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 Abstract— In face recognition and classication, feature extraction

and classication based on insufcient labeled data is a well-known

challenging problem. In this paper, a novel semi-supervised learning

algorithm named deep belief network embedded with Softmax regress

(DBNESR) is proposed to address this problem. DBNESR first learns

hierarchical representations of feature by deep learning and then

makes more efficient classification with Softmax regress. At the same

time, we design many kinds of classifiers based on supervised learning:

BP, HBPNNs, RBF, HRBFNNs, SVM and multiple classification

decision fusion classifier (MCDFC) ——hybrid HBPNNs- HRBFNNs

-SVM classifier. The conducted experiments validate: Firstly, the

proposed semi-supervised deep learning algorithm DBNESR is

optimal for face recognition with the highest and most stable

recognition rates; Second, the semi-supervised learning algorithm has

better effect than all supervised learning algorithms; Third, hybrid

neural networks has better effect than single neural network; Fourth,

the average recognition rate and the variance are respectively shown as

BP<HBPNNs ≈ RBF<HRBFNNs ≈ SVM<MCDFC<DBNESR and

BP>RBF>HBPNNs>HRBFNNs>SVM>MCDFC>DBNESR; At last,

it reflects hierarchical representations of feature by DBNESR in terms

of its capability of modeling hard articial intelligent tasks.

Index Terms—Face recognition, Semi-supervised, Hierarchical

representations, Hybrid neural networks, RBM, Deep belief

network, Deep learning

I. INTRODUCTION

Face recognition (FR) is one of the main areas of

investigation in biometrics and computer vision. It has a wide

range of applications, including access control, information

security, law enforcement and surveillance systems. FR has

caught the great attention from large numbers of research

groups and has also achieved a great development in the past

few decades [1-3]. However, FR suffers from some difculties

because of varying illumination conditions, different poses,

disguise and facial expressions and so on [4-6]. A plenty of FR

algorithms have been designed to alleviate these difculties

[7-9]. FR includes three key steps: image preprocessing, feature

extraction and classication. Image preprocessing is essential

process before feature extraction and also is the important step

in the process of FR. Feature extraction is mainly to give an

effective representation of each image, which can reduce the

computational complexity of the classication algorithm and

enhance the separability of the images to get a higher

recognition rate. While classication is to distinguish those

extracted features with a good classier. Therefore, an effective

face recognition system greatly depends on the appropriate

This work is supported by China National Science Foundation (Project no.

61171141)

representation of human face features and the good design of

classier [10].

To select the features that can highlight classification, many

kinds of feature selection methods have been presented, such as:

spectral feature selection (SPEC) [11], multi-cluster feature

selection (MCFS) [12], minimum redundancy spectral feature

selection (MRSF) [13], and joint embedding learning and

sparse regression (JELSR) [14]. In addition, wavelet transform

is popular and widely applied in face recognition system for its

multi-resolution character, such as 2-dimensional discrete

wavelet transform [15], discrete wavelet transform [16], fast

beta wavelet networks [17], and wavelet based feature selection

[18-19-20].

After extracting the features, the following work is to design

an effective classier. Classification aims to obtain the face

type for the input signal. Typically used classification

approaches include polynomial function, HMM [21-22], GMM

[23], K-NN [23], SVM [24], and Bayesian classifier [25]. In

addition, random weight network (RWN) is proposed in some

articles [26-27] and there are also other kinds of neural

networks used as the classier for FR [28-29].

In this paper, we first make image preprocessing to eliminate

the interference of noise and redundant information, reduce the

effects of environmental factors on images and highlight the

important information of images. At the same time, in order to

compensate the deciency of geometric features, it is well

known that the original face images often need to be well

represented instead of being input into the classier directly

because of the huge computational cost. Therefore, PCA and

2D PCA are used to extract geometric features from

preprocessed images, reduce their dimensionality for

computation and attain a higher level of separability. At last, we

propose a novel semi-supervised learning algorithm called deep

belief network embedded with Softmax regress (DBNESR) as

classier for FR, design many kinds of classifiers based on

supervised learning and make experiments to validate the

effectiveness of the algorithm.

