Abstract
The paper reviewed the results bearing out the deep-seated relation between the parallel computations and learning procedures for the laminated neural networks one of whose formalizations is represented by the theory of committee constructions. Additionally, consideration was given to two combinatorial problems concerned with learning pattern recognition in the class of affine committees—the problem of verifying existence of a three-element affine separating committee and that of element-minimal affine separating committee. The first problem was shown to be N P-complete, whereas the second problem is N P-hard and does not belong to the Apx class.
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References
Mazurov, Vl.D., On the Committee of a System of Convex Inequalities, in Proc. Int. Math. Congr., Moscow: Mosk. Gos. Univ., 1966, no. 14, p. 41.
Mazurov, Vl.D., Metod komitetov v zadachakh optimizatsii i klassifikatsii (Method of Committees in the Problems of Optimization and Classification), Moscow: Nauka, 1990.
Mazurov, Vl.D., Consistent Completion of Systems of Algorithms to Committee Technologies, Pattern Recognit. Image Anal., 1998, vol. 8, no. 4, pp. 501–506.
Ablow, C.M. and Kaylor, D.J., Inconsistent Homogeneous Linear Inequalities, Bull. Am. Math. Soc., 1965, vol. 71, no. 5, p. 724.
Judd, J.S., Neural Network Design and Complexity of Learning, New York: MIT Press, 1990.
Lin, J.H. and Vitter, J.S., Complexity Results on Learning by Neural Nets, Machine Learning, 1991, vol. 6, pp. 211–230.
Blum, A.L. and Rivest, R.L., Training a 3-node Neural Network is NP-complete, Neural Networks, 1992, vol. 5, pp. 117–127.
Mazurov, Vl.D., Committees of Inequality Systems and the Problem of Recognition, Kibernetika, 1971, no. 3, pp. 140–146.
Khachai, M.Yu., On Computational Complexity of the Minimal Committee and Related Problems, Dokl. Ross. Akad. Nauk, 2006, vol. 406, no. 6, pp. 742–745.
Hastad, J., Clique is Hard to Approximate within n 1−ε, Acta Mathematica, 1999, vol. 182, no. 13, pp. 105–142.
Dinur, I., Regev, O., and Smyth, C., The Hardness of 3-uniform Hypergraph Coloring, in Proc. 43rd Ann. IEEE Sympos. Foundat. Comput. Sci., 2002.
Khachai, M.Yu., On the Computational and Approximational Complexity of the Problem of Minimal Separating Committee, Tavrich. Vest. Informatiki i Mat., 2006, no. 1, pp. 34–43.
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Original Russian Text © V.D. Mazurov, M.Yu. Khachai, 2007, published in Avtomatika i Telemekhanika, 2007, No. 5, pp. 182–192.
This work was supported by the Russian Foundation for Basic Research, projects nos. 07-07-00168 and 07-01-399, and the Council for Grants at the President of the Russian Federation, projects nos. NSH-5595.2006.1 and MD-6768.2006.1.