂θ ⟩h −⟨∂Eθ(x,h)
∂θ ⟩x,h, (8)
where ⟨·⟩n denotes the expectation with respect to n. As evident in Eq.8, the gradient has two
phases. The positive phase which tries to lower the energy of the training data xl and the negative
phase which tries to increase the energy of all xin the model.
Assessing the energy on all the data can be an intractable task given the size of the network and the
number of possible conﬁgurations. In order to obtain an approximation Hinton (2006) successfully
proposed the use of Contrastive Divergence (CD). This allows us to sample an approximation of the
expectation over (x,h) using Gibbs sampling at only k steps. Empirically, it has been shown that
setting k = 1 will provide an adequate approximation although it will not follow the theoretical
gradient [? ].
Applying CD on Eq. 8 we can obtain the following update equations for the parameters of the
network.
∆w·j = 1
L
L∑
l=1
xlp(hj = 1|xl) −˜xlp(hj = 1|˜xl) (9)
∆bi = 1
L
L∑
l=1
p(xl
i = 1|h) −p(˜xl
i = 1|h) (10)
∆aj = 1
L
L∑
l=1
p(hj = 1|xl) −p(hj = 1|˜xl), (11)
where the (˜·) deﬁnes the generated distributions obtained by the CD.
In the next section, we introduce a general version of sparse constraints in the learning phase of the
RBM through the use of the mixed norm in an effort to control the activation probabilities of the
hidden units.
3 Mixed Norm RBMs
Several attempts in inducing sparse constraints in the RBM by [ ? ? ] have been successful in
increasing the discriminative power of the models. Examples of these sparse constraints range from
weight decay [? ] to modiﬁed norm penalties [ ? ]. In this paper we focus on the generalized penalty
of the mixed norm (l1,2), but will also provide a theoretical and practical implementation for the use
of overlapping groups. We will refer to this generalized penalty applied to the expectations of the
activation probabilities as the Mixed Norm RBM (MNRBM).
As mentioned before, learning an RBM consists of performing gradient descent on the negative
log-likelihood. We can thus deﬁne the cost function Lto be minimized as L = −logp(x). When
applying the mixed norm regularizer the cost function takes the general form of Eq. 12.
L= −logp(x) + λ(∥p(h = 1|x)∥1,2) (12)
Where λis a regularizer constant. The second term of Eq. 12 deﬁnes the mixed norm penalty on
the expectations of the hidden unit activation probabilities. In order to apply the mixed norm we
assume that the hidden units are divided into groups. These groups can be non-overlapping or
overlapping. As a result, we are able to penalize a whole group and not just individual hidden units.
MNRBM with non-overlapping groups:Given an RBM with J hidden units we deﬁne a partition
of the hidden units into groups Pm where m = 1,2,...M. The groups are non-overlapping and of
equal size to alleviate computational issues. The mixed norm penalty for a data samplexl is deﬁned
in Eq. 14.
∥p(h = 1|xl)∥1,2 =
M∑
m=1
∥p(Pm|xl)∥2
=
M∑
m=1
√ ∑
k∈Pm
p(hk = 1|xl)2
(13)
3
In practice, the desire behind the application of the mixed norm penalty is to set groups of the hidden
units to zero when representing the observed data by forcing their activation probabilities to zero.
As a result, given an observed data sample only a small number of groups of hidden units will be
activated, leading to its sparse representation.
MNRBM with overlapping groups:Given an RBM with J hidden units we deﬁne a partition of
the hidden units into groups Pm where m = 1,2,...M. The groups are overlapping and of equal
size. Depending on the percentage of overlap, a we will obtain a new set of groups P′
k where
k= 1,2,...K.
We can then deﬁne a set of augmented hidden units J′= {h′∈J : ∀Pk,Pk
⋃Pm = Pm,J′⊃J}.
Subsequently, given an RBM with J′hidden units, we can then consider that the set P′
k, deﬁnes
non-overlapping, equally sized groups [? ]. The mixed norm penalty for a data sample xl is deﬁned
in a similar way as in Eq. 14.
∥p(h′ = 1|xl)∥1,2 =
K∑
k=1
∥p(h′m|xl)∥2
=
K∑
k=1
√ ∑
k∈P′
k
p(h′
k = 1|xl)2
(14)
3.1 Training the Mixed Norm RBM
In order to train the MNRBM with non-overlapping groups and obtain the model parameters θwe
need to minimize the cost function presented in Eq. 12. This can be achieved by performing a
coordinate descent once we have obtained the gradients of the regularizers.
The gradient of the mixed norm penalty for the weights, W is as follows:
∂
∂w·j
(∥p(h = 1|xl)∥1,2) =
= 1
2 · 1√ ∑
k∈Pm
p(hk=1|xl)2
·2 ·p(hk = 1|xl) ·∂p(hk=1|xl)
∂w·j
= p(hk=1|xl)
∥p(hm|xl)∥2
·∂p(hk=1|xl)
∂w·j
= p(hk=1|xl)
∥p(hm|xl)∥2
·p(hk = 1|xl)[1 −p(hk = 1|xl)] ·xl
= p(hk=1|xl)2
∥p(hm|xl)∥2
·p(hk = 0|xl) ·xl.
