A performance comparative analysis of some kinds of neural networks
by
Rossella Cancelliere
Department of Mathematics, University of Turin, Italy
Coauthors: Mario Gai (Astronomical Observatory of Turin)

Neural networks are widely used as recognizers and classifiers since the second half of the 80's; this interesting capability is due to their property of solving a nonlinear approximation problem.

Given a set of points [(x_i)\vec],  i = 1, ... , N   distinct and generally scattered, in a domain D subset R^p , and a linear space \Phi(D), spanned by continuous real basis functions o_j([x\vec]), the multivariate approximation problem at scattered data consists in finding a function o([x\vec]) in \Phi(D) such that

å i  ( f( --> xi   )- o( --> xi   ))2 = å i  ( f( --> xi   )- NH å j=1  wj oj( --> xi   ))2\

be minimized.
A neural network achieves this result thanks to the training; this iterative procedure has some very useful features like parallelism, robustness and easy implementation.

Because the choice of the best basis in the space \Phi(D) is often problem dependent, we usually deal with different types of functions o_j([x\vec]), building in this way also different types of neural networks; the most largely used in literature are the radial and the sigmoidal basis functions.
They compute their distances from a point [(x_i)\vec] by two different metrics, respectively the euclidean one and a different metric based on inner product.

In this paper we want to compare performances and properties of a particular class of radial neural networks with the classical models cited before.
We built them using cardinal basis functions with the aim to exploit the interesting feature shown in the similar context of interpolation of multivariate scattered data, that is to exactly solve the problem of the required matrix inversion through a partition and consequently through the resolution of submatrices of smaller dimensions.

Our aim is therefore to check if it is possible to extend this computational property also to the approximation case.

Date received: February 21, 2001