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


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 is defined as:

Figure 1.

Real-valued neuron model. Weights and threshold are all real numbers. The activation function is a real function.

where is the real-valued weight connecting neuron n and m, is the real-valued input signal from neuron , and is the real-valued threshold value of neuron . Then, the output of the neuron is given by . Although several types of activation functions can be used, the most commonly used are the sigmoidal function and the hyperbolic tangent function.

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