(15)
When applied on the expectations of the activation probabilities the mixed norm penalty will follow
their trend while forcing the groups that include members with low activation probabilities towards
zero. The l2 norm in the denominator ensures that the groups with low activations will be pushed
further closer to zero.
Given the gradients of the penalties the update equations for the MNRBM are presented bellow:
∆w·j =
1
L
L∑
l=1
[(p(hj = 1|xl) + λp(hj=1|xl)p(hj=0|xl)√∑p(hm=1|xl)2
) ·xl −p(hj = 1|˜xl)˜xl] (16)
∆aj =
1
L
L∑
l=1
[(p(hj = 1|xl) + λp(hj=1|xl)p(hj=0|xl)√∑p(hm=1|xl)2
−p(hj = 1|˜xl)] (17)
The detailed steps for training the MNRBM are depicted in Algorithm 1.
The general penalty of Eq. 12 allows us through the manipulation of the constant regularizer, λ,
the group size and percentage of overlap to obtain different types of architectures. In this case, the
sparsity is induced at the group level of the hidden units whereby the observed data is represented
by a small number of groups of hidden units. The λconstant is empirically determined [? ] based
on the task at hand.
4
Algorithm 1Mixed Norm RBM learning algorithm
1. Update the parameters θusing CD and Eq. 9- 11
2. Update the parameters again using the gradient of the regularizations as in Eq. 16- 17
3. Repeat steps 1, 2 until convergence
(a) Mixed norm RBM non-overlapping
groups (group size = 20)
(b) Mixed norm RBM overlapping groups
(group size = 50, overlap = 20%)
Figure 1: Sample learned weights W for the mixed norm RBM using the USPS data set
Fig. 1 shows sample weights for the mixed norm RBM when usingλ= 0.1 for both non-overlapping
and overlapping groups. Fig 2 provides the average probability activations for the hidden units given
a batch of the USPS training data. As seen in the ﬁgure, the activation probabilities of the hidden
units appear to be more towards the left-hand side of the ﬁgure which is the desired effect. However,
there appears to be a bimodality whereby a large proportion of the activation probabilities is set to a
high value for the non-overlapping groups MNRBM. This may be attributed to the choice and size
of groups when applying the mixed norm penalty. Given that the activation probabilities are pushed
towards high values one can expect that such a process may have an adverse result for classiﬁcation
tasks since the hidden units will over-represent the observed data. However, in the case of the
overlapping groups most of the activations are pushed towards zero. Although, that is the goal of
adding sparse constraints is to force the activations to zero, in this case we may be dealing with a
biased system that actually under represents the data distribution.
3.2 Data
We have used three different data sets in order to train and test the network.
• MNIST is a popular data set in the community for hand-written digit recognition and is
comprised of 70000, 28 ×28 images (60000 train - 10000 test). It is publicly available at
yann.lecun.com/exdb/mnist.
• The RIMES data set which was created by asking volunteers to write hand written letters
for different scenarios. In this paper we used the digit set of the data base. In total the
set we used was comprised of 37200 images of different sizes ( 29800 train - 7400 test).
Further information can be obtained at www.rimes-database.fr.
• The USPS digit data set that we used is comprised of9280 (7280 train - 2000 test), 16 ×16
images. The extracted images were scanned from mail in working U.S. Post Ofﬁces [? ].
In order to achieve the cross-training and testing all images were resized to have the same size as
the MNIST dataset (28 ×28) given its extensive use in this task. All images were also checked to
ensure that orientations/translations were uniform across the data sets. No other pre-processing was
employed. Example images from the three datasets can be seen in ﬁgure 3
5
(a) Mixed norm RBM non-overlapping
groups (group size = 20)
(b) Mixed norm RBM overlapping groups
(group size = 50, overlap = 20%)
Figure 2: Average of hidden unit activation probabilities for the mixed norm RBM using a batch
of the USPS data set. Y-axis: Hidden unit activation probabilities for mixed norm RBM. X-axis:
Hidden unit activations for vanilla RBM
Figure 3: Examples of images from the three datasets MNIST (left), RIMES (center) and USPS
(right)
3.3 Experimental Results: Pre-training DBNs with MNBMs
RBMs became increasingly popular when Hinton and Salakhudinov [? ] [? ] used them as building
blocks for creating and pre-training efﬁcient DBNs. The proposed MNRBMs can be utilized in the
same manner to initialize DBNs and obtain a sparse and computationally efﬁcient representation of
the observed data.
In order to offer a comparative view between the different architectures we used Hinton’s model for
digit recognition, but we substituted the vanilla RBM with the proposed MNRBM. We pre-trained a
500 −500 −2000 DBN and tested it on three different data sets, MNIST, RIMES and USPS.
