Complex-valued Neural Networks: Utilizing High-dimensional by Tohru Nitta

By Tohru Nitta

Fresh learn shows that complex-valued neural networks whose parameters (weights and threshold values) are all complicated numbers are actually worthwhile, containing features bringing approximately many major purposes.

Complex-Valued Neural Networks: using High-Dimensional Parameters covers the present state of the art theories and functions of neural networks with high-dimensional parameters comparable to complex-valued neural networks, quantum neural networks, quaternary neural networks, and Clifford neural networks, which were constructing lately. Graduate scholars and researchers will simply gather the elemental wisdom had to be on the leading edge of study, whereas practitioners will without problems take up the fabrics required for the applications.

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THE COMPLEX_VALUED BOLTZMANN MACHINES Stochastic Complex-Valued Neurons Boltzmann Machines are stochastic Hopfield Networks and require stochastic neurons. We construct the stochastic discrete and continuous phasor neurons for the complex-valued Boltzmann Machines. First, we define the stochastic discrete phasor neurons. The discrete phasor neurons can take K-types of states pk = exp(2 k −1 / K ) ( k = 1 , 2 , ... , K ). We have to define the probability of the states for the input signal z. We define the probabilities Pr( pk | z ) that the discrete phasor neuron turns to the state pk for the input signal z as follows: Pr(pk | z ) = Kexp(Re( pk z )) .

Consider three points p, q and r in S. Moreover consider two geodesics, the m-geodesic from probability density p to the one q and the e-geodesic from the probability density r to the one q. We know that the m-geodesic is the curve (1 – t) η (p) + t η (q) with expectation parameters and the e-geodesic is the one (1 – t) θ (r) + t θ (q) with natural parameters. Since S is the family of discrete distributions, they are included in S again. The tangent ∂ vector of the m-geodesic on q is ∑ ( i (q) − i ( p) ) ∂ .

USA, World Scientific Pub Co Inc. , & Xibilia M. G. (1998). Neural networks in multidimensional domains: fundamentals and new trends in modelling and control (Lecture notes in control and information sciences), Springer-Verlag. 23 Complex-Valued Boltzmann Manifold Aoki, H. (2003). Applications of complex-valued neural networks for image processing. In A. ), ComplexValued Neural Networks: Theories and Applications, (pp. 181-204). USA, World Scientific Pub Co Inc. , & Hormik, K. (1989). Neural networks and principal component analysis, learning from examples without local minima.

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