Universal approximation bounds for superpositions of a sigmoidalfunction
Barron, A.R.
Information Theory, IEEE Transactions on
Volume 39, Issue 3, May 1993 Page(s):930 - 945
Digital Object Identifier 10.1109/18.256500
Summary:Approximation properties of a class of artificial neural networks
are established. It is shown that feedforward networks with one layer of
sigmoidal nonlinearities achieve integrated squared error of order O
(1/n), where n is the number of nodes. The
approximated function is assumed to have a bound on the first moment of
the magnitude distribution of the Fourier transform. The nonlinear
parameters associated with the sigmoidal nodes, as well as the
parameters of linear combination, are adjusted in the approximation. In
contrast, it is shown that for series expansions with n terms,
in which only the parameters of linear combination are adjusted, the
integrated squared approximation error cannot be made smaller than order
1/n2d/ uniformly for functions satisfying the same
smoothness assumption, where d is the dimension of the input to
the function. For the class of functions examined, the approximation
rate and the parsimony of the parameterization of the networks are shown
to be advantageous in high-dimensional settings
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