Higher-order networks for the optimization of block designs
The existence of a block design with a. given set of parameters is a well-known problem from combinatorial theory. Although there exist some constructive methods, the generation of block designs in the general case is an NP-complete problem.
In this paper we use optimizing neural networks as an heuristic approach to the generation of block designs. First, a cost function is defined and mapped onto a network which has connections of arity four. This network is then used for the generation of some designs that are known to exist. For some designs the results are good, but for some others the system fails to find an optimal solution in a reasonable time. The problem is shown to be a good example to test the performance of optimizing neural networks.
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- [Aarts & Korst,87]Aarts, E.H.L., and Korst, J.H.M., “Boltzmann machines and their applications”, Proc. PARLE. Lecture Notes in Computer Science, Vol 258, p 34–50, 1987Google Scholar
- [Ackley et al., 84]Ackley D.H., Ilinton G.E. and Sejnowsky T.J., “A Learning Algorithm for Boltzmann Machines”, Cognitive Science, Vol 9, 147, 1985.Google Scholar
- [Bofill & Torras, 93a]Bofill P. and Torras C., “Combinatorial Optimization for Block Designs, and Vice Versa”, working paper.Google Scholar
- [Bofill & Torras, 93b]Bofill P. and Torras C., “Maximally Balanced Designs”, working paper.Google Scholar
- [Fisher, 40]Fisher R.A., “An examination of the different possible solutions of a. problem in incomplete blocks”, Ann. Eugen., Vol 10, p 52–57, 1940.Google Scholar
- [Hall, 86]Hall M., Combinatorial Theory, Ed. John Wiley & Sons, Second Edition 1986.Google Scholar
- [Hill, 86]Hill R., A First Course in Coding Theory, Oxford University Press, 1986.Google Scholar
- [Hopfield, 82]Hopfield J.J., “Neural Networks and Physical Systems with Emergent Collective Computational Abilities”, Proc. Nat. Academ. Sciences USA, Vol 79, 2554, 1982.Google Scholar
- [Mathon & Rosa, 90]Mathon R. and Rosa A., “Tables of parameters of BIBD with r≤41 including existence, enumeration and resolvability results: an update”, Ars Combinatoria, Vol 30, December, Winnipeg, Canada, 1990.Google Scholar
- [Peterson & Anderson, 87]Peterson C. and Anderson J.R., “A Mean Field Theory Learning Algorithm for Neural Networks”, Complex Systems, Vol 1, 995–1019, 1987.Google Scholar
- [Sejnowski, 86]Sejnowski T.J., ”Higher-Order Boltzmann Machines”, Proc AIP, Snowbird 1986.Google Scholar
- [Street & Street, 87]Street A. P. and Street D. J., Combinatorics of Experimental Design, Oxford Science Publications, Claredon, Oxford 1987.Google Scholar
- [Torras & Bofill, 89]Torras C., & Bofill, P., “A neural solution to finding optimal multibus interconnection networks”, Proc. IX Conf. de la Sociadad Chilena de Ciencias de la Computaci n, Vol 1, p 446–454, July, 1989.Google Scholar
- [Yates, 35]Yates F., “Complex Experiments (with discussion)”, Journ. of the Royal Statist. Soc., Suppl. 2, 181–247, 1935.Google Scholar