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In general, there is no chance to give an upper bound for the required

number of neurons. The set of continuous function is to general for this

purpose. One should somehow restrict this set requiring some further

smoothness property on the set of approximated functions (or on their

derivates), in terms of, e.g., modulus of continuity, Lipschitz coefficient,

Jacobian derivate, etc.

In such case the number of neurons can be determined as a function of

accuracy. See e.g.

@ARTICLE{BlumLi,

author = "E. K. Blum and L. K. Li",

title = "Approximation theory and feedforward networks",

journal = "Neural Networks",

pages = "511--515",

year = 1991,

volume = 4,

number = 4,

}

@ARTICLE{Kurk92,

author = "V. K\r{u}rkov\'a",

title = "Kolmogorov's theorem and multilayer neural networks",

journal = "Neural Networks",

pages = "501--506",

year = 1992,

}

Regards,

Domonkos

----- Original Message -----

From: "P. Sarma" <psarma@seas.upenn.edu>

To: "Multiple recipients of list" <fuzzy-mail@dbai.tuwien.ac.at>

Sent: Tuesday, December 04, 2001 6:50 PM

Subject: Re: Stupid question

*> It is interesting to note that the continuous model of the binary set, or
*

*> the Heaviside step function S(x) \in [0,1], satisfies the basic criteria
*

for

*> inclusion in the general collection of "sigmoidal" functions, \sigma(x),
*

*> that are used in Cybenko's classic universal function approximator (UFA)
*

*> paper. By definition, S(x) is Lebesgue-integrable; and clearly S(x) ---> 1
*

*> as x ---> \infty, S(x) ---> 0 as x ---> - \infty.
*

*>
*

*> Therefore, according to Cybenko's theorems, S(x) also forms the basis for
*

a

*> UFA. Since Cybenko's paper generates necessary conditions only, there is
*

no

*> specification on the upper bound of number of nodal \sigma(x) units
*

required

*> to actually approximate some smooth function F(x) within a prespecified
*

*> tolerance, \epsilon, except that it is finite, for all such \sigma(x).
*

Thus

*> a finite, though possibly large, collection of binary number-step
*

functions

*> may be used to model any smooth function, including, eg. some fuzzy
*

*> membership function. This generates one interrelationship between binary
*

*> and fuzzy variables.
*

*>
*

*> Corrections, insight, comments are very welcome.
*

*>
*

*>
*

*> Pramit
*

*>
*

*>
*

*>
*

*>
*

*> ----- Original Message -----
*

*> From: "Greg Chien" <gchien@protodesign-inc.com>
*

*> To: "Multiple recipients of list" <fuzzy-mail@dbai.tuwien.ac.at>
*

*> Sent: Wednesday, November 21, 2001 5:42 AM
*

*> Subject: Re: Stupid question
*

*>
*

*>
*

*> > "Ricky" <01900990R@polyu.edu.hk> wrote in message
*

*> > news:9tcggh$1bd3@hkpa05.polyu.edu.hk...
*

*> > > It may be a stupid question, but it really a trouble in my mind
*

*> > these few
*

*> > > days. We use the mathematics expression to formulate the algorithm,
*

*> > and we
*

*> > > use the programme to apply the algorithm, then what is the
*

*> > relationship
*

*> > > between mathematics and computer programme?
*

*> >
*

*> > How about "a computer program is a machine readable/executable
*

*> > implementation of certain logic and mathematics."
*

*> >
*

*> > > In the world of computer programme, binary nos. (1 and 0) are used
*

*> > but as we
*

*> > > all know that there is a lot of things between 1 and 0 in the world
*

*> > of fuzzy
*

*> > > algorithm. Then, contradition is here.
*

*> >
*

*> > A single binary bit may not be fuzzy, but a series of 0's and 1's that
*

*> > forms different patterns may be used to model fuzzy ranges, and,
*

*> > perhaps even, to simulate analog devices.
*

*> >
*

*> > Regards,
*

*> > Greg Chien
*

*> > http://protodesign-inc.com
*

*> >
*

*> >
*

*> >
*

*> >
*

*>
*

############################################################################

*> > This message was posted through the fuzzy mailing list.
*

*> > (1) To subscribe to this mailing list, send a message body of
*

*> > "SUB FUZZY-MAIL myFirstName mySurname" to listproc@dbai.tuwien.ac.at
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*> > (2) To unsubscribe from this mailing list, send a message body of
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*> > "UNSUB FUZZY-MAIL" or "UNSUB FUZZY-MAIL
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*> yoursubscription@email.address.com"
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*> > to listproc@dbai.tuwien.ac.at
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*> > (3) To reach the human who maintains the list, send mail to
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*> > fuzzy-owner@dbai.tuwien.ac.at
*

*> > (4) WWW access and other information on Fuzzy Sets and Logic see
*

*> > http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info
*

*> > (5) WWW archive:
*

*> http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html
*

*> >
*

*>
*

*>
*

*>
*

############################################################################

*> This message was posted through the fuzzy mailing list.
*

*> (1) To subscribe to this mailing list, send a message body of
*

*> "SUB FUZZY-MAIL myFirstName mySurname" to listproc@dbai.tuwien.ac.at
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*> (2) To unsubscribe from this mailing list, send a message body of
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*> "UNSUB FUZZY-MAIL" or "UNSUB FUZZY-MAIL
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yoursubscription@email.address.com"

*> to listproc@dbai.tuwien.ac.at
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*> (3) To reach the human who maintains the list, send mail to
*

*> fuzzy-owner@dbai.tuwien.ac.at
*

*> (4) WWW access and other information on Fuzzy Sets and Logic see
*

*> http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info
*

*> (5) WWW archive:
*

http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html

*>
*

*>
*

############################################################################

This message was posted through the fuzzy mailing list.

(1) To subscribe to this mailing list, send a message body of

"SUB FUZZY-MAIL myFirstName mySurname" to listproc@dbai.tuwien.ac.at

(2) To unsubscribe from this mailing list, send a message body of

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to listproc@dbai.tuwien.ac.at

(3) To reach the human who maintains the list, send mail to

fuzzy-owner@dbai.tuwien.ac.at

(4) WWW access and other information on Fuzzy Sets and Logic see

http://www.dbai.tuwien.ac.at/ftp/mlowner/fuzzy-mail.info

(5) WWW archive: http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html

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