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From: norman@lasspvax.UUCP (Norman Ramsey)
Newsgroups: net.math
Subject: Re: Neural Net COmputing
Message-ID: <713@lasspvax.UUCP>
Date: Fri, 22-Nov-85 16:43:25 EST
Article-I.D.: lasspvax.713
Posted: Fri Nov 22 16:43:25 1985
Date-Received: Sun, 24-Nov-85 05:48:11 EST
References: <663@lasspvax.UUCP> <17@sbcs.UUCP> <55@linus.UUCP>
Reply-To: norman@lasspvax.UUCP (Norman Ramsey)
Organization: LASSP, Cornell University
Lines: 48
Summary: 

In article <55@linus.UUCP> bs@linus.UUCP (Robert D. Silverman) writes:
>The math content of this group has become low to negative recently. Can
>we please have less of this sophistry and more math??? Most of the discussion
>about turing machines vs. humans belongs in net.ai or net.philosophy. Also,
>much of the content of these discussions leaves me wondering whether their
>writers possess any natural intelligence. It certainly sounds as if many
>people simply like to shoot their mouths off concerning a subject which
>they haven't studied. This is not a newsgroup for speculation.
>Bob Silverman   (they call me Mr. 9)


I'm sorry that Mr. Silverman is unhappy about the low math contents of
net.math. I invite him to send his flames out there. If he had read my
original posting carefully, he would have seen that the descriptive material
was background for asking the question, does anyone know anything about the
*mathematics* (there's that word again) of these things...

To be more specific, does anyone out there in net land understand:

  (1) What describes the family of functions minimized by the neuron
computing algorithm? Hopfield's model is nonlinear; the neuron turns on
when its inputs reach a certain threshold, so depending on whether you pick
a single threshold or one for each neuron you have a one- or an N-parameter
family.

  (2) What are the fixed points and basins of attraction of the neural
computing algorithm? In particular, how can altering the firing thresholds
change the (a) number or position of stable fixed points (metastable states,
local minima, pick your own jargon) and (b) the size (and shape?) of the
basins of attraction?

  (3) How does the mathematics of the device change when the range of
possible values for the transfer matrix (neural interconnections) is
restricted?

Hopfield addresses some of these questions briefly in his paper in Natl Acad
Sci USA 79. He also has some nice little estimates of error rates and such.
He doesn't give much discussion to the issue which really interests me,
which is control over the basins of attraction. It's a very nice paper
though.

I can't compare to Perceptrons since I know nothing about them, but I'm told
by those who claim to know that the Perceptron is different.
-- 
Norman Ramsey

ARPA: norman@lasspvax  -- or --  norman%lasspvax@cu-arpa.cs.cornell.edu
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