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**Networks and the best approximation property.**
*(English)*
Zbl 0714.94029

Networks can be considered as approximation schemes. It is well known that multilayer networks of the perceptron type can approximate arbitrarily well continuous functions. A similar result is proven for networks derived from regularization theory and including radial basis functions. From the point of view of approximation theory, however, the property of approximating continuous functions arbitrarily well is not sufficient for characterizing good approximation schemes. More critical is the property of best approximation, that is the presence of an element of minimum distance from the function that has to be approximated. In this paper it is shown that multilayer perceptron networks, of the type used in backpropagation, do not have this property. For regularization networks (in particular radial basis function networks) existence and uniqueness of best approximation immediately derives from the linearity of the theory.

Reviewer: F.Girosi

### MSC:

94C99 | Circuits, networks |

### Keywords:

approximation schemes; multilayer networks; regularization theory; radial basis functions; best approximation; multilayer perceptron
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\textit{F. Girosi} and \textit{T. Poggio}, Biol. Cybern. 63, No. 3, 169--176 (1990; Zbl 0714.94029)

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