FACE VERIFICATION BY SHARING KNOWLEDGE FROM
DIFFERENT SUBJECTS
David Masip
1
,
`
Agata Lapedriza
2
and Jordi Vitri`a
2
1
Department of Applied Mathematics and Analysis (MAiA), University of Barcelona (UB)
Edifici Hist`oric Gran Via de les Corts Catalanes 585, Barcelona, Spain
2
Computer Vision Center, Department of Computer Science
Universitat Aut`onoma de Barcelona, Edifici O, Bellaterra 08193, Spain
Keywords:
Face verification, Computer Vision, Logistic Regression Model, Multi-task Learning.
Abstract:
In face verification problems the number of training samples from each class is usually reduced, making dif-
ficult the estimation of the classifier parameters. In this paper we propose a new method for face verification
where we simultaneously train different face verification tasks, sharing the model parameter space. We use a
multi-task extended logistic regression classifier to perform the classification. Our approach allows to share
information from different classification tasks (transfer knowledge), mitigating the effects of the reduced sam-
ple size problem. Our experiments performed using the publicly available AR Face Database, show lower
error rates when multiple tasks are jointly trained sharing information, which confirms the theoretical approx-
imations in the related literature.
1 INTRODUCTION
Face verification can be defined as a binary classi-
fication problem where we receive as input a high-
dimensional data vector x ∈ R
D
and a claimed iden-
tity for the individual. Then, the goal is to verify the
correctness of the identity using a model of the person
to be verified.
Different facts make that Face Verification is still
nowadays an unsolved problem. For example, the di-
mensionality of the face data is large, making the esti-
mation of the classifier parameters more difficult. On
the other hand, the shortage of samples from a single
person is frequent, what produces classifiers that are
not robust enough.
Most of the face verification algorithms found in
the literature are focused on the classification in high
dimensional spaces problem. The procedure in that
cases is usually divided in two steps. First a fea-
ture extraction step is performed to reduce the data
complexity and second a classifier in the new reduced
space is trained. Thus, the problem of the high dimen-
sional data vectors classification has been addressed
with two complementary methodologies: feature ex-
traction techniques and classification algorithms.
One of the most spread unsupervised feature ex-
traction techniques is PCA, where the goal is to
find the linear projection that minimizes the mean
squared error criterion, obtaining a decorrelated rep-
resentation of the data. Recently, more sophisti-
cated techniques have appeared, imposing extra re-
strictions on the extracted feature space, such as spar-
sity (Lee and Seung, 2000) or independence (Hy-
varinen, 1999), which have been successfully applied
to face classification in presence of occlusions and
strong changes in the illumination. On the other hand,
supervised feature extraction techniques use the data
label in the dimensionality reduction process, find-
ing the subspace that maximizes some separability
criterion on the training data. Linear Discriminant
Analysis (Fisher, 1936) is the most popular super-
vised linear technique (LDA), which seems to out-
perform the PCA ”eigenfaces” approach (Moghad-
dam et al., 1998) (Belhumeur et al., 1997). Neverthe-
less, LDA uses the class-scatter matrices of the train-
ing data as a separability measure, being blind beyond
second order statistics. Recent methods such as Non
Parametric Discriminant Analysis (NDA) (Fukunaga
and Mantock, 1983) or Boosted Discriminant Projec-
tions (Masip and Vitri`a, 2006; Masip et al., 2005)
have been shown to outperform the classic approach.
In this paper we focuss our attention in the short-
286
Masip D., Lapedriza À. and Vitrià J. (2007).
FACE VERIFICATION BY SHARING KNOWLEDGE FROM DIFFERENT SUBJECTS.
In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IU/MTSV, pages 286-289
Copyright
c
 SciTePress

