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for CVNN, in which nonlinearities, uncertainty, and complexities play a major role.
A number of CVNN hardware have also been developed using electromagnetic and
light waves. Further, quantum neural network has also became an emerging field of
investigation in recent researches.
Though complex-values can treat two-dimensional data elements as a single entity,
what we should treat datawithmore than two-dimension in artificial neural networks?
Although this problem can of course be solved by applying several real-valued or
complex-valued neurons, it would be useful to introduce a number systemwith higher
dimensions, the so-called hypercomplex number systems. It is imperative to look
for higher dimensional neuron model that can directly process the high-dimensional
information. It will serve as a building block for a powerful ANNwith fewer neurons.
In the beginning of twenty-first century, researches devoted thier investigation for
neural networkwith 3Dvectors and in the quaternionic domain. They proposed learn-
ing machine consisting of multilayer network of 3D vector-valued and quaternion
neurons. The equivalent error back-propagation training algorithm for 3D vector-
valued and quaternionic neural network is the natural extension of complex-valued
backpropagation (BP) algorithm and has natural ability to learn high dimension
motion compactly and efficiently as C BP learns 2D motion.
1.2.2 ANN Versus High-Dimensional Neural Network
An ANN is a biologically inspired technique that has great abilities of learning,
association, and generalization [ 23 , 24 ]. It relies largely on parallel processing and
is used in decision making based on incomplete data. Extensive studies carried out
during the past several years have revealed that neural networks enjoy numerous
practical advantages over conventional methods. They have proved to be power-
ful mathematical instrument for modeling complex systems. Traditional neural net-
work's parameters are usually real numbers, which deal with real-valued data or
applicabe for single dimension problems. Recently, there has been an increasing
interest in ANN with high-dimensional parameters. They provide an easy imple-
mentation of operations in high dimensions. Complex-valued, vector-valued, and
Quaternion-valued neural networks (CVNN, VVNN, and QVNN) have been intro-
duced to characterize high-dimensional information effectively. Their functionality
depends on mathematical theories that define nonlinear high dimension functions.
The developments in this area presented the second generation of neural network.
Processing multidimensional data is an interestingly important problem for artificial
neural networks. A single conventional neuron can take only one real value as its
input, therefore their network should be configured incorporating several neurons
and typical mesh connections for processing multidimensional data. This type of
configuration is sometimes unnatural and computation itensive in applications of
artificial neural networks. It is also not possible to learn and generalize phase among
different components simultaneously with the magnitude in high-dimensional data
through conventional real-valued neural networks.
 
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