Re: Stupid question

From: Tikk, Domonkos (tikk@ttt.bme.hu)
Date: Mon Dec 10 2001 - 21:31:45 MET

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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
    > >
    > >
    > >
    > >
    >
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