摘要
该文证明具有三角隐层单元的三层前向神经网络逼近多变元周期函数速度的上界估计、下界估计和饱和定理 .揭示该类神经网络之隐层单元数与网络逼近速度、逼近函数结构之间的关系 .特别指出二阶光滑模为该类神经网络的本质逼近阶 ,并且当被逼近函数属于二阶 L ipschitz函数类时 ,该类神经网络的逼近能力完全取决于被逼近函数的光滑性 .文中也证明了该类神经网络的最大逼近能力以及达到最大逼近能力的一个充分必要条件 .该文所获结果对于澄清该类神经网络的函数逼近能力与应用有重要指导意义 .
Function approximation is a key issue in evaluating the computational ability of multi-layer artificial forward neural network. The main results of extensive studies on this subject are that three-layer artificial forward neural network can, with a sufficient number of hidden-layer units, approximate continuous functions and Lebesgue-integrable functions to any degree of accuracy. But most of these results contribute almost nothing to answer such important questions as how we can construct these approximating networks and how many hidden-layer units we need to approximate specific functions within some specified error. To solve these problems, Shin Sazuki [7] constructed a three-layer artificial forward neural network with trigonometric hidden-layer units and proved its constructive approximation theorems. A upper bound on approximation error for periodic functions was estimated by the first order modulus of smoothness. These results, however, cannot precisely characterize the approximation ability of the network and the relationship between the rate of approximation and the topological structure of hidden-layer. In this paper, we first point out that the second order modulus of smoothness is the essential order of approximation for the network. Then, we use the second order modulus of smoothness and give a estimate of upper bounds on approximation rate. As an important result obtained in this paper, an lower bounded estimation on approximation rate for the network is given by means of the theories of K-functional and modulus of smoothness in approximation theory. Since the upper and lower bounds of approximation error obtained by us have the same order, we can use the second modulus of smoothness to characterize the behavior of the network. In particular, when the approximated functions belong to the class of second order Lipschitz, the rate of approximation is fully determined by the smoothness of approximated functions. We also show that there exists the largest capacity of approximation for the network and further prove that the largest capacity is achieved if and only if the approximated functions are the constant functions. So, the relationship among the numbers of hidden-layer units, the rate of approximation for this network and the constructive properties of approximated function is revealed. The results obtained in this paper are very important for understanding the approximation ability of the kind of network and for constructing other networks.
出处
《计算机学报》
EI
CSCD
北大核心
2001年第9期903-908,共6页
Chinese Journal of Computers
基金
国家自然科学基金 ( 6 9975 0 16 )资助
关键词
人工神经网络
函数逼近
光滑模
逆定理
多变无周期函数
逼近阶估计
three layers artificial neural network, function approximation, estimation of lower bounds, modulus of smoothness, converse theorem
