The essential order of approximation for neural networks

• 作 者: XU ZONGBEN Cao Feilong
• 机 构: Institute for Information and System Sciences
• 出 处: 中国科学F辑
• 发表日期: 2004年期
• 关键词: feedforward neural networks approximation order the modulus of smoonthness of a
• 基金项目: 国家自然科学基金;国家高技术研究发展计划(863计划)

There have been various studies on approximation ability of feedforward neural networks (FNNs). Most of the existing studies are, however, only concerned with density or upper bound estimation on how a multivariate function can be approximated by an FNN, and consequently, the essential approximation ability of an FNN cannot be revealed.In this paper, by establishing both upper and lower bound estimations on approximation order, the essential approximation ability (namely, the essential approximation order) of a class of FNNs is clarified in terms of the modulus of smoothness of functions to be approximated. The involved FNNs can not only approximate any continuous or integrable functions defined on a compact set arbitrarily well, but also provide an explicit lower bound on the number of hidden units required. By making use of multivariate approximation tools,it is shown that when the functions to be approximated are Lipschitzian with order up to 2,the approximation speed of the FNNs is uniquely determined by modulus of smoothness of the functions.