Neurocomputing with High Dimensional Parameters - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 34

back-propagation learning algorithm [ 2 , 8 ], and has the ability to learn 3Dmotionas
complex-BP learn 2D motion as its inherent property. In this topic, the author inves-
tigates the characteristics of 3D vector-valued neural networks by various computa-
tional experiments. The experiments suggest that 3DV-BP networks can approximate
3D mapping just by training them only over a part of the domain of the mapping.
Chapter 6 explains the learning rule for 3D vector-valued neural networks. The gen-
eralization ability of 3D neural network in 3D motion interpretation and in 3D face
recognition applications is confirmed through diverse test patterns in Chap. 6 .
2.5 Neurocomputing with Four-Dimensional Parameters
The four-dimensional hypercomplex numbers, the Quaternions, have been exten-
sively employed in several fields, such as modern mathematics, physics, control
of satellites, computer graphics, etc. One of the benefits in graphics provided by
quaternions is that affine transformations (especially spatial rotations) of geomet-
ric constructs in three-dimensional spaces, can be represented compactly and effi-
ciently. How we should treat data with four-dimension in artificial neural networks?
Although this problem can of course be solved by applying several real-valued or
complex-valued neurons. But, a better choice may be to introduce a four-dimensional
hypercomplex number system based neural network, that could be confronted to the
Quaternion in the same way as the complex numbers are confronted to the CVNN.
This hypercomplex number system was introduced by Hamilton [ 15 ], which treat
four-dimensional data elements as a single entity. There has been a growing number
of interests concerning the use of neural networks in the quaternionic domain [ 16 ].
All variables in the multilayered quaternionic-valued neural network (QVNN), such
as input, output, action potential, and connection weights are encoded by quater-
nions. A quaternionic equivalent of error back-propagation algorithm has also been
investigated and theoretically explored by many researchers. The derivation of this
learning scheme adopted a famous Wirtinger calculus [ 17 ] because this calculus
enables a more straightforward derivation of learning rules.
2.5.1 Properties of Quaternionic Space
The quaternionic space is a four-dimensional vector space over the real numbers.
Since the quaternionic algebra is at infancy there are many representations of it,
which leads to variations in operations and properties. Hence, demands for wide
consensus among researchers in quaternionic space. Most of the researchers have
followed Wirtinger calculus in their basic constructions. Representing quaternion
as a vector is more compact as well as intuitively straightforward. The quaternions
may also be used for three-dimensional operations assuming 3D space as being pure
imaginary quaternion.