Sparse `
2
-norm Regularized Regression for Face Recognition
Ahmad J. Qudaimat
1,2
and Hasan Demirel
2
1
Electrical Engineering Departement, Palestine Polytechnic University (PPU), Hebron, Palestine
2
Electrical and Electronic Engineering Departement, Eastern Mediterranean University (EMU), Famagusta, North Cyprus
Keywords:
Sparsifying Transform, Face Recognition, Dictionary Learning, Transform Learning.
Abstract:
In this paper, a new `
2
-norm regularized regression based face recognition method is proposed, with `
0
-norm
constraint to ensure sparse projection. The proposed method aims to create a transformation matrix that
transform the images to sparse vectors with positions of nonzero coefficients depending on the image class.
The classification of a new image is a simple process that only depends on calculating the norm of vectors
to decide the class of the image. The experimental results on benchmark face databases show that the new
method is comparable and sometimes superior to alternative projection based methods published in the field
of face recognition.
1 INTRODUCTION
Face recognition is one of the most challenging ap-
plications of pattern recognition. It has drawn the
attention of researchers for decades where many al-
gorithms have been developed. Among the numer-
ous number of proposed algorithms, regression-based
methods with sparse representation have shown re-
markable results.
In the field of pattern recognition and feature ex-
traction, least-square regression algorithms are known
by thier simplicity and effectivness. Examples of the
traditional featrure extraction algorithms are principal
component analysis (PCA) (Swets and Weng, 1996)
which is not an optimal solution for image recogni-
tion since it based on signal reconstruction. PCA has
been modified many times, for example, regression
methods were used to develop sparse PCA (Zou et al.,
2006) . Another example is linear Fisher discrimi-
nant analysis (LFDA) (Belhumeur et al., 1997). The
idea of LFDA depends on creating a projection ma-
trix that maximizes the ratio of intra-class scatter and
inter-class scatter.
Many other regression based methods that use
`
2,1
−norm regularization were recently developed in-
cluding (Ma et al., 2012), (Ma et al., 2013) and (Lai
et al., 2017). In (Gu et al., 2011), Gu et al proposed
the joint feature selection and subspace learning
(FSSL) which was based on locality-preserving pro-
jection (LPP) (He et al., 2005). Sparsifying transform
learning method has been proposed in (Qudaimat and
Demirel, 2018) for face image classification (STLC).
The main contributions of this paper can be stated
as follows
• A novel regression based image classification
method is proposed. The method uses `
2
− norm
regularized objective function to prevent overlap-
ping of nonzero coefficients that belong to differ-
ent classes.
• The proposed method transform images to sparse
vectors using `
0
− norm constraint.
• Simulation results verifies competitive and supe-
rior performance of the proposed method over the
alternative projection based methods.
The remaining sections of the paper are organized
as follows: Section 2 discusses the details of the pro-
posed method. Simulations and experiments are dis-
cussed in Section 3. Finally, Section 4 summarizes
and concludes the proposed method.
2 PROPOSED METHOD
2.1 Problem Formulation and Objective
Function
Given n training face images for K persons. In matrix
form, the n
k
images from k
th
class can be represented
as
Y
k
= [y
k1
,y
k2
,· ·· , y
kn
k
] ∈ R
N×n
k
, k = 1,2, ...,K
Qudaimat, A. and Demirel, H.
Sparse l2-nor m Regularized Regression for Face Recognition.
DOI: 10.5220/0007355104530458
In Proceedings of the 8th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2019), pages 453-458
ISBN: 978-989-758-351-3
Copyright
c
 2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
453

