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From: dhw@itivax.UUCP (David H. West)
Newsgroups: comp.ai.neural-nets,comp.ai
Subject: Re: Learning arbitrary transfer functions
Message-ID: <380@itivax.UUCP>
Date: 11 Nov 88 22:30:09 GMT
References: <399@uvaee.ee.virginia.EDU>
Sender: dhw@itivax.UUCP
Organization: Industrial Technology Institute
Lines: 34

In article <399@uvaee.ee.virginia.EDU> aam9n@uvaee.ee.virginia.EDU (Ali Minai) writes:
>
>I am looking for any references that might deal with the following
>problem:
>
>y = f(x);         f(x) is nonlinear in x
>
>Training Data = {(x1, y1), (x2, y2), ...... , (xn, yn)}
>
>Can the network now produce ym given xm, even if it has never seen the
>pair before?
>
>That is, given a set of input/output pairs for a nonlinear function, can a
>multi-layer neural network be trained to induce the transfer function
                                                 ^^^
An infinite number of transfer functions are compatible with any
finite data set.  If you really prefer some of them to others, this
information needs to be available in computable form to the
algorithm that chooses a function.  If you don't care too much, you
can make an arbitrary choice (and live with the result); you might 
for example use the (unique) Lagrange interpolation polynomial of
order n-1 that passes through your data points, simply because it's
easy to find in reference books, and familiar enough not to surprise
anyone. It happens to be easier to compute without a neural network,
though :-)
If you want ym for a given xm to be relatively independent of
(sufficiently large) n, however, you need in general to know
something about the domain, and n-1_th (i.e. variable) order polynomial 
interpolation is almost certainly not what you want.

-David West            dhw%iti@umix.cc.umich.edu
		       {uunet,rutgers,ames}!umix!itivax!dhw
CDSL, Industrial Technology Institute, PO Box 1485, 
Ann Arbor, MI 48106