On the computational complexity of binary and analog symmetric Hopfield nets

Neural Computation

Volumn 12, Issue 12, 2000, Pages 2965-2989

  Šíma, Jiří ^a   Orponen, Pekka ^b   Antti Poika, Teemu ^b  

^a INSTITUTE OF COMPUTER SCIENCE   (Czech Republic)

^b UNIVERSITY OF JYVÄSKYLÄ   (Finland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0003640275     PISSN: 08997667     EISSN: None     Source Type: Journal    
DOI: 10.1162/089976600300014791     Document Type: Article

References (51)

1. Aarts, E., & Korst, J. (1989). Simulated annealing and Boltzmann machines. Chichester: Wiley.
2. Ackley, D. H., Hinton, G. E., & Sejnowski, T. J. (1985). A learning algorithm for Boltzmann machines. Cognitive Science, 9, 147-169.
3. Balcázar, J. L., Díaz, J., & Gabarró, J. (1995). Structural complexity (vol. 1) (2nd ed.). Berlin: Springer-Verlag.
4. Balcázar, J. L., Gavaldà, R., & Siegelmann, H. T. (1997). Computational power of neural networks: A characterization in terms of Kolmogorov complexity. IEEE Transactions of Information Theory, 43, 1175-1183.
5. Barahona, F. (1982). On the computational complexity of Ising spin glass models. Journal of Physics A, 15, 3241-3253.
6. Bertoni, A., & Campadelli, P. (1994). On the approximability of the energy function. In Proceedings of the ICANN'94 Conference (pp. 1157-1160). Berlin: Springer-Verlag.
7. Bertoni, A., Campadelli, P., Gangai, C., & Posenato, R. (1997). Approximability of the ground state problem for certain Ising spin glasses. Journal of Complexity, 13, 323-339.
8. Bieche, I., Maynard, R., Rammal, R., & Uhry, J. P. (1980). On the ground states of the frustration model of a spin glass by a matching method of graph theory. Journal of Physics A, 13, 2553-2576.
9. Casey, M. (1996). The dynamics of discrete-time computation, with application to recurrent neural networks and finite state machine extraction. Neural Computation, 8, 1135-1178.
10. Chiueh, T.-D., & Goodman, R. (1988). A neural network classifier based on coding theory. In D. Anderson, (Ed.), Proceedings of the 1987 IEEE Conference on Neural Information Processing Systems - Natural and Synthetic (pp. 174-183). New York: American Institute of Physics.
11. Farhat, N. H., Psaltis, D., Prata, A., & Paek, E. (1985). Optical implementation of the Hopfield model. Applied Optics, 24, 1469-1475.
12. Floréen, P. (1991). Worst-case convergence times for Hopfield memories. IEEE Transactions on Neural Networks, 2, 533-535.
13. Floréen, P., & Orponen, P. (1994). Complexity issues in discrete Hopfield networks (Res. Rep. No. A-1994-4). Helsinki: Department of Computer Science, University of Helsinki.
14. Garey, M. R., Johnson, D. S., & Stockmeyer, L. (1976). Some simplified NP-complete graph problems. Theoretical Computer Science, 1, 237-267.
15. Goemans, M. X., & Williamson, D. P. (1995). Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42, 1115-1145.
16. Gold, B. (1986). Hopfield model applied to vowel and consonant discrimination. In J. Denker (Ed.), Neural networks for computing, AIP Conf. Proc. 151 (pp. 158-164). New York: American Institute of Physics.
17. Goles, E., Fogelman, F., & Pellegrin, D. (1985). Decreasing energy functions as a tool for studying threshold networks. Discrete Applied Mathematics, 12, 261-277.
18. Goles, E., & Martínez, S. (1989). Exponential transient classes of symmetric neural networks for synchronous and sequential updating. Complex Systems, 3, 589-597.
19. Goles, E., & Matamala, M. (1996). Symmetric discrete universal neural networks. Theoretical Computer Science, 168, 405-416.
20. Haykin, S. (1999). Neural networks: A comprehensive foundation (2nd ed.). Upper Saddle River, NJ: Prentice Hall.
21. Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. In Proceedings of the National Academy of Sciences (vol. 79, pp. 2554-2558).
22. Hopfield, J. J. (1984). Neurons with graded response have collective computational properties like those of two-state neurons. In Proceedings of the National Academy of Sciences (vol. 81, pp. 3088-3092).
23. Hopfield, J. J., & Tank, D. W. (1985). "Neural" computation of decisions in optimization problems. Biological Cybernetics, 52, 141-152.
