A Fully Complex-Valued Radial Basis Function Network and its Learning Algorithm

In this paper, a fully complex-valued radial basis function (FC-RBF) network with a fully complex-valued activation function has been proposed, and its complex-valued gradient descent learning algorithm has been developed. The fully complex activation function, sech(.) of the proposed network, satisfies all the properties needed for a complex-valued activation function and has Gaussian-like characteristics. It maps C(n) --> C, unlike the existing activation functions of complex-valued RBF network that maps C(n) --> R. Since the performance of the complex-RBF network depends on the number of neurons and initialization of network parameters, we propose a K-means clustering based neuron selection and center initialization scheme. First, we present a study on convergence using complex XOR problem. Next, we present a synthetic function approximation problem and the two-spiral classification problem. Finally, we present the results for two practical applications, viz., a non-minimum phase equalization and an adaptive beam-forming problem. The performance of the network was compared with other well-known complex-valued RBF networks available in literature, viz., split-complex CRBF, CMRAN and the CELM. The results indicate that the proposed fully complex-valued network has better convergence, approximation and classification ability.

[1]  N. Sundararajan,et al.  Complex-valued growing and pruning rbf neural networks for communication channel equalisation , 2006 .

[2]  Hojjat Adeli,et al.  Wavelet‐Clustering‐Neural Network Model for Freeway Incident Detection , 2003 .

[3]  Narasimhan Sundararajan,et al.  Complex-valued function approximation using a Fully Complex-valued RBF (FC-RBF) learning algorithm , 2009, 2009 International Joint Conference on Neural Networks.

[4]  Marvin Minsky,et al.  Perceptrons: An Introduction to Computational Geometry, Expanded Edition , 1987 .

[5]  Sundaram Suresh,et al.  Lift coefficient prediction at high angle of attack using recurrent neural network , 2003 .

[6]  Hojjat Adeli,et al.  RADIAL BASIS FUNCTION NEURAL NETWORK FOR WORK ZONE CAPACITY AND QUEUE ESTIMATION , 2003 .

[7]  Witold Pedrycz,et al.  Experience-Consistent Modeling for Radial Basis Function Neural Networks , 2008, Int. J. Neural Syst..

[8]  Tülay Adali,et al.  Approximation by Fully Complex Multilayer Perceptrons , 2003, Neural Computation.

[9]  Masaki Kobayashi,et al.  Pseudo-Relaxation Learning Algorithm for Complex-Valued Associative Memory , 2008, Int. J. Neural Syst..

[10]  Daesik Hong,et al.  Nonlinear blind equalization schemes using complex-valued multilayer feedforward neural networks , 1998, IEEE Trans. Neural Networks.

[11]  Xin Wang,et al.  A Fast Method for Implicit Surface Reconstruction Based on Radial Basis Functions Network from 3D Scattered Points , 2007, Int. J. Neural Syst..

[12]  Naotake Kamiura,et al.  Associative Memory in quaternionic Hopfield Neural Network , 2008, Int. J. Neural Syst..

[13]  Hojjat Adeli,et al.  Principal Component Analysis-Enhanced Cosine Radial Basis Function Neural Network for Robust Epilepsy and Seizure Detection , 2008, IEEE Transactions on Biomedical Engineering.

[14]  M. N. Shanmukha Swamy,et al.  Neural methods for antenna array signal processing: a review , 2002, Signal Process..

[15]  Hojjat Adeli,et al.  FUZZY-WAVELET RBFNN MODEL FOR FREEWAY INCIDENT DETECTION , 2000 .

[16]  B. Mulgrew,et al.  Complex-valued radial basis function network, Part II: Application to digital communications channel equalisation , 1994, Signal Process..

[17]  Garimella Rama Murthy,et al.  Global Dynamics of a Class of Complex Valued Neural Networks , 2008, Int. J. Neural Syst..

[18]  Saleem A. Kassam,et al.  Channel Equalization Using Adaptive Complex Radial Basis Function Networks , 1995, IEEE J. Sel. Areas Commun..

[19]  René V. Mayorga,et al.  A Radial Basis Function Network Approach for the Computation of Inverse Continuous Time Variant Functions , 2007, Int. J. Neural Syst..

[20]  Narasimhan Sundararajan,et al.  Communication channel equalization using complex-valued minimal radial basis function neural networks , 2002, IEEE Trans. Neural Networks.

[21]  Simone G. O. Fiori.  Learning by Criterion Optimization on a Unitary Unimodular Matrix Group , 2008, Int. J. Neural Syst..

[22]  Tülay Adali,et al.  Fully Complex Multi-Layer Perceptron Network for Nonlinear Signal Processing , 2002, J. VLSI Signal Process..

[23]  Tohru Nitta,et al.  The uniqueness Theorem for Complex-Valued Neural Networks with Threshold Parameters and the Redundancy of the Parameters , 2008, Int. J. Neural Syst..

[24]  Akira Hirose,et al.  Intelligent beamforming by using a complex-valued neural network , 2004, J. Intell. Fuzzy Syst..

[25]  Narasimhan Sundararajan,et al.  Fully complex extreme learning machine , 2005, Neurocomputing.

[26]  Akira Hirose,et al.  Frequency-Multiplexing Ability of Complex-Valued Hebbian Learning in Logic Gates , 2008, Int. J. Neural Syst..

[27]  Cris Koutsougeras,et al.  Complex domain backpropagation , 1992 .

[28]  Mo Chen,et al.  Online Detection of the Modality of Complex-Valued Real World Signals , 2008, Int. J. Neural Syst..

[29]  Shengrui Wang,et al.  Image classification algorithm based on the RBF neural network and K-means , 1998 .

[30]  Hojjat Adeli,et al.  Comparison of fuzzy-wavelet radial basis function neural network freeway incident detection model with California algorithm , 2002 .

[31]  Nicolas Le Bihan,et al.  Polarized Signal Classification by Complex and quaternionic Multi-Layer Perceptrons , 2008, Int. J. Neural Syst..

[32]  Sheng Chen,et al.  Complex Valued Radial Basis Function Network: Network Architecture and Learning Algorithms (part I) , 1994 .

[33]  Michele Scarpiniti,et al.  Flexible Nonlinear Blind Signal Separation in the Complex Domain , 2008, Int. J. Neural Syst..

[34]  Tohru Nitta,et al.  An Extension of the Back-Propagation Algorithm to Complex Numbers , 1997, Neural Networks.

[35]  Henry Leung,et al.  The complex backpropagation algorithm , 1991, IEEE Trans. Signal Process..

[36]  Narasimhan Sundararajan,et al.  Complex-valued function approximation using an improved BP learning algorithm for feed-forward networks , 2008, 2008 IEEE International Joint Conference on Neural Networks (IEEE World Congress on Computational Intelligence).