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The population of Seattle, WA is 64.3% White , 14.7% Asian , and 6.93% Black . 20.7% of the people in Seattle, WA speak a non-English language, and 90.5% are U.S. citizens.

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Published by: Kowsar
Human Informatics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan

Nitta T. Complex-Valued Neurocomputing and Singular Points, Arch Neurosci. 2015 ;2(4):e27461. doi: 10.5812/archneurosci.27461 .

Context: Recently, the singular points of neural networks have attracted attention from the artificial intelligence community, and their interesting properties have been demonstrated. The objective of this study is to provide an overview of studies on the singularities of complex-valued neural networks.

Evidence Acquisition: This review is based on the relevant literature on complex-valued neural networks and singular points.

Results: Review of the studies and available literature on the subject area shows that the singular points of complex-valued neural networks have negative effects on learning, as do those of real-valued neural networks. However, the nature of the singular points in complex-valued neural networks is superior in quality, and the methods for improving the learning performance have been proposed.

Conclusions: A complex-valued neural network could be a promising learning method from the viewpoint of a singularity.

Keywords: Neural Network; Learning; Non-linear Dynamics

Copyright © 2015, Tehran University of Medical Sciences. This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License ( which permits copy and redistribute the material just in noncommercial usages, provided the original work is properly cited.

An artificial neural network is a machine learning model and has recently attracted attention owing to its human-like intelligence such as learning and generalization ( 1 ).

A complex-valued neural network extends (real-valued) parameters (weights and threshold values) in a real-valued neural network to complex numbers. It is advantageous because the good-natured behavior of a complex number to rotate is assured. Thus, it is suitable for the information processing of complex-valued data and two-dimensional data and has been applied to various fields such as communications, image processing, biological information processing, land-mine detection, wind prediction, and independent component analysis (ICA) ( 2 ). Recently, a complex-valued firing-rate model was presented ( INMONARCH Mens Golden Cream Wedding Shoes MJ0105 kteeq5rPC
), which is an attempt to implement a neural network of an actual brain with complex numbers.

Generally, a hierarchical structure causes singular points. For example, consider a three-layered real-valued neural network. If a weight between a hidden neuron and an output neuron is equal to zero, then no value of the weight vector between the hidden neuron and the input neurons affects the output value of the real-valued neural network. Then, the weight vector is called an unidentifiable parameter, and is a singular point. It has been shown that singular points affect the learning dynamics of learning models and that they can cause a standstill in learning ( 4 - 6 ).

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