age of training samples in the Face Verification field.
We propose to use a face verification scheme where
multiple face verification tasks are simultaneously
learned.
The idea of sharing knowledge by training related
classification tasks was proposed by Caruana (Caru-
ana, 1997). He introduced the term Multi-task learn-
ing to describe a technique that learns a neural net-
work classifier from a set of related tasks, improv-
ing thus the generalization error. This behavior is
justified by the fact that the bias learned in a multi-
ple related tasks environment is likely to be less spe-
cific than in a single task problem. It has been shown
that using a multiple related tasks learning scheme the
number of samples needed decreases with the number
of tasks, achieving also better generalization results
(Thrun and Pratt, 1997).
The multi-task learning paradigm has been re-
cently applied to different classifiers, such as SVM
(Evgeniou et al., 2005), Adaboost (Torralba et al.,
2004), or probabilistic frameworks (Ando and Zhang,
2005). Nevertheless, up to our knowledge it has not
been still applied to face classification tasks.
There no exists in the literature a formal definition
of “related tasks”. Here we consider each verification
problem as a task and suppose that these kind of tasks
are related one to each other.
The paper is organized as follows: in the next sec-
tion we present our method, that is based on applying
the sharing knowledge paradigm to multiple related
logistic regression classifiers, section 3 describes the
experiments performed on a publicly available data
set and section 4 concludes this work.
2 SHARED LOGISTIC
REGRESSION MODEL FOR
CLASSIFICATION
A binary classification task is the problem of assign-
ing to a sample x ∈ R
D
its corresponding label L ∈
{−1, 1}. A common procedure to solve such a task
is to assume that the data follows a specific statistical
model and fix the parameters of the model from a set
of known samples that is called the training data set.
Classic logistic regression model is a statistical
approach for solving a binary classification task and
it assigns to a sample x ∈ R
D
a label L ∈ {−1, 1} as
L = arg max
c∈{−1,1}
P(L = c|x) (1)
where
P(L = 1|x) =
1
1+ e
−βx
T
(2)
and β ∈ R
D
is the parameters vector that is usually
estimated by the maximum likelihood criterion. No-
tice that P(L = −1|x
n
i
, β) = 1 − P(1|x
n
i
, β) given that
P is a probability distribution.
Our proposal is to extend this logistic regression
approach to a multi-task learning framework, where
we have multiple related binary tasks T
1
, . . . , T
M
.
Let be Z
i
= {((x
i1
, L
i1
), . . . , (x
iN
i
, L
iN
i
)
)}
i=1,..,M
training data for each one of the M tasks and Z =
{Z
i
}
i=1,..,M
the entire training samples set. Suppose
that we model each task using the logistic regression
explained above, what yields a parameters matrix B
B =



β
1
1
. . . β
M
1
.
.
.
.
.
.
.
.
.
β
1
D
. . . β
M
D



where each β
i
= (β
i
1
, . . . , β
i
D
) is the parameter vec-
tor for the i-th task. Thus, to assign the label L to an
input x ∈ R
D
according the i-th task, we should follow
the criterion
L = arg max
c∈{−1,1}
P
T
i
(L = c|x) =
1
1+ e
−β
i
x
T
(3)
Given that situation, normally the negated log-
likelihood N(Z, B) is used to estimate the parameters
adding a regularization term, usually
1
σ
2
kBk
2
, to avoid
a complex probability distribution on the parameters
set.
Nevertheless, our goal is to enforce the different
tasks to share some information given that we are as-
suming that they are related. For this aim, we hier-
archically impose a prior distribution on each row of
the matrix B.
Let us define the mean vector
¯
β = (
¯
β
1
, . . . ,
¯
β
D
) as:
¯
β
j
=
∑
M
i=1
β
i
j
M
(4)
and impose a gaussian centered prior to the mean
vector
¯
β. We want to enforce each row of the matrix B
to be gaussian distributed with mean
¯
β
j
. The resulting
optimization function is then G(B) = N(Z, B)+ R(B)
where
R(B) =
1
σ
2
1
k
¯
βk
2
+
1
σ
2
2
M
∑
i=1
kβ
i
j
−
¯
βk
2
(5)
and (σ
2
1
,σ
2
2
) are the corresponding variances of the
imposed priors.
In this work we optimize the criterion G(B) us-
ing the gradient descent algorithm given that this loss

