Enhanced Artificial Neural Networks Using Complex Numbers
Abstract
The model of a simple perceptron using phaseencoded
input and complexvalued weights is proposed. The
aggregation function, activation function, and learning
rule for the proposed neuron are derived and applied to
two and three input Boolean logic functions. An
improvement of 135% over the theoretical maximum of 104
linearly separable problems (of three variables) solvable
by conventional perceptrons is achieved without additional
logic, neuron stages, or higher order terms such as those
required in polynomial logic gates. Such a network is very
attractive for optical implementation since optical
computations are naturally complex.
Introduction
The processing power of an artificial neuron is dependent
on the information representation used in the neuron.
Traditionally, artificial neural networks (ANNs) used to
process realvalued physical data have relied on realvalued
weights. The interconnection weights—which represent
the learned behavior of the ANN—are derived from the
recognition that at a simplified level, a biological neuron’s
firing rate represents the information in the network.
However, some of the limitations of existing ANNs may be
traced to the limitations in the representation of
information.
The objective of this work is to develop a new neuron
model and a new learning paradigm that can encode
information such that largescale problems can be more
easily solved on digital computers. It is hypothesized that
representing real world digitized scalar data as phase and
operating on this data in the complexdomain, might
improve the performance of ANNs.
The idea of using complex number in ANNs, however, is
not new. Various researchers have developed complex
valued ANNs and applied them to complexvalued data,
such as complex signals and Fourier transform [17].
Others have explored optics in the course of finding a
suitable candidate for implementation of neural networks,
which naturally perform calculations in the complex
domain [810]. Complex numbers have also been
exploited for Hopfield type associative memory for
associative retrieval with partial input [11, 12] and for
rotation invariant retrieval using Fourier transform of edge
data [13]. Still others have developed complexvalued
artificial neural networks to solve Boolean logic functions
of n variables by selecting an output state from a complex
plane divided into m regions, with m > n [14].
The work proposed here extends complex numbers for
general ANN architectures and proposes a new learning
paradigm. The representation of the new neuron is shown
to be at least as computationally powerful as, and in many
cases, more powerful than existing ANNs.
Mathematical representation of the proposed
neuron
The proposed complexvalued artificial neuron is similar
in composition to a traditional artificial neuron except all
weights, w
i
, will be represented by complex numbers.
Externally, input data and output data will be real.
Therefore, input mapping and output mappings are
required, along with complexvalued internal neuron
functions. These internal functions will be called
aggregation (f
1
) and activation (f
2
). Each of these
operations is developed in the following sections.
The input mapping defines how the realworld data will
be represented in the ANN calculations. In the complex
Howard E. Michel and A. A. S. Awwal
Computer Science Department
University of Dayton
Dayton, OH 454692160
michel@cps.udayton.edu
Computer Science & Engineering
Wright State University
Dayton, OH 45435
aawwal@cs.wright.edu
valued artificial neuron, this mapping will be from a real
world value—typically defined as a real number, logic
value, or realworld data—into a complex number. It may
be noted that even in the traditional ANNs, realvalued data
from the real world must be mapped into a specified range.
Therefore, the inputmapping is not an additional stage
only required in the complexvalued artificial neuron.
To express the input mapping for the complex valued
artificial neuron, assume that the set of input variables P is
composed of ntuples p
i
, where each of the components p
i
is expressed as equation 1. One possible input mapping for
Boolean data is shown in equation 2. Equation 3 is the
fully developed version of equation 1 for
p
= 1. Discrete
logic levels are thus coded as periodic pulse trains with
unity magnitude and different phases. This is a mapping
from R
n
C
n
.
The complexvalued aggregation function is designed after
the form of a traditional neuron’s aggregation function as
shown in equation 4. Here, p C
n
is column vector of the
input components p
i
, and w C
n
is a row vector of weights
terms w
i
. The aggregation function is thus a mapping
C
n
C.
Unlike traditional neurons, this aggregation function is not
linear, and the resultant output is dependent on the
relationships among the various weights and inputs, as well
as their individual values. These relationships will be
described in detail below.
The aggregation function feeds directly into the activation
function; therefore, the range of the aggregation function is
the domain of the activation function, which we will call
the intermediate space. The range of the activation
function is the output space. Note that the output space is
not the realworld value, but the representation of the
“solution” within the artificial neuron; however, the neuron
must eventually respond with a real valued answer.
The complexvalued neuron will use a perceptronlike
activation function, that is, a hard limiting function.
