The novel aggregation function-based neuron models in complex domain

Abstract

The computational power of a neuron lies in the spatial grouping of synapses belonging to any dendrite tree. Attempts to give a mathematical representation to the grouping process of synapses continue to be a fascinating field of work for researchers in the neural network community. In the literature, we generally find neuron models that comprise of summation, radial basis or product aggregation function, as basic unit of feed-forward multilayer neural network. All these models and their corresponding networks have their own merits and demerits. The MLP constructs global approximation to input-output mapping, while a RBF network, using exponentially decaying localized non-linearity, constructs local approximation to input-output mapping. In this paper, we propose two compensatory type novel aggregation functions for artificial neurons. They produce net potential as linear or non-linear composition of basic summation and radial basis operations over a set of input signals. The neuron models based on these aggregation functions ensure faster convergence, better training and prediction accuracy. The learning and generalization capabilities of these neurons have been tested over various classification and functional mapping problems. These neurons have also shown excellent generalization ability over the two-dimensional transformations.

Exploration and Analysis of Computational Learning Algorithms on the Second Generation Neural Network

“…For supporting the complex domain (second generation neural network) analysis of various representation methods have been developed [1,3,5,6,8,10,12,14]. Further the application of various computational learning algorithms which have been fully proved and introduced by many developers [15,16,20,21,27,29].…”

“…Benvenuoto and Piazza (1992) [8] published a different version of CBP with the extension of real activation function in complex domain. From 2009 to 2012, Tripathi proposed extensive study in the field of novel complex valued neural computing [10]. A. Hirose and S. Yoshida.…”

On the learning machine for three dimensional mapping

“…The higher the complexity of an ANN is the more computations and memory intensive it can be. The number of neurons to be used in an ANN is a function of the mapping or the classifying power of the neuron itself [7,8]. Therefore, in case of high-dimensional problem, it is imperative to look for higher dimensional neuron model that can directly process the high-dimensional information.…”

On the Efficient Machine Learning of the Fundamental ComplexValued Neurons

“…The weights and threshold values used in complex neural networks, are all complex numbers, and the split type output function F C of a complex valued neuron is defined to be ₣c(z) = ₣ R (x) + i ₣ R (y) … Eq. 1 Where z = x + iy is a complex valued input to the complex valued neuron, i denotes the imaginary quantity and F R (u) = 1/(1+ exp(-u)), i.e., the real and imaginary parts of the complex valued output of a complex valued neuron means the sigmoid functions of the real part x and the imaginary part y gives us the net input z to the neuron respectively [7]. The use of sigmoidal activation function separately for the real and imaginary parts ensures that the magnitude of real and imaginary part of F(z) is bounded but now the function is no longer holomorphic (analytic) because the Cauchy-Riemann equation does not hold as in the theory of complex-valued functions, Liouville's theorem states that the complex analytic function cannot be bounded on all of the complex plane unless it is a constant function.…”