Complex-Valued Neural Networks: Distinctive Features

Complex-valued neural networks (CVNNs) deal with information in complex domain with complex-valued parameters and variables. As explained in Section 2.2 in relation to physicality, neural functions including learning and self-organization are influenced by sensorimotor interfaces that connect the neural network with the environment. This characteristic is of great importance also in CVNNs. There exist certain situations where CVNNs are inevitably required or greatly effective. In this Chapter, we list such examples and discuss conditions, which will be helpful for readers to grasp what happens in individual dynamics of the CVNNs illustrated in Chapter 4. However, before listing situations, we first discuss the nature of complex number and its effect on the CVNNs. We look back the mathematical history to elucidate the features of complex number, in particular to confirm the importance of the phase-and-amplitude viewpoint for designing and constructing CVNNs to enhance the advantageous features. This viewpoint is essential in general to deal with waves such as electromagnetic-wave and lightwave. Then we point out that, although we may represent a complex number as an ordered pair of real numbers, CVNNs have dynamics different from that of real-valued neural networks. In short, in CVNNs, we can reduce ineffective degree of freedom in learning or self-organization to achieve better generalization characteristics. This merit is significantly useful not only for wave-related signal processing but also for general processing with frequency-domain treatment through Fourier transform. We also explain a matter specific to CVNNs, i.e., the fact that activation functions of CVNNs cannot be analytic. Additionally, at the end of this chapter, we review CVNN researches reported hitherto.