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From: ram82001@zach.fit.edu (Gary Russel /ADVISOR L. FAUSETT)
Subject: y (approximation) is close to t (target) ?
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Date: Sat, 1 Jun 1996 21:36:04 GMT
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A neural net of the form y = w(1)*z(1) + ... + w(4)*z(4),
where z(j) = z(v(j)) = exp(-v(j))           
           -(x(1) - c(1,j))^2 - (x(2) - c(2,j))^2
and v(j) = -------------------------------------- ,
                                 2 
                           2*s(j)

has approximated t = exp(-(x(1) - .25)^2 - (x(2) - .75)^2)
                   + exp(-(x(1) - .75)^2 - (x(2) - .25)^2).

Of course, y could equal z exactly if
w(1) = 1, w(2) = 1, w(3) = 0, w(4) = 0,
s(1) = sqrt(0.5), s(2) = sqrt(0.5), s(3) <> 0, s(4) <> 0,
c(1,1) = .25, c(1,2) = .75, c(1,3) = ?, c(1,4) = ?,
c(2,1) = .75, c(2,2) = .25, c(2,3) = ?, c(2,4) = ?

But here is the set of weights y learned:
w(1) = -.4493, w(2) = .9760, w(3) = 1.0480, w(4) = -.3181,
s(1) = .5537, s(2) = .6952, s(3) = .8496, s(4) = .0445,
c(1,1) = .9595, c(1,2) = .5839, c(1,3) = .6049, c(1,4) = -.1189,
c(2,1) = .9875, c(2,2) = .5858, c(2,3) = .6161, c(2,4) = 1.0027 

Numerically, one can show how close y is to t
(e.g. t(0,0) = 1.0705, y(0,0) = 1.0814),
but does anyone have some suggestions about what to do analytically?
Does it help that exp(x) = 1 + x + (x^2)/2 + (x^3)/6 + ... ?

Must one investigate optimizing (t - y)^2 ?

GR