function is differentiable and we can compute all the
partial derivatives
∂G(B)
∂β
(s)
k
=
∂N(Z, B)
∂β
(s)
k
+
∂R(B)
∂β
(s)
k
(6)
The first term N(Z, B) only depends on the para-
meter matrix B and can be directly obtained differ-
entiating the negated log-likelihood estimator of the
tasks set Z.
The second term R(B) depends on B and
¯
β, and
can be rewritten as follows
R(B) =
D
∑
j=1
[
¯
β
2
j
σ
2
1
+
1
σ
2
2
M
∑
i=1
(β
i
j
−
¯
β
j
)
2
] (7)
Given that the second term of the sum depends
also on
¯
β, we need to express it as a function of B.
Nevertheless, notice that we we can obtain an expres-
sion of each β
j
depending only (B) since we want to
minimize R(B). Thus, we have
¯
β
j
= argmin
b
(
b
2
σ
2
1
+
1
σ
2
2
M
∑
i=1
(β
i
j
− b)
2
) (8)
and this expression has a global minimum at
¯
β
j
(B) =
σ
2
1
∑
M
i=1
β
i
j
σ
2
2
+ Mσ
2
1
(9)
Given that the
∂
¯
β
j
(B)
∂β
(s)
k
= 0 if j 6= k, we obtain:
∂
¯
β
j
(B)
∂β
(s)
j
=
σ
2
1
σ
2
2
+ Mσ
2
1
(10)
Then, we get a final expression for the derivatives
of the term R(B) substituting here
∂R(B)
∂β
(s)
k
=
2
¯
β
k
σ
2
1
∂
¯
β
k
∂β
(s)
k
+
2
σ
2
2
M
∑
i=1
[(β
(i)
k
−
¯
β
k
)
∂
¯
β
k
∂β
(s)
k
] (11)
the expressions in equations 9 and 10.
3 EXPERIMENTS
To test this proposed shared logistic regression model
in face classification field we have performed dif-
ferent subject verification experiments using images
from the public AR Face Database (Martinez and Be-
navente, 1998).
To perform the experiments we have used only the
internal part of the face images and this fragments
have been resized to be 16 × 16 pixels. All the im-
ages used in the training and test step are aligned by
Figure 1: The corresponding processed training and test im-
ages: resized fragments of the original images including the
internal part of the face.
the center pixel of each eye. Some examples of these
images are shown in figure 1.
The experiments have been repeated 10 times,
and the subjects identifiers, the training images and
the test images have been always randomly selected.
Given that the shared models are specially appropri-
ated when the training set has small size, to train this
verifications we have used only 2 positive samples
(images from the subject we want to verify) and 4
negative samples (images from other subjects). To
test the system we have used 20 positive images and
40 negative samples in each case.
We have considered different task groups that
have from 1 to 10 different tasks, using the classical
single-task logistic regression model and the shared
presented model to train all the tasks of each group at
a time. The results are shown in table 1, considering
that a correct verification is the correct classification
of a subject in its corresponding positive or negative
label.
The subject classification results obtained show
that when more than 4 tasks are simultaneously
trained sharing information, the general accuracies
increase. The performance of our proposal progres-
sively increases as new tasks are added to the sys-
tem. However, when there are a few tasks to train
our shared logistic regression model, the optimization
method fails in obtaining the proper model parame-
ters.
4 CONCLUSIONS
In this paper we introduce a shared logistic regres-
sion classifier applied to a face verification problem.
We show that the theoretic benefits of the inductive
transfer knowledge in the machine learning process
stated in the recent literature can be practically ap-
plied in a probabilistic modelling of a real life prob-
lem. The results obtained in the face verification ap-
plication, show that considering the training of the
different subject classification tasks sharing the model
information yields improved accuracies. Moreover,

Table 1: Mean accuracies and 95% confidence intervals of the logistic regression method trained separately (first row) and
following our shared logistic approach (second row). When more than 4 verification tasks are simultaneously trained, the
error rates of the shared approach become lower.
1 2 3 4 5
Logistic 68.1± 8.2 65.5± 6.4 69.5± 5.3 68.2± 4.2 69.8± 3.9
Shared Logistic 59.4± 4.2 64.2± 5.4 67.9± 5.2 71.3± 5.2
6 7 8 9 10
Logistic 68.2± 3.6 68.6± 3.3 70.4± 3.3 69.8± 3.0 70.4± 2.9
Shared Logistic 72.8± 4.2 76.4± 3.1 78.2± 2.8 82.5± 2.4 84.6± 2.3
the improvement is more significant when the number
of jointly trained verification tasks increases, being a
15% higher in the case of 10 simultaneous verifica-
tions.
The probabilistic modelling presented in this pa-
per suggests new lines of future research. In our
first formulation, the sharing knowledge property is
imposed by constraining the parameter space of the
classifiers along the multiple tasks. Other approaches
could be followed, such as a more complex modelling
based on a hidden model that generates the parameter
space.
Moreover, the addition of extra related tasks from
different domains could be studied. For example, a
gender or ethnicity recognition problem. The enlarge-
ment of the task pool should benefit the amount of
shared information between the related tasks, miti-
gating the effects of the small sample size problem
in face verification.
ACKNOWLEDGEMENTS
This work is supported by MEC grant TIN2006-
15308-C02-01, Ministerio de Ciencia y Tecnologia,
Spain.
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