Journal of Computer and Communications, 2014, 2, 172-181
Published Online March 2014 in SciRes. http://www.scirp.org/journal/jcc
http://dx.doi.org/10.4236/jcc.2014.24023
How to cite this paper: Wu, S.-F., Chiou, Y.-S. and Lee, S.-J. (2014) Multi-Valued Neuron with Sigmoid Activation Function
for Pattern Classification. Journal of Computer and Communications, 2, 172-181. http://dx.doi.org/10.4236/jcc.2014.24023
Multi-Valued Neuron with Sigmoid
Activation Function for Pattern
Classification
Shen-Fu Wu, Yu-Shu Chiou, Shie-Jue Lee
Department of Electrical Engineering, National Sun Yat-sen University
Received December 2013
Abstract
Multi-Valued Neuron (MVN) was proposed for pattern classification. It operates with complex-va-
lued inputs, outputs, and weights, and its learning algorithm is based on error-correcting rule. The
activation function of MVN is not differentiable. Therefore, we can not apply backpropagation
when constructing multilayer structures. In this paper, we propose a new neuron model, MVN-sig,
to simulate the mechanism of MVN with differentiable activation function. We expect MVN-sig to
achieve higher performance than MVN. We run several classification benchmark datasets to com-
pare the performance of MVN-sig with that of MVN. The experimental results show a good poten-
tial to develop a multilayer networks based on MVN-sig.
Keywords
Pattern Classification; Mu lt i -Valued Neuron (MVN); Differentiable Activation Function;
Backpropagation
1. Introduction
The discrete multi-valued neuron (MVN) was proposed by N. Aizenberg and I. Aizenberg in [1] for pattern
classification. The neuron operates with complex-valued inputs, outputs, and weights. Its inputs and outputs are
mapped onto the complex plane. They are located on the unit circle, and are exactly the
th
k
roots of unity. The
activation function of MVN
k
-valued logic maps a set of the
th
k
roots of unity on itself. Two discrete-valued
MVN learning algorithms are presented in [2]. They are based on error-correcting learning rule and are deriva-
tive-free. This makes MVN have higher functionality than sigmoidal or radial basis function neurons.
The multilayer feedforward neural network based on MVN (MLMVN) was introduced in [3,4]. This model
can achieve good performance using simpler structures. MLMVN learning rule is heuristic error backpropa-
gation due to the fact that the activation function of MVN is not differentiable. The error with certain neuron is
retrieved from next layer and evenly shared among the neurons connected from the former layer and itself. We
can not apply function optimization methods to this model because of the activation function. This property led
us to develop a multi-valued neuron with a differentiable activation function.

