High-Dimensional Mapping - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 104

Chapter 5
High-Dimensional Mapping
Abstract The complex plane is the geometric representation of complex numbers
established by the real axis and the orthogonal imaginary axis. A point on the com-
plex plane can be viewed as a complex number with X and Y coordinates regarded as
real and imaginary parts of the number. It can be thought of as a modified Cartesian
plane, where real part is represented by a displacement along the X -axis and imagi-
nary part by a displacement along the Y -axis. The set of complex numbers is a Field
equipped with basic algebraic properties of addition and multiplication operations
[ 1 ], and hence gives a perfect platform of operation. The properties of the complex
plane are different from those of the real line. A complex number have a nonnegative
modulus and an argument (Arg) associated with it that locates the complex number
uniquely on the plane. It is natural to represent a nonzero complex number with a
directed line segment or vector on the complex plane. The extension of traditional
real-valued neuron on complex plane has varied its structure from single dimension to
two dimensions. Real-valued neuron administers motion on real line, while learning
with a complex-valued neuron applies a linear transformation, called 2D motion,
[ 2 , 3 ] to each input signal (complex number on plane). Thus, learning in a complex-
valued neural network (CVNN) is characterized with the complex-valued signals
flowing through the network, and has ability to capture two dimension patterns nat-
urally. Therefore, the concept of complex plane allows a geometric interpretation
of complex numbers in CVNN. The present chapter investigates and explores the
mapping properties of the CVNN through some problems of mapping to bring forth
the differences between CVNN and ANN, where the stress was on problems that
CVNN solves and ANN does not.
5.1 Mapping Properties of Complex-Valued Neural Networks
A function is an association that maps points in a domain to points in the range.
Many properties are associated with functional maps—continuity, differentiability,
and compactness among many others. In most problems of practical interest, the
actual function that governs the input-output behavior is unknown as it becomes
increasingly difficult to treat the differential equations that govern the system as