Path: utzoo!attcan!uunet!zaphod.mps.ohio-state.edu!sdd.hp.com!ucsd!sdcc6!zaius!pluto From: pluto@zaius.ucsd.edu (Mark Plutowski) Newsgroups: comp.ai.neural-nets Subject: Re: Searching for 13th Hilbert Problem article Message-ID: Date: 8 Nov 90 21:42:00 GMT References: <39365@ut-emx.uucp> Sender: news@sdcc6.ucsd.edu Lines: 33 Nntp-Posting-Host: zaius.ucsd.edu phbd641@ccwf.cc.utexas.edu (David Chao) writes: >In Richard Lippmann's review article in IEEE ASSP (4,April 1987) magazine, >he makes reference to "Mathematical developments Arising from Hilbert >Problems," American Mathematical Society (ed. Browder), 1976. >In the article on Hilbert's 13th problem, reference is made to >Kolmogorov's proof concerning the approximation of any continous function of >N variables using only linear summations and nonlinear but >continously increasing functions of only one variable. David, could you clarify this last sentence for me? Do you mean a monotonically increasing sequence of continuous functions of one variable? Or, do you mean that each function of one variable is continuous, with its value monotonically increasing as its argument tends to infinity? I expect the latter to be the correct interpretation, but am not certain, since the statement here does not say how many functions are required. (BTW: the version I heard required on the order of 2N+1 functions -- is this correct? This would be equivalent to requiring 2N+1 hidden units, and a single linear output unit. However, as I understand it, in Kolmogorov's theorem each hidden unit may be required to have a different, and unknown, transfer function. Hence, the theorem proves the existence of such a neural network, but not constructively.) Thanks, =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- M.E. Plutowski INTERNET: pluto%cs@ucsd.edu UCSD, Computer Science and Engineering 0114 9500 Gilman Drive La Jolla, California 92093-0114 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-