S.-F. Wu et al.
173
In this paper, we propose a new neuron model, MVN-sig, to simulate the mechanism of MVN with dif-
ferentiable activation function. We consider the activation function of MVN as a function of the argument of a
weighted sum. We stack multiple sigmoid functions to approximate this multiple step function. Hence, we can
obtain a differentiable input/output mapping and we apply a naive gradient descent method as its learning rule.
We expect MVN-sig to achieve better performance than MVN.
The rest of the paper is organized as follows. Section 2 briefly describes MVN and its activation function.
Section 3 presents MVN-sig and the sigmoid activation function. The learning algorithm of MVN-sig is des-
cribed in detail. Section 4 presents the results of three experiments. Finally, conclusions and discussions are
given in Section 5.
2. Multi-Valued Neuron
2.1. Discrete MVN
A discrete-valued MVN is a function mapping from a
n
-feature input onto a single output. This mapping is
described by a multiple-valued (
k
-valued) function of
n
-feature instances,
),,( 1n xxf 
, which uses
1+n
complex-valued weights:
)(=),,( 1101 nnn xxPxxf ωωω +++ 
(1)
where
1
x
,
n
x
are the features of an instance, on which the performed function depends, and
0
ω
,
1
ω
,
n
ω
are the weights. The values of the function and of the features are complex. They are the
th
k
roots of unity:
)/2( =kj iexp
j
π ε
,
1, 0,1, −∈ kj 
, and
i
is an imaginary unity.
P
is the activation function of the neuron:
k
j
zarg
k
j
if
k
ji
zP1)(2 <)(
2
),
2
(exp=)( + ≤
πππ
(2)
where
1,0,1,=−kj 
are values of the
k
-valued logic,
nn
xxz ωωω +++ 
110
=
is the weighted sum, and
)(zarg
is the argument of the complex number
z
. Equation (2) is illustrated in Figure 1.
Equation (2) divides the complex plane into
k
equal sectors and maps the whole complex plane onto a
subset of points belonging to the unit circle. This subset corresponds exactly to a set of the
th
k
roots of unity.
The MVN learning is reduced to the movement along the unit circle and is derivative-free. The movement is
determined by the error which is the difference between the desired and actual output. The error-correcting
learning rule and the corresponding learning algorithm for the discrete-valued MVN were described in [5 ] and
modified by I. Aizenberg and C. Moraga [3]:
,)(
||1)(
=
1i sq
r
r
r
i
r
i
x
zn
C
ε εωω −
+
+
+
(3 )
for
ni ,0,1,= 
, where
i
x
is the input of
th
i
feature with the components complex-conjugated,
n
is the
number of the input features,
q
ε
is the desired output of the neuron,
)( =zP
s
ε
is the actual output of the
Figure 1. Geometrical interpretation
of the discrete-valued MVN activation
function.

S.-F. Wu et al.
174
neuron (see Figure 2),
r
is the number of the learning epoch,
r
i
ω
is the current weighting of the
th
i
feature,
1 +r
i
ω
is the following weighting of the
th
i
feature after correction,
r
C
is the constant part of the learning rate
(it may always equal to 1), and
||
r
z
is the absolute value of the weighted sum obtained on the
th
r
epoch. The
factor
| |
1
r
z
is useful when learning non-linear functions with a number of high irregular jumps. Equation (3)
ensures that the corrected weighted sum moves from sector
s
to sector
q
(see Figure 2(a)). The direction of
this movement is determined by the error
sq
εεδ −=
. The convergence of the learning algorithm was proven
in [6].
2.2. Continuous MVN
The activation function Equation (2) is piece-wise discontinuous. This function can be modified and generalized
for the continuous case in the following way. When
∞→k
in Equation (2), the angle value of the sector (see
in Figure 1) will approach to zero. The activation function is transformed as follows:
||
=))((exp=)( z
z
ziargzP
(4)
where
z
is the weighted sum,
)(zarg
is the argument of complex number
z
, and
|| z
is the modulus of
the complex number
z
. The activation function Equation (4) maps the weighted sum into the whole unit circle
(see Figure 2(b)). Equation (2) maps only to the discrete subsets of the points belonging to the unit circle.
Equation (2) and Equation (4) are both not differentiable, but their differentiability is not required for MVN
learning. The Learning rule of the continuous-valued MVN is shown as follows:
1
=()=(),
(1) ||(1) ||||
rrqsrq
rr
i iiii
rr
CC z
xx
nznz z
ωωεεωε
+
+−+−
++
(5)
for
ni , 0,1,= 
.
3. MVN with Sigmoid Activation Function
The learning algorithm of MVN is reduced to the movement along the unit circle on the complex plane. They
are based on error-correcting learning rule and are derivative-free. Therefore, we can not use chain rule for error
backpropagation to construct multilayer networks using MVN. In [3], I. Aizenberg and C. Moraga introduced a
heuristic approach to construct the multilayer feedforward neural networks based on MVN. This inspired us to
develop a differentiable activation function to which we can apply backpropagation learning rule. We expect the
new learning rule to have more execution time but it can achieve better performance than MVN on a single
neuron.
(a) (b)
Figure 2. Geometrical interpretation of the MVN learning rule: (a) Discrete-
valued MVN; and (b) Continuous-valued MVN.