Let Y = [Y
1
,Y
2
,· ·· ,Y
K
] ∈ R
N×n
denotes the matrix
representation of all subjects.
To train the dictionary W , we have formulated the
following problem
(P1) min
W,X
kWY − X k
2
F
− λ log
h
K
∑
i=1
K
∑
j=1
i6= j
kW
i
Y
i
−W
j
Y
j
k
2
F
i
s.t. kX
i
k
0
≤ s
i
∀i
(1)
where λ is a regularization parameter. W
i
is a sub-
matrix of W such that W = [W
T
1
,W
T
2
,· ·· ,W
T
K
]
T
. For
simplicity, we assume here that each submatrix W
i
is
a representation of consecutive number of rows in W .
But in the real implementation of the algorithm, W
i
represent distributed rows in W where we keep the
largerst s coefficients in each column as explained in
the next section.
The rational of this objective function is to in-
crease the inner product between the transformations
of the images that belong to the same class and to de-
crease the inner product between the transformations
of the images that belong to different class. This claim
becomes obvious when we expand second term in
the objective function as kW
i
Y
i
−W
j
Y
j
k
2
F
= kW
i
Y
i
k
2
F
+
kW
j
Y
j
k
2
F
− 2(W
i
Y
i
)
T
(W
j
Y
j
). In other words, ideally,
W will transform images to sparse vectors where the
positions of nonzero coefficients for image transfor-
mation will not overlap with image transformation of
other classes.
2.2 Solution Procedure
The solution of the nonconvex optimization problem
(P1) can be achieved iteratively by alternating be-
tween two steps; sparse coding and dictionary update.
step 1) sparse coding: solve (P1) by updating the
vector X while keeping the dictionary W fixed. The
given problem turns to the following equation
min
X
kWY − X k
2
F
, s.t. kX
i
k
0
≤ s ∀i (2)
The solution of the above equation for X can be found
by zeroing all except the largest s coefficients in each
column of matrix WY . Indices of these coefficients
are kept in a database to be used in classification pro-
cess.
step 2) dictionary update: solve (P1) by updating W
while keeping X fixed. In this step (P1) is switched to
following minimization problem
min
W
kWY − X k
2
F
− λ log
h
K
∑
i=1
K
∑
j=1
i6= j
kW
i
Y
i
−W
j
Y
j
k
2
F
i
(3)
optimization problem (3) has a convex differentiable
objective function in W . Minimizing this objective
function can be achieved by computing its derivative
with respect to W and then solve for W that makes the
gradient zero. The gradient of the first part is
∇
W
kWY − X k
2
F
= 2WYY
T
− 2XY
T
(4)
To find the derivative of the second part of the objec-
tive function, consider
g(W ) =
K
∑
i=1
K
∑
j=1
i6= j
kW
i
Y
i
−W
j
Y
j
k
2
F
(5)
and
f (W) = log

g(W )

(6)
Now, the derivative of f (W ) can be computed using
the partial derivative
∂g
∂W
k
as follows
∇
W
f =
1
g(W )
h
∂g
∂W
1
,
∂g
∂W
2
,. .. ,
∂g
∂W
K
i
T
(7)
To compute
∂g
∂W
k
, We first expand (5) to find the
terms that includes W
k
g =
K
∑
j=1
i6= j
kW
k
Y
k
−W
j
Y
j
k
2
F
+
K
∑
i=1
i6= j
kW
i
Y
i
−W
k
Y
k
k
2
F
+
K
∑
i=1
i6=k
K
∑
j=1
j6=k
i6= j
kW
i
Y
i
−W
j
Y
j
k
2
F
= 2
K
∑
j=1
j6=k
kW
k
Y
k
−W
j
Y
j
k
2
F
+
K
∑
i=1
i6=k
K
∑
j=1
j6=k
i6= j
kW
i
Y
i
−W
j
Y
j
k
2
F
(8)
The second part of this equation does not depend on
W
k
, so its derivative is zero. Hence
∂g
∂w
k
= 4
K
∑
j=1
j6=k
(W
k
Y
k
−W
j
Y
j
)Y
T
K
= 4

(K − 1)W
k
Y
k
−
K
∑
j=1
j6=k
W
j
Y
j

Y
T
K
(9)
To find the minimum value of the objective function
we solve the following equation for every W
k
W
k

2YY
T
− 4λ(K − 1)Y
k
Y
T
K

− 2XY
T
+ 4λ

K
∑
j=1
j6=k
W
j
Y
j

Y
T
K
= 0
(10)
ICPRAM 2019 - 8th International Conference on Pattern Recognition Applications and Methods
454

