Complex-valued neural networks: The merits and their origins

This paper discusses what the merits of complex-valued neural networks (CVNNs) arise from. First we look back the mathematical history to elucidate the features of complex numbers, in particular to confirm the importance of the phase-and-amplitude viewpoint for designing and constructing CVNNs to enhance the features. The viewpoint is essential in general to deal with waves such as electromagnetic-wave and lightwave. Then we point out that, although we represent a complex number as an ordered pair of real numbers for example, we can reduce ineffective degree of freedom in learning or self-organization in CVNNs to achieve better generalization characteristics. This wave-oriented merit is useful widely for general signal processing with Fourier synthesis or in frequency-domain treatment through Fourier transform.