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From: jtoth@handel.Princeton.EDU (Gabor J.Toth)
Subject: Re: Hopfield Storage Capacity
Message-ID: <1994Nov11.195217.11524@Princeton.EDU>
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References: <1994Oct17.230541.9773@Princeton.EDU> <1994Oct19.231519.3819@Princeton.EDU> <38vkos$bkt@xpat.postech.ac.kr> <39ssos$da@agate.berkeley.edu>
Date: Fri, 11 Nov 1994 19:52:17 GMT
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In article <39ssos$da@agate.berkeley.edu>,
Atul Joshi <vonjoshi@cyclone.Berkeley.EDU> wrote:

>: Everything about Hopfield storage capacity is well written in the
>: book "Introduction to the theory of neural networks", by Hertz,
>: Krogh, Palmer(Addison-Wesley).
> The physcists contribution to the question of hopefield networks
> comes from their study of something called spinglasses whch has been
> described by well formulated mathematics. The appropriateness of
> this analogy I cant comment on because i have no clue as to what the
> hell a spin glass is.  However it seems that the crossover has
> proved very fruitful.

The spin-glasses model a kind  of magnetism.  It  is supposed that the
material contains small magnets which  interact with each other.  Each
magnet $i$ is associated with a number $s_i$ which is  either +1 or -1
meaning that   the magnet is pointing  either  upward or downward. The
interaction is taken into account by writing the  energy of the system
as

      +--
       >  J_ij s_i s_j,
      +--
      i,j

where   $J_ij$  are treated  as random   numbers that characterize the
system.  In low  temperature the system   will try to  find the lowest
energy state, hence the correspondence to the Hoppfield net. There are
elaborate techniques  to  calculate different characteristics  of spin
glasses, and some of  them must be used in  the quoted work.  However,
no mathematician would _ever_ say these methods are (as yet) rigorous,
though they do tend to work.

--Gabor

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