By Akira Hirose (auth.)
This publication is the second one enlarged and revised variation of the 1st winning monograph on complex-valued neural networks (CVNNs) released in 2006, which lends itself to graduate and undergraduate classes in electric engineering, informatics, regulate engineering, mechanics, robotics, bioengineering, and different suitable fields.
In the second one version the hot traits in CVNNs study are integrated, leading to e.g. virtually a doubled variety of references. The parametron invented in 1954 is additionally noted with dialogue on analogy and disparity. additionally numerous extra arguments at the benefits of the complex-valued neural networks bettering the variation to real-valued neural networks are given in quite a few sections.
The ebook comes in handy for these starting their stories, for example, in adaptive sign processing for hugely sensible sensing and imaging, regulate in unknown and altering surroundings, robotics encouraged by means of human neural structures, and brain-like info processing, in addition to interdisciplinary experiences to achieve cozy society. it's also invaluable to people who perform study and improvement relating to new services and products at businesses. the writer wrote this booklet hoping specifically that it presents the readers with significant tricks to make strong use of neural networks in absolutely sensible functions. The publication emphasizes uncomplicated principles and methods of pondering. Why can we have to reflect on neural networks that take care of advanced numbers? What merits do the complex-valued neural networks have? what's the starting place of the benefits? In what components do they boost vital functions? This ebook solutions those questions by means of describing info and examples, with the intention to motivate the readers with new rules.
The e-book comes in handy for these starting their reports, for example, in adaptive sign processing for hugely useful sensing and imaging, regulate in unknown and altering setting, robotics encouraged via human neural platforms, and brain-like details processing, in addition to interdisciplinary reviews to gain cozy society. it's also priceless to people who perform study and improvement relating to new services and products at businesses. the writer wrote this booklet hoping specifically that it offers the readers with significant tricks to make reliable use of neural networks in totally useful purposes. The ebook emphasizes simple rules and methods of considering. Why can we have to examine neural networks that take care of advanced numbers? What benefits do the complex-valued neural networks have? what's the beginning of the benefits? In what parts do they strengthen significant functions? This publication solutions those questions through describing info and examples, on the way to encourage the readers with new rules.
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Additional resources for Complex-Valued Neural Networks
2 Comparison of Complex- and Real-Valued Feedforward Neural Networks 25 The complex least mean square (LMS) algorithm is the most widely-used basis of adaptive processing of complex signals . The introduction of nonlinearity into the neuron activation function once seemed to have a serious problem in the diﬀerentiability in the complex domain. Liouville’s theorem in complex analysis states that every entire function (holomorphic function) must be constant. It follows that, if we introduce some nonlinearity, we have to abandon the diﬀerentiability.
1. 34 3 Complex-Valued Neural Networks: Distinctive Features The amplitude-phase-type activation function is also suitable for applications where we ﬁnd essence in point symmetry concerning origin, or in polar coordinate expression of the environment. 6. 2 in Chapter 1 are compatible with the amplitude-phase-type activation function. , winner determination and weight update, both of which sub-processes may consist of only addition and subtraction in its arithmetics without any multiplication that utilizes the complex nature of phase rotation.
This manner is practically eﬀective. At the same time, it is true that we discard the conformal mapping nature of the holomorphic function. However, when we utilize a conformal mapping function, we often concentrate upon the mapping structure itself, rather than a combination with some nonlinearity. Additional nonlinearity should rather be a hindrance. Accordingly, the non-holomorphy is not a big problem again. In complex-valued associative memories, researchers investigated the requirements necessary in the nonlinearity to determine an eﬀective energy function .