Neurocomputing in Complex Domain - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 43

Chapter 3
Neurocomputing in Complex Domain

Abstract There are many areas of applications which involve signals that are
inherently complex-valued. The characteristics of these applications can be
effectively realized if they are operated with the complex-valued neural networks
(CVNNs). Apart from that it is also widely observed in researches that the real-
valued problems can be solved far efficiently if they are represented and operated
in the complex domain. Therefore, CVNNs have emerged a very good alternative
in second generation of neurocomputing. The CVNNs to preserve and process the
data (signals) in the complex domain itself are gaining more attention over their
real-valued counterparts. The use of neural networks is naturally accompanied by
the trade-off between issues such as the overfitting, generalization capability, local
minima problems, and stability of the weight-update system. The main obstacle in
the development of a complex-valued neural network (CVNN) and its learning algo-
rithm is the selection of an appropriate activation function and error function (EF).
It can be said that the suitable error function-based training scheme with a proper
choice of activation function can substantially decrease the epochs and improve the
generalization ability for the problem in question. This chapter presents prominent
functions as a basis for making these choices and designing a learning scheme. The
choice of EF and activation function in the training scheme also circumvents some
of the existing lacunae such as error getting stuck and not progressing below a cer-
tain value. This chapter further introduces a novel approach to improve resilient
propagation in complex domain for fast learning.

3.1 Complex Domain Neuron

We may recall that complex domain neurons are attractive due to the “reliable theo-
retical results for their universal approximation abilities and for their generalization
power measured by series of researchers” [ 1 ]. The complex variable-based neuron
differs from the conventional real-valued neuron in almost all respects except of
course the architecture. Neurons get activated when signal is impinged, the signals
along with the weights and biases are complex numbers. The input to the bias neuron
is set to 1 j which operates to enable an offset on complex plane, as is not with

the real domain neuron. The Neuron fires according to the function of activation,
which in again a complex valued. The properties of the complex plane are much

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