THE UNIQUENESS THEOREM FOR COMPLEX-VALUED NEURAL NETWORKS WITH THRESHOLD PARAMETERS AND THE REDUNDANCY OF THE PARAMETERS

2008 ◽  
Vol 18 (02) ◽  
pp. 123-134 ◽  
Author(s):  
TOHRU NITTA
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Finite Group ◽  
Uniqueness Theorem ◽  
A Finite Group ◽  
Hidden Neurons ◽  
Complex Valued ◽  
Layered Complex ◽  
The Uniqueness Theorem

This paper will prove the uniqueness theorem for 3-layered complex-valued neural networks where the threshold parameters of the hidden neurons can take non-zeros. That is, if a 3-layered complex-valued neural network is irreducible, the 3-layered complex-valued neural network that approximates a given complex-valued function is uniquely determined up to a finite group on the transformations of the learnable parameters of the complex-valued neural network.

Orthogonality of Decision Boundaries in Complex-Valued Neural Networks

2004 ◽  
Vol 16 (1) ◽  
pp. 73-97 ◽  
Author(s):  
Tohru Nitta
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Neural Nets ◽  
Decision Boundary ◽  
Generalization Ability ◽  
The Real ◽  
Learning Speed ◽  
Decision Boundaries ◽  
Complex Valued ◽  
Layered Complex

This letter presents some results of an analysis on the decision boundaries of complex-valued neural networks whose weights, threshold values, input and output signals are all complex numbers. The main results may be summarized as follows. (1) A decision boundary of a single complex-valued neuron consists of two hypersurfaces that intersect orthogonally, and divides a decision region into four equal sections. The XOR problem and the detection of symmetry problem that cannot be solved with two-layered real-valued neural networks, can be solved by two-layered complex-valued neural networks with the orthogonal decision boundaries, which reveals a potent computational power of complex-valued neural nets. Furthermore, the fading equalization problem can be successfully solved by the two-layered complex-valued neural network with the highest generalization ability. (2) A decision boundary of a three-layered complex-valued neural network has the orthogonal property as a basic structure, and its two hypersurfaces approach orthogonality as all the net inputs to each hidden neuron grow. In particular, most of the decision boundaries in the three-layered complex-valued neural network inetersect orthogonally when the network is trained using Complex-BP algorithm. As a result, the orthogonality of the decision boundaries improves its generalization ability. (3) The average of the learning speed of the Complex-BP is several times faster than that of the Real-BP. The standard deviation of the learning speed of the Complex-BP is smaller than that of the Real-BP. It seems that the complex-valued neural network and the related algorithm are natural for learning complex-valued patterns for the above reasons.

Learning Transformations with Complex-Valued Neurocomputing

2012 ◽  
Vol 3 (2) ◽  
pp. 81-116
Author(s):  
Tohru Nitta
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Affine Transformations ◽  
Generalization Ability ◽  
Practical Applications ◽  
Optical Flows ◽  
Fractal Images ◽  
Inherent Nature ◽  
The Relationship ◽  
Complex Valued

The ability of the 1-n-1 complex-valued neural network to learn 2D affine transformations has been applied to the estimation of optical flows and the generation of fractal images. The complex-valued neural network has the adaptability and the generalization ability as inherent nature. This is the most different point between the ability of the 1-n-1 complex-valued neural network to learn 2D affine transformations and the standard techniques for 2D affine transformations such as the Fourier descriptor. It is important to clarify the properties of complex-valued neural networks in order to accelerate its practical applications more and more. In this paper, first, the generalization ability of the 1-n-1 complex-valued neural network which has learned complicated rotations on a 2D plane is examined experimentally and analytically. Next, the behavior of the 1-n-1 complex-valued neural network that has learned a transformation on the Steiner circles is demonstrated, and the relationship the values of the complex-valued weights after training and a linear transformation related to the Steiner circles is clarified via computer simulations. Furthermore, the relationship the weight values of the 1-n-1 complex-valued neural network learned 2D affine transformations and the learning patterns used is elucidated. These research results make it possible to solve complicated problems more simply and efficiently with 1-n-1 complex-valued neural networks. As a matter of fact, an application of the 1-n-1 type complex-valued neural network to an associative memory is presented.

scholarly journals Ensemble of single-layered complex-valued neural networks for classification tasks

2009 ◽  
Vol 72 (10-12) ◽  
pp. 2227-2234 ◽  
Author(s):  
Md. Faijul Amin ◽  
Md. Monirul Islam ◽  
Kazuyuki Murase
Keyword(s):  
Neural Networks ◽  
Classification Tasks ◽  
Complex Valued ◽  
Layered Complex

Storage Capacity of Quaternion-Valued Hopfield Neural Networks with Dual Connections

2021 ◽  
pp. 1-15
Author(s):  
Masaki Kobayashi
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Stochastic Analysis ◽  
Storage Capacity ◽  
Hopfield Neural Network ◽  
Activation Function ◽  
Hopfield Neural Networks ◽  
Noise Tolerance ◽  
Complex Valued ◽  
Projection Rule

Abstract A complex-valued Hopfield neural network (CHNN) is a multistate Hopfield model. A quaternion-valued Hopfield neural network (QHNN) with a twin-multistate activation function was proposed to reduce the number of weight parameters of CHNN. Dual connections (DCs) are introduced to the QHNNs to improve the noise tolerance. The DCs take advantage of the noncommutativity of quaternions and consist of two weights between neurons. A QHNN with DCs provides much better noise tolerance than a CHNN. Although a CHNN and a QHNN with DCs have the samenumber of weight parameters, the storage capacity of projection rule for QHNNs with DCs is half of that for CHNNs and equals that of conventional QHNNs. The small storage capacity of QHNNs with DCs is caused by projection rule, not the architecture. In this work, the ebbian rule is introduced and proved by stochastic analysis that the storage capacity of a QHNN with DCs is 0.8 times as many as that of a CHNN.

A neural network architecture for data classification

2001 ◽  
Vol 11 (01) ◽  
pp. 33-42 ◽  
Author(s):  
Olivier Lezoray ◽  
Hubert Cardot
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Network Architecture ◽  
Neural Network Architecture ◽  
Classification Rate ◽  
Improved Performance ◽  
Global Classification ◽  
Hidden Neurons ◽  
Number Of Classes

This article aims at showing an architecture of neural networks designed for the classification of data distributed among a high number of classes. A significant gain in the global classification rate can be obtained by using our architecture. This latter is based on a set of several little neural networks, each one discriminating only two classes. The specialization of each neural network simplifies their structure and improves the classification. Moreover, the learning step automatically determines the number of hidden neurons. The discussion is illustrated by tests on databases from the UCI machine learning database repository. The experimental results show that this architecture can achieve a faster learning, simpler neural networks and an improved performance in classification.

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