S.-F. Wu et al.
175
3.1. Multi-Valued Sigmoid Activation Function
The basic idea of the multi-valued sigmoid activation function is to approximate the functionality of MVN using
multiple sigmoid functions. In [7], E. Wilson presented a similar model for robot thruster control and farther
applications. The original activation function of MVN is written in Equa tion (2). This activation function
operates with the argument of weighted sum. Therefore, we can transfer this activation function into a function
of argument which is illustrated in Figu re 3. The corresponding MVN presentation is in Figure 4. This form of
original activation function of MVN is a combination of multiple step functions, which can be approximated by
stacking multiple sigmoid functions:
1()
1
11
11 ()
1
=1 =1
2/
( )=1exp
2/
( )=( )=1exp
ici
kk
ici
ii
k
f
k
Ff
θτ
θτ
π
θ
π
θθ
−−
−−
−−
+
+
∑∑
(6)
where
)(
1
θ F
is the multi-valued sigmoid activation function,
1
θ
is the argument of weighted sum,
i
τ
is the
argument of
th
i
root of unity and
)(
1
θ
i
f
is the sigmoid function setting on
i
τ
. The activation function
211
:)( θθθ →F
in Equation (6) is a mapping from the argument of weighted sum onto the argument of certain
root of unity. Hence, we use Euler’s formula in Equation (7) to transfer the outcome to a complex value.
)() ( =
2 2
2
θ θ
θ
isincos e
i
+
(7)
where
2
θ
is the outcome of activation function
)(
1
θ F
. We can approach the functionality of MVN activation
function in Equation (2) by using Equations (6) and (7). Therefore, we can build a new model architecture which
Figure 3. MVN activation function
11
{0, ,,}
k
ττ
−

represents
the set of roots of unity by their arguments.
Figure 4. Geome trical interpretation
of MVN activation function by
i
τ
.

S.-F. Wu et al.
176
can not only solve multi-valued logic problems but also be differentiable. We call this model “MVN-sig” and its
model architecture is illustrated in Figure 5.
Since the activation function is no longer discrete-valued, we need to determine how to obtain the cor-
responding label for classification problems. The most intuitive method is to equally divide the mapped range of
the activation function. This label judgement method is illustrated in F i g ure 6.
3.2. Learning Single Neuron Using Gradient Descent Method
Because the MVN-sig a complex-valued neuron, the output of MVN-sig can be derived from input:
,,, ,
= ,=
jj rejimjj rejim
xxxww w ++
(8)
where
rej
x
,
and
imj
x,
indicate the real and imaginary parts of
th
j
input variable, respectively. Likewise,
rej
w
,
and
imj
w
,
indicate the real and imaginary parts of
th
j
weight. Thus, the weighted sum of a given
n-variable instance can be written as:
,, ,,,, ,,
=0 =0=0
==() ()
nn n
jjj rej rej imj imj imj rej rej im
jj j
zwxwxwxi wxwx−+ +
∑∑ ∑
(9)
Let the output of MVN-sig be
2
22 =)()(==
ˆ
θ
θθ
i
ttt eisincosibay ++
. For an instance
t
,
t
a
and
t
b
are
the real and imaginary parts of MVN-sig output, respectively.
22 11
ˆ== ()()= (())(())
=(( ()))(( ()))
ttt
yaibcosisincos Fisin F
cosF arg zisinF arg z
θθ θθ ++ +
+
(10)
Figure 5. MVN-sig model architecture.
Figure 6. Label judgement.