Continuing, to obtain classiﬁcation error rates we added 10 softmax layers to get the posterior prob-
abilities for the different classes. The network was ﬁne-tuned using conjugate gradient as described
in [? ]. The constant regularizer was empirically set to λ = 0 .1 for all different models [ ? ].
Continuing, for the mixed norm architecture with non-overlapping groups we used different group
sizes for the hidden units, 5, 10, 20 and 100 respectively. In the case of overlapping groups we used
group sizes of 20, and 50 with a = {20%,50%}. Results on the classiﬁcation accuracy and the
computational cost of the models can be seen in Table 1 and Table 2 respectively. All experiments
were performed on a 24 core server (AMD Opteron processor 8435) with a core CPU of 2.6GHz
and a cache of 512KB.
6
(a) Average Probability density function
for activation probabilities, p(h = 1|x)
(b) Classiﬁcation Accuracy for the differ-
ent penalties using the USPS data set
Figure 4: Average log pdf for activation probabilities for the vanilla RBM and mixed norm RBM
using a batch of the USPS data set and classiﬁcation accuracy for the USPS data set using the
different architectures
Table 1: Classiﬁcation accuracies for the different architectures based on the general sparse penalty.
ARCHITECTURE MNIST RIMES UPS
DBN 98.83% 99.30 % 94.85%
MN DBN (5) 97.28 % 99.24 % 92.90 %
MN DBN (10) 98.83% 99.33 % 94.70 %
MN DBN (20) 98.77 % 99.38 % 94.65 %
MN DBN (100) 98.80 % 99.40% 94.35 %
MN W/O DBN (20/20%) 95.10 % 95.70 % 85.05 %
MN W/O DBN (20/50%) 93.50 % 93.62 % 80.90 %
MN W/O DBN (50/20%) 96.50 % 97.60 % 92.95 %
MN W/O DBN (50/50%) 95.84 % 96.27 % 91.35 %
From Table 1 we can infer that our proposed mixed norm penalty can offer the ﬂexibility of creating
architectures that will be able to matchthe classiﬁcation accuracy of the models depending on the
underlying distributions. It appears that for the task of hand-written digit recognition the distribu-
tion of the observed data favors the use of larger non-overlapping group sizes for the mixed norm
architectures.
In order to get a better understanding of the impact of the different sparse constraints and architec-
tures, Figure 4 depicts the average probability density functions of the expectations of the activation
probabilities for the MNIST training data.
It is interesting to note that the proposed architectures that utilize the mixed norm penalty
(MNDBN)with the overlapping groups tend to aggressively push their activation probabilities to
zero. However, these architectures also tend to offer lower accuracy rates which can be attributed
to an inability of the models to concisely capture the underlying data. A possible way for exploring
this phenomenon further may be to constrain the penalty of the expectations as seen in [? ].
3.4 Conclusions
In this work we provided some ﬁrst insights for the use of the mixed norm sparse constraint in
DBNs. We performed experiments using three different data sets for the task of hand written digit
recognition and offered a practical approach for the use of overlapping groups with the mixed norm
constraint. Although, our initial experiments were limited in the use of equal size overlapping
groups, one could easily extended to non-symmetric overlapping groups, using a similar methodol-
ogy. Inducing sparse constraints based on speciﬁc geometries may also provide better results in the
case of digit recognition and offer more interesting results for tasks such as scene categorization.
7
Table 2: CPU times for the different architectures based on the general sparse penalty.
ARCHITECTURE MNIST RIMES UPS
DBN 167.90 H >60H 31.15 H
MN DBN (5) 62.14 H 33.70 H 8.62 H
MN DBN (10) 66.10 H 40.70 H 10.00 H
MN DBN (20) 70.10 H 69.80 H 12.75 H
MN DBN (100) 71.50 H 85.80 H 15.85 H
MN W/O DBN (20/20%) >60H 39.27 H 10.40 H
MN W/O DBN (20/50%) >60H >45H 22.90 H
MN W/O DBN (50/20%) >60H 35.60 H 9.56 H
MN W/O DBN (50/50%) >70H >45H 24.00 H
8


Peer review:
[Review]:
Since the last version of the paper (v2) is incomplete my following comments are mainly based on the first version.

This paper proposes using $l_{1,2}$ regularization (for both non-overlapping and overlapping groups) upon the activation possibilities of hidden units in RBMs. Then DBNs pretrained by the resulting mixed Norm RBMs are applied the task of digit recognition.    

My main concern is the mistakes in equation 16 (and 17), the core of this paper. The sign of the term of $lambda$ should be minus. There is also a missing $P(h_j=1|x^l)$ in that term. Since these mistakes could explain why the results are worse than the baseline and why the bigger non-overlapping groups (which can make the regularization term smaller) are preferred very well, I do not think they are merely typos.