L p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R +1 → R 1 and $g(\parallel x\parallel _{R^n } )$g(\parallel x\parallel _{R^n } ) ∈ L loc p (R n ) with 1 ≤ p < ...L p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R +1 → R 1 and $g(\parallel x\parallel _{R^n } )$g(\parallel x\parallel _{R^n } ) ∈ L loc p (R n ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L p (K) with any accuracy for any compact set K in R n , if and only if g(x) is not an even polynomial.展开更多
This paper studies approximation capability to L^2（Rd） functions of incremental constructive feedforward neural networks （FNN） with random hidden units. Two kinds of therelayered feedforward neural networks are co...This paper studies approximation capability to L^2（Rd） functions of incremental constructive feedforward neural networks （FNN） with random hidden units. Two kinds of therelayered feedforward neural networks are considered： radial basis function （RBF） neural networks and translation and dilation invariant （TDI） neural networks. In comparison with conventional methods that existence approach is mainly used in approximation theories for neural networks, 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 （Rd） to any accuracy. Our result shows given any non-zero activation function g ： R＋ → R and g（｜｜x｜｜R^d） ∈ L^2（Rd） for RBF hidden units, or any non-zero activation function g（x） ∈ L^2（R^d） for TDI hidden units, the incremental network function fn with randomly generated hidden units converges to any target function in L2 （R^d） with probability one as the number of hidden units n → ∞, if one only properly adjusts the weights between the hidden units and output unit.展开更多
基金the National Natural Science Foundation of China (10471017)
文摘L p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R +1 → R 1 and $g(\parallel x\parallel _{R^n } )$g(\parallel x\parallel _{R^n } ) ∈ L loc p (R n ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L p (K) with any accuracy for any compact set K in R n , if and only if g(x) is not an even polynomial.
基金Supported by the National Nature Science Foundation of China (Grant No10871220)"Mathematics+X" of DLUT (Grant No842328)
文摘This paper studies approximation capability to L^2（Rd） functions of incremental constructive feedforward neural networks （FNN） with random hidden units. Two kinds of therelayered feedforward neural networks are considered： radial basis function （RBF） neural networks and translation and dilation invariant （TDI） neural networks. In comparison with conventional methods that existence approach is mainly used in approximation theories for neural networks, 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 （Rd） to any accuracy. Our result shows given any non-zero activation function g ： R＋ → R and g（｜｜x｜｜R^d） ∈ L^2（Rd） for RBF hidden units, or any non-zero activation function g（x） ∈ L^2（R^d） for TDI hidden units, the incremental network function fn with randomly generated hidden units converges to any target function in L2 （R^d） with probability one as the number of hidden units n → ∞, if one only properly adjusts the weights between the hidden units and output unit.