S.-F. Wu et al.
177
The error function for an instance
t
is defined as:
1ˆ
= ,=
2
tttttt
E yy δδ δ −
(1 1 )
where
t
y
is the desired output and
t
y
ˆ
is the actual output of MVN-sig neuron.
t
δ
is the error between the
desired value and the output.
t
δ
signifies the complex conjugate of
t
δ
. Let the desired output be
ttt
iy
βα +=
.
)()(=
ˆ
=
ttttttt
biayy −+−− βαδ
. Hence, the error function
t
E
is a real scalar function:
})(){(
2
1
=
2
1
=
22ttttttt
baE −+− βαδδ
(12)
In order to use the chain rule to find the gradient of error function
t
E
, we have to calculate both real and
imaginary parts independently. In Equation (12), we observe that
t
E
is a function of both
)(
tt
a− α
and
)( tt b− β
, and
t
a
and
t
b
are both functions of
rej
w
,
and
imj
w
,
. We defined the gradient of the error func-
tion
t
E
with respect to the complex-valued weight
j
w
as follows:
,,
= ()=()
def
t tt
jjj rej im
E EE
wi
w ww
ηη
∂ ∂∂
∆−− +
∂ ∂∂
(13 )
The gradient of the error function with respect to the real and imaginary parts of
j
w
can be written as:
,,
,
( ())()
=( )
( ())()
( ())()
()
( ())()
tt t
j retj re
tt
tj re
EEa Farg zarg z
waF argzarg zw
Eb
Farg zargz
bF arg zarg zw
∂∂∂ ∂∂
∂ ∂∂∂∂
∂∂
∂∂
+∂∂∂ ∂
(14)
,,
,
( ())()
=( )
( ())()
( ())()
()
( ())()
tt t
j imtjim
tt
tj im
EEa Farg zarg z
waFarg zarg zw
Eb
Farg zarg z
bF arg zarg zw
∂∂∂ ∂∂
∂ ∂∂∂∂
∂∂
∂∂
+∂∂∂ ∂
(15)
We can combine Equation (14) and Equation (15) into Equation (13):
,,
={ }
( ())( ())
( ())()()
{ }{}
()
tt tt
jtt
j rej im
Ea Eb
waF arg zbF arg z
Farg zarg zarg z
i
arg zww
η
∂∂ ∂∂
∆− +
∂∂ ∂∂
∂ ∂∂
+
∂ ∂∂
(16)
Firstly, we derive each term within the first braces in Equation (16) from Equation (12) and Equatio n (10):
)(=)(=
ttt
t
t
Rea
a
E
δα −−−
∂
∂
(17)
)(=) (=
t tt
t
t
Imb
b
E
δβ −− −
∂
∂
(18)
)))(((=
))((
)))(((
=
))(( zargFsin zargF
zargFcos
zargF
at−
∂
∂
∂
∂
(19)
)))(((=
))((
))) (((
=
))(( zargFcos zargF
zargFsin
zarg F
b
t
∂
∂
∂
∂
(20)
where
)(
t
Re δ
and
)(
t
Im δ
represent the real and imaginary parts of
t
δ
. Secondly, the second braces in