Finally, we obtain the following closed form solution
for each W
k
W
k
=
h
2YY
T
− 4λ(K − 1)Y
k
Y
T
K
i
−1
h
2XY
T
− 4λ

K
∑
j=1
j6=k
W
j
Y
j

Y
T
K
i
(11)
2.3 Classification
Given a new face image y
new
to be classified to one
of the classes, a new test image will be transformed
using the dictionary W as
x = Wy
new
(12)
Consider a function δ
k
(x) : R
L
→ R
s
for each class
k, which only selects the coefficients in x that cor-
respond to class k. Then based on the maximum
`
2
-norm of the selected coefficients, the test image y
will be classified to its class as
class(y
new
) = argmax
k
kδ
k
(x)k
2
2
(13)
Figure 1 shows an example of image transformation.
In this example, the test image in figure 1a is selected
from the first class of AR database. The normalized
sparse vector x that is obtained is shown in figure 1b.
Figure 1c shows the normalized norms of coefficeints
of vector x for each class, it is obvious that the first
class has the maximum norm.
3 EXPERIMENTAL VALIDATION
To show the performance and validity of the proposed
algorithm, we test our algorithm with the benchmark
face databases ORL (Samaria and Harter, 1994), AR
(Martinez and Benavente, 1998), the extended-YaleB
face database (Georghiades et al., 2001) and LFW
databases (Huang et al., 2007). The performance of
the proposed algorithm was compared with other pro-
jection based methods in litreature, i.e., PCA, LFDA
(Belhumeur et al., 1997), LPP (He et al., 2005), SPP
(Qiao et al., 2010), and CRP (Yang et al., 2015). near-
est neighbor classifier was used.
In the experiments on LFW database, 10-fold
cross validation are used. In the experiments on other
databases, l face images for each subject are randomly
selected to train the dictionary and the remaining im-
ages are used for testing. The experiments are re-
peated for the following number of training images
l={2, 6} for ORL facedatabase, l={4, 8, 16} for AR
face database and l={8, 16} for extended YaleB face
database.
(a)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Cofficient index
-1
-0.5
0
0.5
1
Normalized value
(b)
0 10 20 30 40 50 60 70 80 90 100
Class Number
0
0.2
0.4
0.6
0.8
1
Nomalized Norm
(c)
Figure 1: Example of an image transformation (a) Test im-
age image (b) The sparse coefficient vector x. (c) Norm of
coefficeints for each subject.
The regularization parameter λ is set to 1 ×10
−9
.
The number of the iterations is set to 50. The dictio-
nary is randomely initialized. The ratio of nonzero co-
efficients is selected to be 10% of the total coefficients
. The simulations are carried out with Intel i77500U
CPU at 2.7GHz and 2.9 GHz and 12GB memory.
3.1 ORL Face Database
ORL face database (Samaria and Harter, 1994) con-
tains 400 facial-images captured for 40 persons, 10
sample images for each one. They were taken with
variances in illumination, facial expressions and fa-
cial details. Each image was cropped to size 92 ×112
pixels. Simulation results listed in table 1 shows that
our method achieves the highest recognition rate.
Sparse l2-norm Regularized Regression for Face Recognition
455

