L~2 Approximation Capability Of Radial-basis-function Networks With Random Hidden Units

This paper studies approximation capability to square integrable functions on compact sets of incremental constructive radial-basis-function(RBF) neural networks with random hidden units.In conventional neural network theories,the approximation capability theorems only confirm the existence of suitable RBF networks as desired approximators.Besides,all the parameters of the networks have to be chosen through learning,which causes the learning process complicated and inefficient.Unlike conventional neural network theories,we follow a constructive approach to prove that one may simply randomly choose parameters of hidden units and then adjust the weights between the hidden units and the output unit to make the neural network approximate any function in L²(K) to any accuracy,where K is a compact subset in Rn.Our result shows that for any given activation function g R₊¹â†'R¹ such that g is not an even polynomial and g(‖x‖)∈L_loc²(Rⁿ),the incremental RBF network function f_m with randomly generated hidden nodes can converge to any target function in L²(K) with probability one,provided that one is free to properly adjust the weights between the hidden units and output unit.This thesis is organized as follows.Some background information about neural networks is reviewed and some popular results on RBF networks are introduced in Chapter 1.Some fundamental properties of random sequence,distributions and functional analysis are introduced in Chapter 2,including the relationship between fundamental space and distributions,convolutions and so on.The third chapter mainly deals with the approximation capability of RBF neural networks with random hidden units.The result we obtained presents an automatic and efficient way to construct an incremental three-layered feedforward network for approximation.