S.-F. Wu et al.
178
Equation (16) contains the derivative of the multi-valued sigmoid activation function. We can derive this acti-
vation function in Equation (6):
(() )
11
(() )(() )
=1 =1
(( ))2/2exp
==
()() 11
exp exp
c argz
kk
i
c argzc argz
ii
ii
F arg zkc
arg zarg zk
τ
ττ
ππ
−−
−−
−− −−
∂∂
∂∂++
∑∑
( 21 )
Finally, the third braces in Equation (16) contains the derivatives of the argument of weighted sum
z
. The
derivation is shown as follows:
122
()()()()() ()
= ()=
() () ()
jj
argzIm zRe z ImzRez Im z
tan
wwRe zRe zIm z
−
′′
∂∂ −
∂∂ +
(22)
where
) (z
Re
and
)(zIm
represent the real and imaginary parts of weighted sum
z
and
)(zeR ′
and
)(zmI ′
represent their derivatives, respectively. From Equation (9), we can simplify this equation:
),(=
)(
),(=
)(
,, j rej
j
rej
xIm
w
zIm
xRe
w
zRe
∂
∂
∂
∂
(23)
)(=
)(
),(=
)(
,, j imj
j
imj
xRe
w
zIm
xIm
w
zRe
∂
∂
−
∂
∂
(24)
where
)(
j
xRe
and
)(
j
xIm
represent the real and imaginary parts of
th
j
input variable
x
. Thus, the
derivatives of the argument of weighted sum with respect to the real (
rej
w
,
) and imaginary
imj
w
,
parts is
shown as follows:
))()()()((
||
1
=
)(
2
,jj rej
xRezImxImzRe
z w
zarg −
∂
∂
(25)
))()()()((
||
1
=
)(
2
,jj imj
xImzImxRezRe
z w
zarg +
∂
∂
(26)
j
imjrej
x
z
z
i
w
zar g
i
w
zar g
2
,,
||
=
)()(
∂
∂
+
∂
∂
(27)
where
j
x
represents the complex conjugate of
th
j
input variable
x
. From Equations (17)-(21) and (27), we
can generate the learning rule of
th
j
input weighting
j
w∆
:
()
11
22 () 2
1
=1
2exp
={( )()( )( )}{}{}
||
1exp
c
ki
jt tj
ci
i
cz
wRe sinIm cosix
kz
θτ
θτ
π
ηδ θδθ
−−
−
−−
∆− −+
∑
(28)
where
1
θ
is the argument of the weighted sum and
2
θ
is the output of multi-valued sigmoid activation func-
tion (
)(
1
θ F
).
3.3. Stopping Criteria
We combine two stopping criteria to make sure the learning process stops when the output is accurate and stable.
Firstly, we continue to iterate until the difference between the network response and the target function reaches
some acceptable level. In Equation (29 ), we keep tracking the mean squared error until it reaches a given value
λ
. Secondly, we check the relative change in the training error is small using Equation (30).
λδδ ≤
∑
tt
N
t
N
1=
1
(29)
Λ≤
−
−
−
−
)}()({
2
1|)()(|
1
1
tt
tt
wEwE
wEwE
(30)

S.-F. Wu et al.
179
where
N
is the total number of learning instances,
t
δ
is the error between desired value and estimation, and
)(
t
wE
and
) (
1− t
w E
represent the mean squared errors at
th
t
and
th
t1)(−
learning epochs.
Figure 7 shows two examples of error convergence. Vertical and horizontal axis represent mean squared error
and time epoch, respectively. In Figure 7(a), if we only use Equation (30) as the stopping criterion, the learning
process will stop early and has worse training and testing accuracy. On the other hand, if we do not use Equation
(30), it will take too much time to converge. In Figure 7(b), we can see the learning process is relatively stable
after 300 epochs. We monitored the testing accuracy at 300 epoch and at the end of learning process. They did
not change much during this interval and that is why we use Equation (29) and Equation (30) as our stopping
criteria.
3.4. MVN-Sig Learning Algorithm
We develop the learning algorithm based on the multi-valued sigmoid function and the gradient descent method.
The convergence of the learning algorithm can be proven based on the convergence of the gradient descent
method in [8]. The implementation of the proposed learning algorithm in one iteration consists of the following
steps:
procedure MVN-sig
This is one learning epoch with N learning instances
Let
1=i
represents the
th
i
instance
Let z be the current value of weighted sum.
s
zP ε =) (
/

Equation (10)

/
sq
εεδ −←
while
Ni≤
do
Check equation with activation function.
for j = 0 to n do /

n
-variable instance

/
j
r
j
r
j
www ∆+←
+1
/

Equation (28)

/
end for
n
r
n
rr
x wx w w z
1
1
1
1
1
0
=
~ + ++
++ +
)
~
(z P
q
− ← εδ
1
=+ii
end while
Iterates until the outcome meets stopping criteria. /

Equation (29) and Equation (30)

/
end procedure
4. Simulation Results
The proposed strategies and learning algorithms are implemented and checked over a given three benchmark
(a) (b)
Figure 7. MSE-epoch plots.

