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Published in:
2009 | OriginalPaper | Chapter
Lower Bounds for Approximation of Some Classes of Lebesgue Measurable Functions by Sigmoidal Neural Networks
Authors : José L. Montaña, Cruz E. Borges
Published in: Bio-Inspired Systems: Computational and Ambient Intelligence
Publisher: Springer Berlin Heidelberg
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We propose a general method for estimating the distance between a compact subspace
K
of the space
L
1
([0,1]
s
) of Lebesgue measurable functions defined on the hypercube [0,1]
s
, and the class of functions computed by artificial neural networks using a single hidden layer, each unit evaluating a sigmoidal activation function. Our lower bounds are stated in terms of an invariant that measures the oscillations of functions of the space
K
around the origin. As an application we estimate the minimal number of neurons required to approximate bounded functions satisfying uniform Lipschitz conditions of order
α
with accuracy
ε
.