24. Horne, B. G., & Hush, D. R. (1996). Bounds on the complexity of recurrent neural network implementations of finite state machines. Neural Networks, 9, 243-252.
25. Indyk, P. (1995). Optimal simulation of automata by neural nets. In Proceedings of the 12th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comp. Sci. 900 (pp. 337-348). Berlin: Springer-Verlag.
26. Koiran, P. (1994). Dynamics of discrete time, continuous state Hopfield networks. Neural Computation, 6, 459-468.
27. Komlós, P. and Paturi, R. (1988). Convergence results in an associative memory model. Neural Networks, 1, 239-250.
28. Kosko, B. (1988). Bidirectional associative memories. IEEE Transactions on Systems, Man, and Cybernetics, 18, 49-60.
29. Liao, L.-Z. (1999). A recurrent neural network for N-stage optimal control problems. Neural Processing Letters, 10, 195-200.
30. Maass, W., & Orponen, P. (1998). On the effect of analog noise in discrete-time analog computations. Neural Computation, 10, 1071-1095.
31. Mahajan, S., & Ramesh, H. (1995). Derandomizing semidefinite programming based approximation algorithms. In Proceedings of the 36th Annual Symposium on Foundations of Computer Science (pp. 162-163). Los Alamitos, California: IEEE.
32. Mańdziuk, J. (2000). Optimization with the Hopfield network based on correlated noises: Experimental approach. Neurocomputing, 30, 301-321.
33. Orponen, P. (1996). The computational power of discrete Hopfield nets with hidden units. Neural Computation, 8, 403-415.
34. Parberry, I. (1994). Circuit complexity and neural networks. Cambridge, MA: MIT Press.
35. Park, J., Cho, H., & Park, D. (1999). On the design of BSB neural associative memories using semidefinite programming. Neural Computation, 11, 1985-1994.
36. Pawlicki, T., Lee, D.-S., Hull, J., & Srihari, S. (1988). Neural networks and their application to handwritten digit recognition. In Proceedings of the IEEE International Conference on Neural Networks (Vol. II, pp. 63-70).
37. Poljak, S., & Su̇ra, M. (1983). On periodical behaviour in societies with symmetric influences. Combinatorica, 3, 119-121.
38. Rojas, R. (1996). Neural networks: A systematic introduction. Berlin: Springer-Verlag.
39. Sheu, B., Lee, B., & Chang, C.-F. (1991). Hardware annealing for fast-retrieval of optimal solutions in Hopfield neural networks. In Proceedings of the IEEE International Joint Conference on Neural Networks (Vol. II, pp. 327-332).
40. Siegelmann, H. T. (1999). Neural networks and analog computation: Beyond the Turing limit. Boston: Birkhäuser.
41. Siegelmann, H. T., & Sontag, E. D. (1994). Analog computation via neural networks. Theoretical Computer Science, 131, 331-360.
42. Siegelmann, H. T., & Sontag, E. D. (1995). Computational power of neural networks. Journal of Computer System Science, 50, 132-150.
43. Siu, K.-Y., Roychowdhury, V., & Kailath, T. (1995). Discrete neural computation: A theoretical foundation. Englewood Cliffs, NJ: Prentice Hall.
44. Šíma, J. (1995). Hopfield languages. In Proceedings of the 22nd SOFSEM Seminar on Current Trends in Theory and Practice of Informatics, Lecture Notes in Comp. Sci. 1012 (pp. 461-468). Berlin: Springer-Verlag.
45. Šíma, J. (1997). Analog stable simulation of discrete neural networks. Neural Network World, 7, 679-686.
46. Šíma, J., & Orponen, P. (2000). A continuous-time Hopfield net simulation of discrete neural networks. In Proceedings of NC'2000: Second International ICSC Symposium on Neural Computing (pp. 36-42). Wetaskiwin, Canada: ICSC Academic Press.
47. Šíma, J., Orponen, P., & Antti-Poika, T. (1999). Some afterthoughts on Hopfield networks. In Proceedings of the 26th SOFSEM Seminar on Current Trends in Theory and Practice of Informatics, Lecture Notes in Comp. Sci. 1725 (pp. 459-469). Berlin: Springer-Verlag.
48. Šíma, J., & Wiedermann, J. (1998). Theory of neuromata. Journal of the ACM, 45, 155-178.
49. Wegener, I. (1987). The complexity of boolean functions. Chichester: Wiley/Teubner.
50. Wiedermann, J. (1994). Complexity issues in discrete neurocomputing. Neural Network World, 4, 99-119.
51. Wu, A., & Tam, P. K. S. (1999). A neural network methodology of quadratic optimization. International Journal of Neural Systems, 9, 87-93.