Lower estimation of approximation rate for neural networks  

作　　者：CAO FeiLong ZHANG YongQuan XU ZongBen 

机构地区：[1]College of Science, China Jiliang University, Hangzhou 310018, China [2]Institute of Information and System Sciences, Xi'an Jiaotong University, Xi'an 710049, China

出　　处：《Science in China(Series F)》2009年第8期1321-1327,共7页中国科学（F辑英文版）

基　　金：Supported by the National Natural Science Foundation of China (Grant No. 60873206);the National Basic Research Program of China(Grant No. 2007CB311002)

摘　　要：Let SFd and ∏φ,n,d ={∑j^n=1bjφ（wj.x＋θj）：bj,θj∈R,wj∈R^d} be the set of periodic and Lebesgue＇s square-integrable functions and the set of feedforward neural network （FNN） functions, respectively. Denote by dist （SFd , ∏φ,n,d） the deviation of the set SFd from the set ∏φ,n,d. A main purpose of this paper is to estimate the deviation. In particular, based on the Fourier transforms and the theory of approximation, a lower estimation for dist （SFd and ∏φ,n,d） is proved. That is, dist（SFd and ∏φ,n,d） ≥C/（nlog2n）1/2. The obtained estimation depends only on the number of neuron in the hidden layer, and is independent of the approximated target functions and dimensional number of input. This estimation also reveals the relationship between the approximation rate of FNNs and the topology structure of hidden layer.Let SFd and ∏φ,n,d ={∑j^n=1bjφ（wj.x＋θj）：bj,θj∈R,wj∈R^d} be the set of periodic and Lebesgue＇s square-integrable functions and the set of feedforward neural network （FNN） functions, respectively. Denote by dist （SFd , ∏φ,n,d） the deviation of the set SFd from the set ∏φ,n,d. A main purpose of this paper is to estimate the deviation. In particular, based on the Fourier transforms and the theory of approximation, a lower estimation for dist （SFd and ∏φ,n,d） is proved. That is, dist（SFd and ∏φ,n,d） ≥C/（nlog2n）1/2. The obtained estimation depends only on the number of neuron in the hidden layer, and is independent of the approximated target functions and dimensional number of input. This estimation also reveals the relationship between the approximation rate of FNNs and the topology structure of hidden layer.

关 键 词：feedforward neural networks APPROXIMATION topology structure of hidden layer RATE LOWER 

分 类 号：TP18[自动化与计算机技术—控制理论与控制工程] O212.2[自动化与计算机技术—控制科学与工程]