The main contributions of this paper can be concluded as

follows:

1) A novel semi-supervised learning algorithm called deep

belief network embedded with Softmax regress (DBNESR) is

proposed. DBNESR first learns hierarchical representations [30]

of feature by deep learning and then makes more efficient

classification with Softmax regress.

2) Many kinds of classifiers based on supervised learning: BP,

HBPNNs, RBF, HRBFNNs, SVM and multiple classification

Learning Hierarchical Representations for Face

Recognition using Deep Belief Network Embedded with

Softmax Regress and Multiple Neural Networks

Hai-jun Zhang, Nan-feng Xiao

School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China

decision fusion classifier (MCDFC) ——hybrid HBPNNs-

HRBFNNs-SVM classifier are designed.

3) The analysis and experiments are performed on the precise

rate of face recognition. The conducted experiments validate:

Firstly, the proposed semi-supervised deep learning algorithm

DBNESR is optimal for face recognition with the highest and

most stable recognition rates; Second, the semi-supervised

learning algorithm has better effect than all supervised learning

algorithms; Third, hybrid neural networks has better effect than

single neural network; Fourth, the average recognition rate and

the variance are respectively shown as BP<HBPNNs ≈ RBF

HBPNNs>HRBFNNs>SVM>MCDFC>DBNESR; At last, it

reflects hierarchical representations of feature by DBNESR in

terms of its capability of modeling hard articial intelligent

tasks.

The remainder of this paper is organized as follows. Section

2 reviews the images preprocessing. Section 3 introduces the

feature extraction methods. Section 4 designs the classifiers of

supervised learning. Section 5 gives and designs the classifier

of semi-supervised learning proposed by us. Experimental

results are presented and discussed in Section 6. Section 7 gives

the concluding remarks.

II. IMAGES PREPROCESSING

Images often appear the phenomenon such as low contrast,

being not clear and so on in the process of generation,

acquisition, input, etc. of images due to the influence of

environmental factors such as the imaging system, noise and

light conditions so on. Therefore it needs to make images

preprocessing. The purpose of the preprocessing is to eliminate

the interference of noise and redundant information, reduce the

effects of environmental factors on images and highlight the

important information of images [31]. Images preprocessing

usually includes gray of images, images filtering, gray

equalization of images, standardization of images, compression

of images (or dimensionality-reduced) and so on [32]. The

process of images preprocessing is as following.

A. Face images filtering

We use median filtering to make smoothing denoising for

images. This method not only can effectively restrain the noise

but also can very well protect the boundary. Median filter is a

kind of nonlinear operation, it sorts a pixel point and all others

pixel points within its neighborhood as the size of grey value,

sets the median of the sequence as the gray value of the pixel

point, as shown in Eq.(1).





( , ) ( , )

f i j Med f i j

(1)

where,

is the filter window. Using the template of 3×3 makes

median filtering for the experiment in the back.

B. Histogram equalization

The purpose of histogram equalization is to make images

enhancement, improve the visual effect of images, make

redundant information of images after preprocessing less and

highlight some important information of images.

Set the gray range of image

( , )A x y

[0, ]L

, image

histogram for

( )

H r

, Therefore, the total pixel points are

0 0

( )

A H r dr 

(2)

Making normalization processing for the histogram, the

probability density function of each grey value can be obtained:

( )

H r

p r



(3)

The probability distribution function is

0 0

( ) ( ) ( )

L L

P r p r dr H r dr

   

(4)

Set the gray transformation function of histogram equalization

as the limited slope not reduce continuously differentiable

function

( )s T r

, input it into

( , )A x y

to get the output

( , )B x y

( )

H r

is the histogram of output image, it can get

( ) ( )

B A

H s ds H r dr

(5)

( ) ( )

( )

A A

H r H r

H s

ds dr T r

 

(6)

where,

( )T r ds dr

. Therefore, when the difference

between the molecular and denominator of

( )

H r

is only a

proportionality constant,

( )

H r

is constant. Namely

( ) ( )

T r H r



(7)

( ) ( ) ( )

s T r H r dr CP r

   

(8)

In order to make the scope of

for

[0, ]L

, can get

C L

. For

discrete case the gray transformation function is as following:

( ) ( ) ( )

k k

k i

i i o

s T r CP r C p r C

 

   

 

(9)

where,

is the

kth

grayscale,

is the pixel number of

the total pixels number of images, the scope of

for

[0, 1]L 

We make the histogram equalization experiment for the images

in the back.

