RE: Need proofs of approximation power of the 1st-order Sugeno systems

From: Ulrich Bodenhofer (ulrich.bodenhofer@scch.at)
Date: Mon Sep 03 2001 - 13:50:53 MET DST

  • Next message: Vitaliy Kolodyazhniy: "Re: Need proofs of approximation power of the 1st-order Sugeno systems"

    Dear Vitaliy,

    I would also like to refer to works stating that the universal
    approximation property is lost when you restrict to predefined,
    but arbitrarily large, maximal number of rules (even if you admit
    any shape of the membership functions). In fact, the sets of
    Sugeno (or first-order TSK models) are nowhere dense in the
    set of continuous real-valued functions (just in case that
    you are not familiar with the topological notion of "nowhere
    denseness": it means that the opening of the closure is empty,
    which means that a set is, nonchalantly speaking, "almost
    discrete"):

    @article{Moser99,
      author = {B. Moser},
      title = {Sugeno controllers with a bounded number of rules
                      are nowhere dense},
      journal = {Fuzzy Sets and Systems},
      volume = {104},
      number = {2},
      pages = {269--277},
      year = {1999}
    }

    @article{Tikk99,
      author = {D. Tikk},
      title = {On nowhere denseness of certain fuzzy controllers
                      containing prerestricted number of rules},
      journal = {Tatra Mt. Math. Publ.},
      year = {1999},
      volume = {16},
      pages = {269--277}
    }

    I know this was not in the core of your question, but I thought
    that it could be interesting to look at these results which demonstrate
    the limitations of the universal approximation property.

    Best regards,
    Ulrich

    P.S.: If you have problems accessing the papers, please drop me
          an e-mail and I will send you e-mail addresses of the authors.

    > -----Original Message-----
    > From: fuzzy-mail@dbai.tuwien.ac.at
    > [mailto:fuzzy-mail@dbai.tuwien.ac.at]On Behalf Of Vitaliy Kolodyazhniy
    > Sent: Friday, August 31, 2001 8:24 PM
    > To: Multiple recipients of list
    > Subject: Need proofs of approximation power of the 1st-order Sugeno
    > systems
    >
    >
    > Dear all,
    >
    > I am a 2-nd year PhD student. Currently, I am summarizing all my research
    > and working on my thesis. The title of the thesis is "Control of
    > stochastic
    > systems using fuzzy-neural models". I have developed a number of adaptive
    > control schemes based on the 1st-order Sugeno models, and now I need
    > mathematical proofs of approximation power of such models. As it seems to
    > me, most existing proofs deal with the 0-th order Sugeno-type or
    > alike fuzzy
    > systems, among them are the results obtained by L.-X. Wang, B. Kosko, V.
    > Kreinovich, and many others. Intuitively, it is quite clear that the
    > 1st-order Sugeno systems possess much better approximation capability than
    > the 0th-order ones, but so far I haven't found any mathematical
    > work on this
    > issue.
    >
    > Another important thing (in close connection with the first one) is
    > estimation of the approximation accuracy with a given number of clusters
    > (rules) with given parameters.
    >
    > Could you give me any links to downloadable papers? Though I would
    > appreciate any help.
    >
    > Thanks in advance.
    >
    > Vitaliy Kolodyazhniy
    > kolodyazhniy@ieee.org
    >
    >
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