论文信息 - Degree of Approximation Results for Feedforward Networks Approximating Unknown Mappings and Their Derivatives

Degree of Approximation Results for Feedforward Networks Approximating Unknown Mappings and Their Derivatives

Recently Barron (1993) has given rates for hidden layer feedforward networks with sigmoid activation functions approximating a class of functions satisfying a certain smoothness condition. These rates do not depend on the dimension of the input space. We extend Barron's results to feedforward networks with possibly nonsigmoid activation functions approximating mappings and their derivatives simultaneously. Our conditions are similar but not identical to Barron's, but we obtain the same rates of approximation, showing that the approximation error decreases at rates as fast as n1/2, where n is the number of hidden units. The dimension of the input space appears only in the constants of our bounds.

[1] W. Rudin. Real and complex analysis , 1968 .

[2] Frances Chevarley Edmonds. Enumeration of arrays of a given size , 1977, Discret. Math..

[3] D. Edmunds,et al. Fourier approximation and embeddings of Sobolev spaces , 1977 .

[4] D. Cox. MULTIVARIATE SMOOTHING SPLINE FUNCTIONS , 1984 .

[5] W. Rudin. Real and complex analysis, 3rd ed. , 1987 .

[6] H. White,et al. There exists a neural network that does not make avoidable mistakes , 1988, IEEE 1988 International Conference on Neural Networks.

[7] B. Irie,et al. Capabilities of three-layered perceptrons , 1988, IEEE 1988 International Conference on Neural Networks.

[8] Kurt Hornik,et al. Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks , 1990, Neural Networks.

[9] Kurt Hornik,et al. Approximation capabilities of multilayer feedforward networks , 1991, Neural Networks.

[10] L. Jones. A Simple Lemma on Greedy Approximation in Hilbert Space and Convergence Rates for Projection Pursuit Regression and Neural Network Training , 1992 .

[11] Andrew R. Barron,et al. Universal approximation bounds for superpositions of a sigmoidal function , 1993, IEEE Trans. Inf. Theory.

[12] Daniel F. McCaffrey,et al. Convergence rates for single hidden layer feedforward networks , 1994, Neural Networks.

[13] Halbert White,et al. Sup-norm approximation bounds for networks through probabilistic methods , 1995, IEEE Trans. Inf. Theory.