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TR00-002 | 23rd December 1999 00:00
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#### Lower Bounds on the Complexity of Approximating Continuous Functions by Sigmoidal Neural Networks

**Abstract:**
We calculate lower bounds on the size of sigmoidal neural networks

that approximate continuous functions. In particular, we show that

for the approximation of polynomials the network size has to grow

as $\Omega((\log k)^{1/4})$ where $k$ is the degree of the polynomials.

This bound is valid for any input dimension, i.e. independently of

the number of variables. The result is obtained by introducing a new

method employing upper bounds on the Vapnik-Chervonenkis dimension

for proving lower bounds on the size of networks that approximate

continuous functions.