S.-F. Wu et al.
180
datasets [9]. The software simulator is written in Matlab R2011b (64-bit) running on a computer with AMD
Phenom II X4 965 3.4 GHz CPU and 8 GB RAM.
4.1. Wine Dataset
This dataset is downloaded from the website of UC Irvine Machine Learning Repository [9]. It contains 178
instances and, for each instance, there are 13 real-valued input variables. The instances belongs to any of the 3
classes that indicates 3 different types of wines.
To solve this classification problem, we use 5-fold cross-validation. The first fold has 136 instances in the
training set and 42 instances in the testing set. The other 4 folds have 144 instances in the training set and 34
instances in the testing set, respectively. For MVN, we used the testing set to compute their average classi-
fication performance after learning with the training set completely with no training error, i.e ., training accuracy
being 100%. For MVN-sig, we computed their average classification performance after learning process met the
stopping criteria. The testing accuracy obtained by MVN and MVN-sig, respectively, for this dataset, is shown
in Table 1. From this table, we can see that MVN-sig achieves a better testing accuracy than MVN.
4.2. Iris Dataset
This well known dataset is also downloaded from the website of UC Irvine Machine Learning Repository [9]. It
contains 150 instances and, for each instance, there are 4 real-valued input variables. The instances are belong-
ing to any of the 3 classes (Setosa: 0, Versicolour: 1 and Virginica: 2).
To solve this classification problem, we use 5-fold cross-validation. The dataset is randomly divided into 120
instances of training set and 30 instances of testing set. For MVN, the learning process will not stop after 10,000
iterations if we want to learn whole instances with no error, i.e., training accuracy being 100%. The training
accuracy of our MVN-sig algorithm is from 96% to 99%, and we choose 96% to be the stopping criterion for
MVN learning. For MVN-sig,we computed their average classification performance after learning process met
the stopping criteria. The testing accuracy obtained by MVN and MVN-sig, respectively, for this dataset, is
shown in Table 1. From this table, we can see that MVN-sig achieves a better testing accuracy than MVN.
4.3. Breast Cancer Wisconsin (Diagnostic) Dataset
This is also downloaded from the website of UC Irvine Machine Learning Repository [9]. It contains 569
instances and, for each instance, there are 32 real-valued input variables. The instances are belonging to 2
classes (malignant: 0 and benign: 1).
To solve this classification problem, we use 5-fold cross-validation. The first fold has 452 instances in the
training set and 117 instances in the testing set. The other 4 folds have 456 instances in the training set and 113
instances in the testing set, respectively. For MVN, we used continuous learning rule in Equation (5) since the
discrete one is not suitable for binary classification. We used the testing set to compute their average classification
performance after learning with the training set completely with no training error, i.e ., training accuracy being
100%,. For MVN-sig,we computed their average classification performance after learning process met the stop-
ping criteria. The testing accuracy obtained by MVN and MVN-sig, respectively, for this dataset, is shown in
Table 1. From this table, we can see that MVN-sig achieves a better testing accuracy than MVN.
5. Conclusions and Discussions
The parameter
c
in Equation (6) affects the converging time and performance. We can develop a parameterfree
model since the activation function of MVN-sig is differentiable. But if we make this model parameter-free, the
Table 1. Accuracy comparison of MVN and MVN-sig.
MVN MVN-sig
Wine 94.404 94.980
Iris 93.200 95.870
Diagnostic 86.936 87.480

S.-F. Wu et al.
181
execution time is increased tremendously. In this paper, we select suitable parameters for each experiment.
The simulation results obtained with benchmark datasets show MVN-sig meets our expectation of original
thoughts. From the result of Iris dataset, we can see if we loosen the MVN stopping criterion, MVN-sig can still
achieve better testing accuracy. But the critical downside is the much higher time complexity. In this paper, we
just apply a simple gradient descent method to the neuron. More complicated techniques for reducing back-
propagation time can be applied.
There are three main concerns about the future work of the MVN-sig neuron. Firstly, we have to develop a
multilayer neural network using MVN-sig and investigate its performance. Secondly, to reduce its execution
time we need to do more research on the learning algorithm. Finally, we have to find its pros and cons in the
real-world applications.
References
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