Because the magnitude of complex number is easy to
compute, and easy to measure optically and electronically,
and because it captures the effects of angle differences
and individual component magnitudes, it was chosen as
the domain variable for the activation function. That is,
the range of the complex valued activation function will
be the magnitude of the values in the intermediate space,
q. The activation function is shown in equation 5,where a
and T are real numbers, and q is complex.
As opposed to the linear threshold used in conventional
neuron, this is equivalent to a circular threshold. Thus the
value lies either inside or outside of the decision circle.
The activation function mapping is thus of the form
C R.
In a traditional neuron, the output mapping from an
internal representation to the physical representation is
required. This is a mapping of the form R R that
typically is concerned with scaling and/or numerical
accuracy. Because the complexvalued artificial neuron’s
activation function is of the form C R, the output
mapping in the complexvalued neuron is of the form
R R and is identical to the traditional neuron’s output
mapping.
The aggregation function of a twoinput complexvalued
neuron is shown in equation 6. The variables , and
p
were defined earlier by equation 1. The variables
w
and
i
correspond to the magnitude and angle of the weight
term respectively. For simplification purposes in the
present discussion, we will assume
w
is 1, and the
learned weights are represented in
i
Since only the magnitude of the resultant vector (and not
the phase) will effect the outcome of the activation
function, and the fact that all ’s = 1, equation 6 can be
replaced by a simplified formula. Equation 7 expresses
the magnitude squared of the intermediate result in terms
of the magnitudes and phases of the inputs and weights
for the simple 2tuple neuron.
i
i
i
pi
ep
(1)
TRUE data if
2
FALSEdata if0
:m
i
(2)
TRUEdata if e
FALSE data if e
:m
2
i
0
ron_inputcv_percept
i
p (3)
wp
q
(4)
Tq
Tq
a
if1
if0
(5)
222222
111111
sincos
sincos
i
iq
pw
pw
(6)
2211
2
cos22 qr (7)
The effective change in output in response to a weight
change depends on the relationship of that weight to the
other weights and all inputs. A weight term is not simply
associated with only its corresponding input. This issue
will be considered further as a new learning rule is
developed for complexvalued artificial neurons.
Assume that, the weight change is w = w
new
 w
old
. As
discussed above, the relevant part of the weight term is its
angle, , therefore, w = =
new

old
. The required
change in the resultant is expressed as r = r
new
 r
old
.
Equation 8 relates the change in the weight, w, to the
change in the resultant vector, r. Equation 9 is thus
selected as the training rule for the complexvalued neuron.
However, equation 9 assumes that the desired change in
the resultant r, that is, r, is known. In actuality, the
desired change in the output, the error d – a, (desired –
actual) is known. To arrive at the resultant r, this error
must be brought back across the activation function,
defined by equation 5. If the activation function was
continuous, the partial derivatives in equations 8 and 9
could be extended back to the output. However, it is not; it
is discontinuous at the threshold point. Therefore, it is not
mathematically correct to take this derivative. As an
approximation, it will be assumed that a correction of r in
the direction toward the threshold of the activation
function, on either side of the threshold, will satisfy the
training goal in a “local” manner. Specifically, if d – a is
positive, r should be positive, and vice versa if r is
negative. Therefore, by replacing r in equation 9 with a
proportion of the output error d – a, a final training rule for
the 2input complexvalued artificial neuron is shown in
equation 10. The proportionality constant, ,is also known
as the “learning” rate.
The complexvalued 2inputplusbias neuron
Bias in a traditional neuron can be viewed in two ways.
The first is that the bias shifts the threshold point for the
activation function. In this context, an equivalent bias term
in the complexvalued neuron is a shift in the decision
threshold of the activation function. This is equivalent to a
shrinking or expanding of the decision circle. The second
view of the bias term in a traditional neuron is that it adds
an inputindependent value to the summation performed
by the aggregation function. This view of bias can be
accommodated in the new complex valued neuron by
adding an inputindependent complex number to the
complex summation performed by the aggregation
function.
The incorporation of complex number in the complex
valued neuron is equivalent to a vectorlike shift of the
resultant vector in the intermediate space before
thresholding. Note that in a traditional neuron, the
threshold is the additive inverse of the bias, thus providing
one additional “degree of freedom.” In the complex
valued neuron, the bias and threshold provide three
additional degrees of freedom—one for the threshold, and
one for each of the magnitude and angle of the bias.
