Nature of complex number and complex-valued neural networks

Abstract

We discuss the nature of complex number and its effect on complex-valued neural networks (CVNNs). After we review some examples of CVNN applications, we look back at the mathematical history to elucidate the features of complex number, in particular to confirm the importance of the phaseand-amplitude viewpoint for designing and constructing CVNNs to enhance the features. This 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 merit is significantly useful not only for waverelated signal processing but also for general processing with frequency-domain treatment through Fourier transform.Keywords electromagnetic wave, lightwave, coherence, adaptive processing in sensing and imaging, learning logic, neural hardware IntroductionComplex-valued neural networks (CVNNs) [1-6] extend the application fields steadily. We have various application systems employing CVNNs in the field of, for example, ultrasonic fault detection to find defects in metals and other materials [7], blind separation based on principal component analysis (PCA) in sonar [8] and voice processing [9], radars including ground penetrating

Complex-Valued Neural Networks Training: A Particle Swarm Optimization Strategy

“…Also, multi-valued neural network is a special type of CVNN, its has a threshold function of multi-valued logic and complexvalued weights is considered [12], [13], [14]. The CVNN which has complex number processing structure that made the network have stronger learning ability, better generalization ability [15], superior reducing power [16], faster convergence [17], lower computational complexity and less data is needed for network training. There are many previously successful attempts to implement the PSO in generating QSAR models.…”

“…PROBLEM FORMULATION QSAR models are in essence a mathematical function that relates features and descriptors generated from small molecule structures to some experimental determined activity or property [15]. The structure-activity study can indicate which features of a given molecule correlate with its activity, thus making it possible to synthesize new and more potent compounds with enhanced biological activities.…”

Dissipativity and passivity analysis for discrete-time complex-valued neural networks with time-varying delay

“…The most notable feature of complex-valued neural networks (CVNNs) is the compatibility with wave phenomena and wave information related to, for example, electromagnetic wave, light wave, electron wave, and sonic wave (Hirose, 2011). Furthermore, CVNNs are widely applied in coherent electromagnetic wave signal processing.…”

Multistability of delayed complex-valued recurrent neural networks with discontinuous real-imaginary-type activation functions

“…Although a complex number is presented as an ordered pair of two real numbers, the most significant advantage of complex-valued neural networks originates not from the two-dimensionality of the complex plane, but from the fact that the multiplication of the synaptic weights, which is the elemental process at the synapses of neurons that construct the whole network. [21] This fact reduces the de-gree of freedom in learning and self-organization, in comparison with a double-dimensional real-valued neural network. Complex-valued recurrent neural networks extend the application fields steadily.…”