Table 1: Recognition Accuracy (%) on ORL face database.
Training
samples
PCA LFDA LPP SPP CRP
Proposed
method
l=2 74.6 92.6 88.5 90.1 89.8 93.1
l=6 82.4 97.8 95.7 97.4 96.6 98.3
3.2 AR Database
Asubset of AR face databse has has been randomly
selected. The selected subset consist of a 100 persons
for 50 women and 50 men. For each one, 26 gray
scale face images of dimension 165 × 120 pixels are
used for training and testing the proposed algorithm.
Figure 2 shows sample face images of one person.
Simulation results are shown in table 2. As shown
in the table, the proposed method achieves the highest
rate among all methods. The recognition rates for the
proposed method and CRP are very close when the
number of trainig images is 16.
Figure 2: Sample images of AR database.
Table 2: Recognition Accuracy (%) on AR face database.
Training
samples
PCA LFDA LPP SPP CRP
Proposed
method
l=4 55.6 69.7 67.1 66.2 74.3 75.1
l=8 61.7 78.6 69.4 79.6 79.5 81.4
l=16 80.7 94.7 94.4 97.2 98.0 98.1
3.3 Extended Yale B Face Database
Extended YaleB (Georghiades et al., 2001) face
Database comprises 38 persons with 2414 frontal per-
son’s face images. They were captured from different
angles under different ligthining conditions. The im-
ages were cropped to the size 192 × 168 pixels. Each
person has about 60 samples. 30 image samples are
used for training and 30 image samples for testing. In
these experiments, PCA was used to reduce the fea-
ture dimensions to 120. Figure 3a shows subset of
face images of one person. table 3 shows the simula-
tion results for different number of randomly selected
training images.
The proposed method is also tested under occlu-
sion and corruption as shown in figure 3b and figure
(a)
(b)
(c)
Figure 3: Sample images of Extended Yale B databasse of
(a) Samples for one peron. (b) Samples with 20% occlusion.
(c) Samples with 20% corruption.
3c, respectively. Table 4 shows the recognition rates
under 20% occlusion, where babbon image is added
to the original images, as shown in figure 3b. Results
of experiments under 20% corruption are shown in ta-
ble 5.
All simulation results on Extended Yale B Face
Database in tables 3, 4 and 5 show the superiority
of the proposed method over other methods. The
proposed method achieves 97.3% recognition rate at
l = 16. Even though the performance degrade to
90.7% in case of occluded and and to 91.6% in case of
corrupted images, The proposed method still have the
highest recognition rate compared to other methods
under the same conditions.
Table 3: Recognition Accuracy (%) on YaleB database.
Training
samples
PCA LFDA LPP SPP CRP
Proposed
method
l=8 63.6 78.2 70.3 81.4 80.6 82.2
l=16 72.3 95.4 95.3 97.2 96.3 97.3
Table 4: Recognition Accuracy (%) on Extended YaleB face
database with 20% block occlusion.
Training
samples
PCA LFDA LPP SPP CRP
Proposed
method
l=8 52.4 60.7 69.4 69.8 71.6 71.1
l=16 60.2 85.9 86.6 82.3 90.4 90.7
ICPRAM 2019 - 8th International Conference on Pattern Recognition Applications and Methods
456

Table 5: Recognition Accuracy (%) on Extended YaleB face
database with 20% corruption.
Training
samples
PCA LFDA LPP SPP CRP
Proposed
method
l=8 53.6 73.8 71.8 72.8 72.3 73.1
l=16 64.3 84.1 83.9 82.3 91.1 91.6
3.4 LFW Database
For this database, 100 subjects of LFWa database (Liu
et al., 2015) were used. For each person, 6 images
are selected. The images are cropped to eliminate the
background, and resized to 64 × 64. Samples of one
subject are shown in figure 4.
As in experiments with the privious databases, our
method gets the highest recognition rates among all
methods with LFW database as shown in table 6.
(a)
(b)
(c)
Figure 4: Sample images of LFW databasse of (a) Samples
for one peron. (b) Samples with 20% occlusion . (c) Sam-
ples with 20% corruption.
Table 6: Recognition Accuracy (%) on LFW database.
Database PCA LFDA LPP SPP CRP
Proposed
method
LFW 66.9 94.1 92.5 93.8 95.1 95.3
Occluded
LFW
60.9 86.8 81.3 86.1 83.2 89.1
Corrupted
LFW
61.5 88.7 85.8 87.4 84.1 89.4
4 CONCLUSION
A new regression based face recognition algorithm
is proposed. The method uses `
0
-norm to transform
the image into a sparse vector. It uses `
2
-norm regu-
larization to prevent the overlapping of nonzero cof-
fecients that belongs to different subjects. The pro-
posed method is tested with different face databases.
It is compared with other well known face recognition
methods. The experimental results show the superior-
ity of accuracy of the proposed method. They also
show the robustness of the method under occlusion
and corruption. Another advantage of the proposed
method is the low computational cost, since it only
contains matrix vector multiplication and norm com-
putation to achieve the classification task.
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