The addition of a bias term to an artificial neuron can be
expressed by incorporating a bias element into the input
and weight vectors to create extended vectors resulting in
an increase of their dimensionality by one. Therefore, the
2inputplusbias complexvalued neuron uses a 3tuple
input set instead of a 2tuple input set. This added term is
a constant, independent of the input. This additional term
should not be confused with the transformation from a
2input threshold logic gate (TLG) to a 3input
polynomial logic gate (PLG), in which the additional term
is a function of the other two inputs. The
2inputplusbias artificial neuron is still a single level
operation. Changing the weight vector to an extended
weight vector involves adding an additional weight term
to be applied to the bias term. The extended p
i
is defined
by equation 11, with its components, p
i
, defined by
equation 1. The component b can be either real or
complex. For simplicity, and without loss of generality, it
will be assumed that b = 1. The extended w is defined by
equation 12.
By applying equations 11 and 12 to the aggregation
function defined by equation 4, an expression for the
resultant q is obtained similar to equation 6. This is
shown in equation 13. By making similar simplifying
assumptions about the magnitudes of the input and weight
terms,
p
and
w
respectively, an expression for the
magnitude squared of the resultant, r, similar to equation 7
is obtained. It is shown in equation 14. Note that no
assumptions were made about the magnitude of the bias
rw
r
1
(8)
21
11
2
1
old
2
old
1
new
2
new
1
rr
r (9)
21
11
old
2
old
1
new
2
new
1
rr
ad
(10)
T
21i
bppp (11)
b2
2
1
1
iii
eee
ww b
w (12)
term,
b
, and it is included in expression 14. This issue
will be covered further below.
Following the development, a learning rule similar to
equation 10 is developed. Those equations are shown
formally as equations 15 through 18.
Up to this point, the development of the 2inputplusbias
complexvalued artificial neuron has followed the
development of the 2input version. All input and
aggregation equations developed for the more complex
neuron have similar counterparts in the simpler neuron.
The parameters
b
and T not addressed will now be
discussed
The actual output a
i
is related to the threshold T through the
hardlimiting function depicted in equation 5. That is, if
the magnitude of the intermediatespace resultant q
i
is less
than the threshold T, the actual output will be set to 0;
otherwise, the actual output will be set to 1. Two error
conditions can exist. First, the desired output d is 1 and the
actual output is 0, and second, the desired output is 0 and
the actual output is 1. These errors can be corrected as
follows. If the desired output is 1, and the actual output is
zero, the threshold should be reduced. Conversely, if the
desired output is 0 and the actual output is 1, the threshold
should be increased.
If the error is defined as the difference between the
magnitude of the desired output and the actual output, it
can be seen that by subtracting the error (which can be –1,
0 or 1) from the threshold, the threshold moves in the
correct direction. By paralleling the perceptron learning
rule, a new learning rule for the complex valued neuron
was developed. The learning rule for the threshold T is
shown in equation 19. In equation 19, is a learning
constant similar to equation 18 above, but the values need
not be equivalent.
Changing
b
, the bias term magnitude, changes which
intermediatevalue terms q
i
will exceed the threshold
magnitude T, and therefore, their corresponding output
values a
i
. There is a very complex relationship between
these terms however. It can be seen that the effect of
changes in the magnitude of the bias term on the output is
related to amount of change, and the angle of the bias
term as a component of the total angle of the intermediate
term, q.
Equation 14 expresses the relationship between the
intermediate resultant squared, r, and the magnitude of the
bias term
b
. The partial derivative of r, with respect to
b
mathematically captures the effect of changing
b
has on
r. The goal of this learning rule is to change
b
in such a
manner as to effect a desired change in r. That is, given a
desired change in r, what should be the change in
b
?
Equation 20 expresses the relationship. Again, is a
learning constant, not necessarily equivalent to the other
learning constants used in equations 18 and 19.
Computer Simulation result
Applying the learning rules developed above, the 2input
plusbias complexvalued artificial neuron is capable of
learning all 16 possible functions of two Boolean
variables, x
1
and x
2
. Traditional perceptrons are capable of
learning only 14 of those functions.
