Complex-Valued Neurons

Complex-Valued Neural Networks

The primarily CIL research area is Complex-Valued Neural Networks (CVNNs), mainly Multi-Valued Neurons and neural networks based on them.

Complex-Valued Neural Networks become increasingly popular. The use of complex-valued inputs/outputs, weights and activation functions make it possible to increase the functionality of a single neuron and of a neural network, to improve their performance and to reduce the training time.

The history of complex numbers shows that although it took a long time for them to be accepted (almost 300 years from the first reference to "imaginary numbers" by Girolamo Cardano in 1545 to Leonard Euler's and Carl Friedrich Gauss' works published in 1748 and 1831, respectively), they have become an integral part of engineering and mathematics. It is difficult to imagine today how signal processing, aerodynamics, hydrodynamics, energy science, quantum mechanics, circuit analysis, and many other areas of engineering and science could develop without complex numbers. It is a fundamental mathematical fact that complex numbers are a necessary and absolutely natural part of numerical world. Their necessity clearly follows from the Fundamental Theorem of Algebra, which states that every non-constant single-variable polynomial of degree n with complex coefficients has exactly n complex roots, if each root is counted up to its multiplicity.

Answering a question frequently asked by some "conservative" people, what one can get using complex-valued neural networks ("twice more" parameters, more computations, etc.), we may say that one may get the same as using the Fourier transform, but not just the Walsh transform in signal processing. There are many engineering problems in the modern world where complex-valued signals and functions of complex variables are involved and where they are unavoidable. Thus, to employ neural networks for their analysis, approximation, etc., the use of complex-valued neural networks is natural. However, even in the analysis of real-valued signals (for example, images or audio signals) one of the most frequently used approaches is frequency domain analysis, which immediately leads us to the complex domain. In fact, analyzing signal properties in the frequency domain, we see that each signal is characterized by magnitude and phase that carry different information about the signal. This fundamental fact was deeply discovered by A.V. Oppenheim and J.S. Lim in their paper "The importance of phase in signals", IEEE Proceedings, v. 69, No 5, 1981,pp.: 529- 541. They have shown that the phase in the Fourier spectrum of a signal is much more informative than the magnitude: particularly in the Fourier spectrum of images, just phase contains the information about all shapes, edges, orientation of all objects.

This property can be illustrated by the following example. Let us consider two popular test images “Lena” and “Bridge”.

 
 
 
Lena
 
Bridge
 

Let us take their Fourier transforms and then let us swap magnitude and phase of their Fourier spectra combining the phase of “Lena” with the magnitude of “Bridge” and wise-versa. After taking the inverse Fourier transform we clearly realize that those images were restored whose phases were combined with the counterpart magnitude:

 
 
 
Restored from Lena Phase + Bridge Magnitude
 
Restored from Bridge phase + Lena Magnitude
 

Thus, in fact, phase contains information of what is represented by the corresponding signal. To use this information properly, the most appropriate solution is movement to the complex domain. Hence, one of the most important characteristics of Complex-Valued Neural Networks is the proper treatment of amplitude and phase information, e.g., the treatment of wave-related phenomena such as electromagnetism, light waves, quantum waves and oscillatory phenomenon.

There are different specific types of complex-valued neurons and complex-valued activation functions. But it is important to mention that all Complex-Valued Neurons and Complex-Valued Neural Networks have a couple of very important advantages over their real-valued counterparts. The first one is that they have much higher functionality. The second one is their better plasticity and flexibility: they learn faster and generalize better. The higher functionality means first of all the ability of a single neuron to learn those input/output mappings that are nonlinearly separable in the real domain. This means the ability to learn them in the initial space without creating higher degree inputs and without moving to the higher dimensional space, respectively. To illustrate this ability of a complex-valued neuron, let us consider how it can easily solve the XOR problem, which is a classical nonlinearly separable problem. Let us take the Universal Binary Neuron (UBN), which is comprehensively observed by I. Aizenberg et al. in the monograph Multi-valued and universal binary neurons: theory, learning, applications, Kluwer Academic Publishers, Boston/Dordrecht/London, 2000. UBN is a neuron with binary inputs and output and with the complex-valued weights. Let us consider UBN with the activation function PB defined on the complex plane as follows:

It is very easy to check that the weighting vector (0, 1, i) where i is an imaginary unity, implements the XOR function on a single UBN. This is illustrated by the following table:

This example clearly shows that a complex-valued neuron is more functional than any traditional real-valued neuron. Complex-Valued Neural Networks are respectively more functional than their real-valued counterparts.

Complex-Valued Neural Networks is a rapidly growing area. Its popularity is confirmed by successful organization of a number of special sessions in the most representative international conferences in the area over last 7-8 years (ICONIP 2002, Singapore; ICANN/ICONIP 2003, Istanbul; ICONIP 2004, Calcutta; WCCI-IJCNN 2006, Vancouver; "Fuzzy Days 2006"; Dortmund, ICANN 2007, Porto; WCCI-IJCNN 2008, Hong Kong; IJCNN 2009, Atlanta; WCCI-IJCNN 2010, Barcelona). Everywhere these sessions had large audience, which is growing continuously. There were many interesting presentations and very productive discussions. In 2010, the CVNN Task Force group was established by the IEEE Computational Intelligence Society Technical Committee on Neural Networks.

There are several new directions in CVNNs development: from formal generalization of the commonly used algorithms to the complex-valued case to the use of original complex-valued activation functions that can increase significantly the neuron and network functionality. There are also many interesting applications of CVNNs in pattern recognition and classification, image processing, time series prediction, bioinformatics, robotics, etc.