Neurocomputing with High Dimensional Parameters - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 23

pendent real-valued signals. Such a real-valued neural network is unable to perform
mapping in higher dimensions because corresponding learning algorithms cannot
preserve each input point's angle in magnitude as well as in sense. Besides that the
huge network topology is another limiting factor, which enhances storage memory
requirement, If conventional ANN is applied for high dimensional problems, they
also require a large number of training iterations for the acceptable solutions and pro-
vide poor class distinctiveness in classification. There may be cases in which learning
speed is a limiting factor in practical applications of neural networks to problems
that require high accuracy. The most acceptable solution resulted from researches is
to consider neural network designed with high dimensional parameters. High dimen-
sion neural networks have been found worth while in recent researches to overcome
from these problems considerably. This topic is an attempt to further improve many
issues through consideration of theoretical and practical aspects of high dimensional
neurocomputing.

2.2 Neurocomputing in High Dimension

In our day to day life, we come across many quantities that involve only one value
(magnitude), which is a real number. However, there are also many quantities that
involve magnitude and direction. Such quantities are generally called vectors, which
may be represented by hypercomplex number system and/or real-valued vectors. It
is a matter of universal incidence that a vector represents a cluster of particles in the
factual world. The recent researches in neurocomputing are dedicated to formulate a
model neuron that can deal with N signals as one cluster, called N-dimensional vector
neuron. In this topic, high dimensional neurons are defined through these vectors,
and high dimensional neural networks such as complex neural networks, quater-
nary neural networks, and three-dimensional exterior neural networks are unified in
terms of a vector representation. These vectors (signals), which are supposed to flow
through a high dimensional neural network, are the unit of learning. Therefore, they
are capable of learning high-dimensional motions as its inherent property, which is
not possible in real domain neural networks.
In science and engineering, we frequently come across with both types of
quantities. In mathematics, a hypercomplex number is a traditional term for an
element of algebra over a Field. 1 The hypercomplex numbers are the generalization
of the concept of real numbers to n dimensions which come up with an unexpected
outcome. Our idea of “number like” behavior in R n 1

(real numbers R ), 2 (complex numbers C ) that we already know. In trying to gener-
alize the real number to higher dimensions, we find only four dimensions where the

is motivated by the cases n =

1 In abstract algebra, a field is a set F, together with two associative binary operations, typically
referred to as addition and multiplication.