IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 10, NO. 4, JULY 1999 925
A Geometrical Representation of McCulloch–Pitts
Neural Model and Its Applications
Ling Zhang and Bo Zhang
Abstract— In this paper, a geometrical representation of
McCulloch-Pitts neural model is presented. From the represen-
tation, a clear visual picture and interpretation of the model can
be seen. Two interesting applications based on the interpretation
are discussed. They are 1) a new design principle of feedforward
neural networks and 2) a new proof of mapping abilities of
three-layer feedforward neural networks.
Index Terms—Feedforward neural networks, measurable func-
tions, neighborhood covering.
I. INTRODUCTION
I
N 1943, McCulloch and Pitts [1] first presented a mathe-
matical model (M-P model) of a neuron. Since then many
artificial neural networks have developed from the well-known
M-P model [2], [3].
An M-P neuron is an element with
inputs and one output.
The general form of its function is
where
—an input vector
—a weight vector
—a threshold
(1)
Rumelhart et al. [12] presented the concept of feedforward
neural networks and their corresponding learning algorithm,
back propagation (BP), which provided the means for neural
networks to be practicable. In essence, the BP is a gradient
descent approach. In BP, the node function is replaced by
a class of sigmoid functions, i.e., infinitely differentiable
functions, in order to use mathematics with ease. Although
BP is a widely used learning algorithm it is still limited
by some disadvantages such as low convergence speed, poor
performance of the network, etc.
In order to overcome the learning complexity of BP and
other well-known algorithms, several improvements have been
presented. Since the node function
of the M-P
Manuscript received March 11, 1999; revised January 30, 1998, December
15, 1998, and March 11, 1999. This work was supported by the National
Nature Science Foundation of China and the National Key Basic Research
Program of China.
L. Zhang is with The State Key Lab of Intelligent Technology and Systems,
Artificial Intelligence Institute, Anhui University, Anhui, China.
B. Zhang is with the Department of Computer Science, Tsinghua University,
Beijing, China.
Publisher Item Identifier S 1045-9227(99)05482-X.
model can be regarded as a function of two functions: a
linear function
and a sign (or characteristic) function
, generally, there are two ways to reduce the learning
complexity. One is to replace the linear function by a quadratic
function or a distance function. For example, the polynomial-
time-trained hyperspherical classifier presented in [7] and [8]
and the restricted Coulomb energy algorithm presented in [6]
and [14] are neural networks using some distance function as
their node functions. A variety of classifiers such as those
based on radial basis functions (RBF) [9]–[13] and fuzzy
classifiers with hyperboxes [15] and ellipsoidal regions based
on Gaussian basis function [16], etc., use the same idea,
i.e., replacing the linear function by a quadratic function.
Although the learning capacity of a neural network can be
improved by making the node functions more complicated,
the improvement of the learning complexity would be limited
due to the complexity of functions. Another way to enhance
the learning capacity is by changing the topological structure
of the network. For example, in [17] and [18], the number of
hidden layers and/or the number of connections between layers
are increased. Similarly, the learning capacity is improved
at the price of increasing the complexity of the network. A
detailed description of the above methods can be found in
[19].
Now the problem is whether we can reduce the learning
complexity of a neural network and still maintain the simplic-
ity of the M-P model and its corresponding network. Some
researchers tried to do so by directly using the geometrical
interpretation of the M-P model, however, they have been
unsuccessful. Note that
can be interpreted as a
hyperplane
in an -dimensional space. When
input vector falls into the positive half-space of the hy-
perplane
. Meanwhile, . When
, input vector falls into the negative half-space
of
, and . In summary, the function of an M-P neuron
can geometrically be regarded as a spatial discriminator of an
-dimensional space divided by the hyperplane . Rujan et al.
[4], [5] intended to use such a geometrical interpretation
to analyze the behavior of neural networks. Unfortunately,
when the dimension
of the space and the number of
the hyperplanes
(i.e., the number of neurons) increase, the
mutual intersection among these hyperplanes in
-dimensional
space will become too complex to analyze. Therefore, so far
the geometrical representation has still rarely been used to
improve the learning capacity of complex neural networks.
In order to overcome this difficulty, a new representation
is presented as follows. First, assume that each input vector
has an equal length (norm). Thus all input vectors will be
restricted to an
-dimensional sphere . (In general cases, the
1045–9227/99$10.00 1999 IEEE
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