Soft Computing

, Volume 17, Issue 2, pp 265–273 | Cite as

Hebbian and error-correction learning for complex-valued neurons



In this paper, we observe some important aspects of Hebbian and error-correction learning rules for complex-valued neurons. These learning rules, which were previously considered for the multi-valued neuron (MVN) whose inputs and output are located on the unit circle, are generalized for a complex-valued neuron whose inputs and output are arbitrary complex numbers. The Hebbian learning rule is also considered for the MVN with a periodic activation function. It is experimentally shown that Hebbian weights, even if they still cannot implement an input/output mapping to be learned, are better starting weights for the error-correction learning, which converges faster starting from the Hebbian weights rather than from the random ones.


Complex-valued neural networks Derivative-free learning Multi-valued neuron Hebbian learning Error-correction learning XOR problem 


  1. Aizenberg I (2008) Solving the XOR and parity n problems using a single universal binary neuron. Soft Comput 12(3):215–222MathSciNetMATHCrossRefGoogle Scholar
  2. Aizenberg I (2010) A periodic activation function and a modified learning algorithm for a multi-valued neuron. IEEE Trans Neural Netw 21(12):1939–1949CrossRefGoogle Scholar
  3. Aizenberg I (2011) Complex-valued neural networks with multi-valued neurons. Springer, HeidelbergMATHCrossRefGoogle Scholar
  4. Aizenberg NN, Aizenberg IN (1992) CNN based on multi-valued neuron as a model of associative memory for gray-scale images. In: Proceedings of the second IEEE international workshop on cellular neural networks and their applications. Technical University Munich, Germany, October 14–16, pp 36–41Google Scholar
  5. Aizenberg I, Moraga C (2007) Multilayer feedforward neural network based on multi-valued neurons (MLMVN) and a backpropagation learning algorithm. Soft Comput 11(2):169–183CrossRefGoogle Scholar
  6. Aizenberg NN, Ivaskiv YL, Pospelov DA (1971) About one generalization of the threshold function. Doklady Akademii Nauk SSSR (Rep Acad Sci USSR) 196(6):1287–1290 (in Russian)Google Scholar
  7. Aizenberg I, Aizenberg N, Vandewalle J (2000) Multi-valued and universal binary neurons theory, learning and applications. Kluwer, BostonCrossRefGoogle Scholar
  8. Aizenberg I, Moraga C, Paliy D (2005) Feedforward neural network based on multi-valued neurons. In: Reusch B (ed) Computational intelligence, theory and applications. Advances in soft computing, vol XIV. Springer, Berlin, pp 599–612CrossRefGoogle Scholar
  9. Aoki H, Watanabe E, Nagata A, Kosugi Y (2001) Rotation-invariant image association for endoscopic positional identification using complex-valued associative memories. In: Mira J, Prieto A (eds) Bio-inspired applications of connectionism. Lecture notes in computer science, vol 2085. Springer, Berlin, pp 369–374CrossRefGoogle Scholar
  10. Fiori S (2003) Extended Hebbian learning for blind separation of complex-valued sources. IEEE Trans Circuits Syst Part II 50(4):195–202CrossRefGoogle Scholar
  11. Fiori S (2005) Non-linear complex-valued extensions of Hebbian learning: an essay. Neural Comput 17(4):779–838MathSciNetMATHCrossRefGoogle Scholar
  12. Goh SL, Chen M, Popovic DH, Aihara K, Obradovic D, Mandic DP (2006) Complex-valued forecasting of wind profile. Renew Energy 31:1733–1750CrossRefGoogle Scholar
  13. Haykin S (1998) Neural networks: a comprehensive foundation, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  14. Hebb DO (1949) The organization of behavior. Wiley, New YorkGoogle Scholar
  15. Hirose A (2006) Complex-valued neural networks. Springer, BerlinMATHCrossRefGoogle Scholar
  16. Jankowski S, Lozowski A, Zurada JM (1996) Complex-valued multistate neural associative memory. IEEE Trans Neural Netw 7:1491–1496CrossRefGoogle Scholar
  17. Mandic D, Su Lee Goh V (2009) Complex valued nonlinear adaptive filters noncircularity. Widely linear and neural models. Wiley, New YorkCrossRefGoogle Scholar
  18. Muezzinoglu MK, Guzelis C, Zurada JM (2003) A new design method for the complex-valued multistate Hopfield associative memory. IEEE Trans Neural Netw 14(4):891–899CrossRefGoogle Scholar
  19. Rosenblatt F (1960) on the convergence of reinforcement procedures in simple perceptron. Report VG 1196-G-4. Cornell Aeronautical Laboratory, Buffalo, NYGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Texas A&M University-Texarkana7101 University AveTexarkanaUSA

Personalised recommendations