C. Compression of images (or dimensionality-reduced)

It is well known that the original face images often need to be

well represented instead of being input into the classier

directly because of the huge computational cost. As one of the

popular representations, geometric features are often extracted

to attain a higher level of separability. Here we employ multi-

scale two-dimensional wavelet transform to generate the initial

geometric features for representing face images.

We make the multi-scale two-dimensional wavelet transform

experiment for the images in the back.

III. FEATURE EXTRACTION

There are two main purposes for feature extraction: One is to

extract characteristic information from the face images, the

feature information can classify all the samples; The second is

to reduce the redundant information of the images, make the

data dimensionality being on behalf of human faces as far as

possibly reduce, so as to improve the speed of subsequent

operation process. It is well known that image features are

usually classied into four classes: Statistical-pixel features,

visual features, algebraic features, and geometric features (e.g.

transform-coefcient features).

A. Extract features with PCA

Suppose that there are

facial images

{ }

i i



column vector of

dimension. All samples can be expressed

as following:

1 2

( , , , )

X X X X 

(10)

Calculate the average face of all sample images as following:

X X





(11)

Calculate the difference of faces, namely the difference of each

face with the average face as following:

, 1, 2, ,

i i

d X X i N   

(12)

Therefore, the images covariance matrix

can be represented

as following:

1 2 N

1 1

A=(d ,d , ,d )

T T

i i

C d d AA

N N



 





(13)

Using the theorem of singular value decomposition (SVD) to

calculate the eigenvalue



and orthogonal normalization

eigenvector



A A

, through Eq.(14) the eigenvalues of

covariance matrix

can be calculated.

,( 1, 2, , )

i i

u Av i N



  

(14)

Making all the eigenvalues

1 2

[ , , , ]

  



order in descend

according to the size, through the formula as following

min ,

t k t











 

  

 

 







(15)

where, usually set

90%a 

, can get the eigenvalues face

subspace

 

1 2

, , ,

U u u u 

. All the samples project to

subspace

, as following:

Z U X

(16)

therefore, using front

principal component instead of the

original vector

, not only make the facial features parameter

dimension is reduced, but also won't loss too much feature

information of the original images.

B. Extract features with 2D-PCA

Suppose sample set is





, 1, 2, , ; 1, 2, ,

i m n

S R i N j M



    

is the category,

is the sample of the

ith

category,

is the total number of

category,

is the total number of samples of each category,

K N M 

is the number of all samples.

Let

be average of all samples as follows:

1 1

N M

i j

S S

 





(17)

Therefore, the images covariance matrix

can be represented

as follows:

1 1

( ) ( )

N M

i T i

j j

i j

G S S S S

 

  



(18)

and the generalized total scattered criterion

( )J X

can be

expressed by:

( )

J X X GX

(19)

Let

opt

be the unitary vector such that it maximizes the

generalized total scatter criterion

( )J X

, that is:

arg max ( )

opt

X J X

(20)

In general, there is more than one optimal solution. We usually

select a set of optimal solutions

{ , , }

X X

subjected to the

orthonormal constraints and the maximizing criterion

( )J X

where,

is smaller than the dimension of the coefcients matrix.

In fact, they are those orthonormal eigenvectors of the

matrix

corresponding to

largest eigenvalues.

Now for each sub-band coefcient matrix

, compute the

principal component of the matrix

as follows:

, 1, 2, ,

ij i j

y A x j t  

(21)

Then we can get its reduced features matrix

[ , , ]

i i it

Y y y 

1,2, ,i m 

We extract features respectively with PCA and 2D-PCA and

compare their effects for the images in the back experiment.

IV. DESIGNING THE CLASSIFIERS OF SUPERVISED LEARNING

Usually the classifiers based on supervised learning are often

used for FR, in the paper we design two types of classifiers.

One is the type of supervised learning classifiers and the other

is semi-supervised learning classifiers [33].

A. Single BP neural network

The BP neural network is a kind of multilayer feed-forward

network according to the back-propagation algorithm for errors,

is currently one of the most widely used neural network models

[34]. The recognition and classification of the face images is an

important application for the BP neural network in the field of

pattern recognition and classification. The network consists

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