The learned weightvalues for all 16 functions are shown
in table 1. In table 1, the Y column represents all 16
possible functions of 2 Boolean variables. Each Y entry
represents a function by specifying which of the four
minterms are included in the output. That is, Y = y
1
y
2
y
3
y
4
, with
y
1
= x
1
x
2
, y
2 =
x
1
x
2
, y
3
= x
1
x
2
and
y
4
= x
1
x
2
. A 1 associated with a particular minterm is
interpreted to mean that minterm is included in the output
function, a 0 means the minterm is not included. For
example, Y = 0001 includes only minterm y
4
and is thus
bbb
pw
pw
i
i
iq
sincos
sincos
sincos
222222
111111
(13)
bb
bb
b
qr
22
11
2211
2
2
cos2
cos2
cos22
(14)
bbbb
b
r
2211
sin2sin2
(15)
bb
r
112211
1
sin2sin2
(16)
bb
r
222211
2
sin2sin2
(17)
b
oldold
old
rrr
bb
ad
111
21
newnew
new
2121
(18)
oldoldnew
TadTT (19)
b
r
bb
ad
oldnew
(20)
the AND function, while Y = 0110 includes minterms y
2
and y
3
, and is thus the XOR function.
The learned weights in table 1 represent solutions when all
i
= 1 and T = 1.2. The input values are encoded as
specified in equation 3. The correctness of the learned
values can be verified by applying equation 13 to the
vectors. For example, minterm y
1
of the AND function is
calculated
7154.0
9035.11151.005425.10
iiiii
eeeeeq and
minterm y
4
is calculated
4431.1
9035.101151.0
2
5425.1
2
iii
i
i
i
eeeeeeq
Applying equation 5, it can be seen that q
1
< 1.2 and q
4
>
1.2, therefore, the actual output is 0 for minterm y
1
(as it
would be for minterms y
2
and y
3
if they were shown) and is
1 for minterm y
4
.
The 2inputplusbias complexvalued artificial neuron has
been extended to 3input plus bias neuron. The 3input
plusbias complexvalued artificial neuron was trained to
learn all threevariable Boolean functions. Based on these
simulations, the neuron was able to effectively compute
solutions to 245 of the 256 possible functions. Since only
104 of these are linearly separable, the complexvalued
neuron is at least 135% more powerful that a conventional
perceptron.
Cost Issues
Those modes of implementation that are inherently more
powerful—i.e. optical computing, software implemented
on parallel computers, or software implemented on
computers with coprocessors—will benefit more than
implementations on standard serial computers. Expected
benefits include reduced network size, reduced delay
when operating in the recall phase, and quicker learning.
These benefits will arise because the complexvalued
representation will be computationally more powerful
than the existing representations. For example, a single
complexvalued neuron constructed using the new
representation can solve problems that are not linearly
separable. Conventional neurons require at least two
layers to solve this problem; therefore, ANNs can be
constructed with fewer artificial neurons. Although each
individual neuron will be more complex, the overall ANN
will require less hardware or use fewer mathematical
operations to solve existing problems, therefore, speed of
operation will be increased and cost will be lowered.
These expected benefits are implementation dependent.
The cost of complexvalued neuron is less in all cases
than the traditional neuron when implemented optically.
Therefore, all the benefits the complexvalued artificial
neuron can be obtained without additional cost.
Additionally, the complexvalued neuron should be
equally superior in those implementations that provide
hardware support for complex arithmetic, for example
computers with neuralnetwork coprocessors based on
digital signal processing chips. On those implementations
dependent on standard serial computers, the complex
valued neuron will be more cost effective only in those
applications where its increased power can offset the
requirement for additional neurons.
Conclusion
The complexvalued neuron was shown to demonstrate
higher computational capability for a large class of
problems involving Boolean functions. The complex
valued neuron is able to solve all 16 functions of 2input
Boolean logic, and 245 of the 256 functions of the 3input
Boolean logic.
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Y =
y
1
y
2
y
3
y
4
1
2
b
0000 0.3246 1.4096 1.7286
0001 1.5425 0.1151 1.9035
0010 1.0964 1.9306 1.0902
0011 0.0973 2.8959 3.0582
0100 2.4710 0.2429 2.3096
0101 0.6664 2.0151 0.7127
0110 2.3024 1.0335 1.7290
0111 0.6707 0.4818 2.2412
1000 1.9473 1.9035 0.1437
1001 2.4268 3.0646 1.1171
1010 1.3255 3.1392 0.7240
1011 2.1486 2.3042 1.7459
1100 0.5393 1.6445 2.0235
1101 0.6765 0.4642 0.6235
1110 1.1095 0.8900 0.0889
1111 0.2856 0.4488 0.6283
Table 1. Learned weights, in radians, for 2inputplus
bias complexvalued perceptron
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