Learning Transformations with Complex-Valued Neurocomputing

Learning Transformations with Complex-Valued Neurocomputing

Tohru Nitta (National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan)
DOI: 10.4018/joci.2012040103

Abstract

The ability of the 1-n-1 complex-valued neural network to learn 2D affine transformations has been applied to the estimation of optical flows and the generation of fractal images. The complex-valued neural network has the adaptability and the generalization ability as inherent nature. This is the most different point between the ability of the 1-n-1 complex-valued neural network to learn 2D affine transformations and the standard techniques for 2D affine transformations such as the Fourier descriptor. It is important to clarify the properties of complex-valued neural networks in order to accelerate its practical applications more and more. In this paper, first, the generalization ability of the 1-n-1 complex-valued neural network which has learned complicated rotations on a 2D plane is examined experimentally and analytically. Next, the behavior of the 1-n-1 complex-valued neural network that has learned a transformation on the Steiner circles is demonstrated, and the relationship the values of the complex-valued weights after training and a linear transformation related to the Steiner circles is clarified via computer simulations. Furthermore, the relationship the weight values of the 1-n-1 complex-valued neural network learned 2D affine transformations and the learning patterns used is elucidated. These research results make it possible to solve complicated problems more simply and efficiently with 1-n-1 complex-valued neural networks. As a matter of fact, an application of the 1-n-1 type complex-valued neural network to an associative memory is presented.
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Neural Network Model

A brief overview of neural networks is given. In the early 1940s, the pioneers of the field, McCulloch and Pitts, proposed a computational model based on a simple neuron-like element (McCulloch & Pitts, 1943). Since then, various types of neurons and neural networks have been developed independently of their direct similarity to biological neural networks. They can now be considered as a powerful branch of present science and technology.

Neurons are the atoms of neural computation. Out of those simple computational neurons all neural networks are build up. An illustration of a (real-valued) neuron is given in Figure 1. The activity of neuron joci.2012040103.m01 is defined as:

joci.2012040103.m02
(1)
Figure 1.

Real-valued neuron model. Weights joci.2012040103.m10 and threshold joci.2012040103.m11 are all real numbers. The activation function joci.2012040103.m12 is a real function.

joci.2012040103.f01
where joci.2012040103.m03 is the real-valued weight connecting neuron n and m, joci.2012040103.m04 is the real-valued input signal from neuron joci.2012040103.m05, and joci.2012040103.m06 is the real-valued threshold value of neuron joci.2012040103.m07. Then, the output of the neuron is given by joci.2012040103.m08. Although several types of activation functions joci.2012040103.m09 can be used, the most commonly used are the sigmoidal function and the hyperbolic tangent function.

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