Journal
NEURAL NETWORKS
Volume 16, Issue 7, Pages 995-1002Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/S0893-6080(03)00006-6
Keywords
three-layer binary neural network; Boolean function; set covering algorithm; unit sphere covering; Hamming space; weighted distance sphere
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This paper investigates an important problem concerning the complexity of three-layer binary neural networks (BNNs) with one hidden layer. The neuron in the studied BNNs employs a hard limiter activation function with only integer weights and an integer threshold. The studies are focused on implementations of arbitrary Boolean functions which map from {0,1}(n) into {0,1}. A deterministic algorithm called set covering algorithm (SCA) is proposed for the construction of a three-layer BNN to implement an arbitrary Boolean function. The SCA is based on a unit sphere covering (USC) of the Hamming space (HS) which is chosen in advance. It is proved that for the implementation of an arbitrary Boolean function of n-variables (n greater than or equal to 3) by using SCA, [3L/2] hidden neurons are necessary and sufficient, where L is the number of unit spheres contained in the chosen USC of the n-dimensional HS. It is shown that by using SCA, the number of hidden neurons required is much less than that by using a two-parallel hyperplane method. In order to indicate the potential ability of three-layer BNNs, a lower bound on the required number of hidden neurons which is derived by using the method of estimating the Vapnik-Chervonenkis (VC) dimension is also given. (C) 2003 Elsevier Science Ltd. All rights reserved.
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