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A Discrete-Time Quantized-State Hopfield Neural Network

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Abstract

A discrete-time quantized-state Hopfield neural network is analyzed with special emphasis in its convergence, complexity and scalability properties. This network can be considered as a generalization of the Hopfield neural network by Shrivastava et al. [27] into the interior of the unit hypercube. This extension allows its use in a larger set of combinatorial optimization problems and its properties make of it a good candidate to build hybrid algorithms along with other heuristics such as the evolutive algorithms. Finally, the network is illustrated in some instances of the linear assignment problem and the frequency assignment problem.

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References

  1. S. Abe, Global convergence and suppression of spurious states of the Hopfield Neural Networks, IEEE Trans. Circuits Syst. 40(4) (1993) 246–257.

    Google Scholar 

  2. S.V.B. Aiyer, M. Niranjan and F. Fallside, A theoretical investigation into the performance of the Hopfield model, IEEE Trans. Neural Networks 1(2) (1990) 204–216.

    Google Scholar 

  3. J. Balicki, A. Stateczny and B. Zak, Genetic algorithms and Hopfield neural networks for solving combinatorial optimization problems, Appl. Math. Comput. Sci. 7(3) (1997) 567–592.

    Google Scholar 

  4. F. Barahona, On the computational complexity of Ising spin glass models, J. Phys. A: Math. General 15(10) (1982) 815–826.

    Google Scholar 

  5. J. Bruck and J.W. Goodman, A generalized convergence thorem for neural networks, IEEE Trans. Inform. Theory 34(5) (1988) 1089–1092.

    Google Scholar 

  6. M.A. Cohen and S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Systems, Man Cybernetic 13(5) (1983).

  7. F. Fabris and G.D. Riccia, An application of the Hopfield model to huffman codes, IEEE Trans. Inform. Theory 39(3) (1993) 1071–1076.

    Google Scholar 

  8. N. Funabiki, N. Okutani and S. Nishikawa, A three-stage heuristic combined neural-network algorithm for channel assignment in cellular mobile systems, IEEE Trans. Vehicular Technol. 49(2) (2000) 397–403.

    Google Scholar 

  9. N. Funabiki and Y. Takefuji, A neural network parallel algorithm for channel assignment problems in cellular radio networks, IEEE Trans. Vehicular Technol. 41(4) (1992) 430–437.

    Google Scholar 

  10. S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Machine Intell. 6(6) (1984) 721–741.

    Google Scholar 

  11. R.M. Golden, Mathematical Methods for Neural Network Analysis and Design(MIT Press, Cambridge, MA, 1996).

    Google Scholar 

  12. A. Haken and M. Luby, Steepest descent can take exponential time for symmetric connectionist networks, Complex Systems 2(2) (1988) 191–196.

    Google Scholar 

  13. J.K. Hao, R. Dorne and P. Galinier, Tabu search for frequency assignment in mobile radio networks, J. Heuristics 4(1) (1998) 47–62.

    Google Scholar 

  14. N. Hartsfield and G. Ringel, Pearls in Graph Theory. A Comprehensive Introduction(Academic Press, San Diego, CA, 1994).

    Google Scholar 

  15. J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA Biophys. 81 (1984) 3088–3092.

    Google Scholar 

  16. J.J. Hopfield, Neurons with graded response have collective computational properties like those of two-states neurons, Proc. Natl. Acad. Sci. USA Biophys. 79 (1982) 2554–2558.

    Google Scholar 

  17. J.C. Juang, Stability analysis of Hopfield-type neural networks, IEEE Trans. Neural Networks 10(6) (1999) 1366–1374.

    Google Scholar 

  18. H.W. Khun, The hungarian method for the assignment problem, Naval Research Logistics Quarterly 2(83) (1955).

  19. P.Koiran, Dynamics of discrete time, continuous state Hopfield networks, Neural Comput. 6(3) (1994) 459–468.

    Google Scholar 

  20. W.K. Lai and G.C. Coghill, Channel assignment through evolutionary optimization, IEEE Trans. Vehicular Technol. 45(1) (1996) 91–96.

    Google Scholar 

  21. S. Matsuda, “Optimal” Hopfield network for combinatorial optimization with linear cost function, IEEE Trans. Neural Networks 9(6) (1998) 1319–1330.

    Google Scholar 

  22. S. Matsuda, Quantized Hopfield networks for integer programming, Systems Comput. Japan 6 (1999) 1–12.

    Google Scholar 

  23. S. Matsuda, Theoretical analysis of quantized Hopfield network for integer programming, in: Proc. of the IEEE International Joint Conference on Neural Networks(1999) pp. 568–571.

  24. Neurocomputing 8, Special Issue on Optimization and Combinatorics (1995).

  25. I. Parberry, Scalability of a neural network for the knight's tour problem, Neurocomputing 12 (1996) 19–34.

    Google Scholar 

  26. S. Salcedo-Sanz, C. Bousoño-Calzón and A.R. Figueiras-Vidal, A mixed neural-genetic algorithm for the broadcast scheduling problem, IEEE Trans. Wireless Commun. 2(2) (2003) 277–283.

    Google Scholar 

  27. Y. Shrivastava, S. Dasgupta and S.M. Reddy, Guaranteed convergence in a class of Hopfield networks, IEEE Trans. Neural Networks 3(6) (1992) 951–961.

    Google Scholar 

  28. J. Sima, P. Orponen and T. Antti-Poika, On the computational complexity of binary and analog symmetric Hopfield nets, Neural Comput. 12(12) (2000) 2965–2989.

    Google Scholar 

  29. K.Y. Siu, V. Roychowdhury and T. Kailath, Discrete Neural Computation. A Theoretical Foundation(Prentice Hall, Englewood Cliffs, NJ, 1995).

    Google Scholar 

  30. A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis(Cambridge University Press, Cambridge, UK, 1998).

    Google Scholar 

  31. C. Voudouris and E.P.K. Tsang, Solving the radio link frequency assignment problem using guided local search proceedings, in: Proc. of NATO Symposium on Radio Length Frequency Assignment, Sharing and Conservation Systems (Aerospace), Aalborg, Denmark (October 1998).

    Google Scholar 

  32. Y. Watanabe, N. Mizuguchi and Y. Fujii, Solving optimization problems by using a Hopfield neural network and genetic algorithm combination, Systems Comput. Japan 29(10) (1998) 68–74.

    Google Scholar 

  33. W.J. Wolf, Inhibitory grids and the assignment problem, IEEE Trans. Neural Networks 4(2) (1993) 319–331.

    Google Scholar 

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Authors and Affiliations

  1. Department of Signal Theory and Communications, Universidad Carlos III de Madrid, Spain

    Carlos Bousoño-Calzón & Sancho Salcedo-Sanz

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  1. Carlos Bousoño-Calzón
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  2. Sancho Salcedo-Sanz
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Correspondence to Sancho Salcedo-Sanz.

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Bousoño-Calzón, C., Salcedo-Sanz, S. A Discrete-Time Quantized-State Hopfield Neural Network. Annals of Mathematics and Artificial Intelligence 42, 345–367 (2004). https://doi.org/10.1023/B:AMAI.0000038311.03614.f0

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