From: dstamp@watserv1.waterloo.edu (Dave Stampe-Psy+Eng)
Subject: Re: Sources of the concept of "information as space"? 
Date: Fri, 10 Jan 1992 05:21:21 GMT
Message-ID: <1992Jan10.052121.29447@watserv1.waterloo.edu>
Organization: University of Waterloo



gbnewby@uxh.cso.uiuc.edu writes:

>>object-object similarity is often regarded as some
>>sort of inverse to distance.  The greater the similarity between objects is
>>judged to be, the "closer" we say they are.
>>
>>One of the empirical results from the research I and others did at New Mexico
>>State some years ago was that if you examined people's pairwise similarity
>>judgements for a set of objects and converted the similarities into dis-
>>tances, you found widespread violations of the metric axioms that define the >>nature of euclidean space.  The axioms violated most often were symmetry
>>(distance(A->B) == distance(B->A)) and the triangle inequality, although 
>>there
>>were a smaller number of cases where the "distance" from an object to itself
>>could reasonably be interpreted as nonzero, which violated the third axiom.
>>Examples of symmetry violations are:
>>
>>       1) A woman was judged to be more like a rose than a rose like a woman
>>       2) the word "orange" is more like "fruit" than "fruit" is like "orange"
>
>Another aspect Chris didn't explicitly mention is that of the non-euclidean
>nature of conceptual space in the first place.  That is, if you give
>people a bunch of paired-comparisons to make, the outcome usually
>can NOT be plotted in a 'real' (euclidean) space.  Galileo methodology
>uses a Riemann space (which includes 'imaginary' dimensions).  This
>is the rule, not the exception.

Have you ever looked at Kohenen's paper on self-organizing semantic
"maps" with neural networks?  Supposedly it manages to organize
concepts (based on frequency of pairing) into a 2D map of distances.
Now, obviously this is still a Euclidian space because you've used
pairs, so distance makes sense as a measure.  The higher dimensions
are "folded" down to 2, though.

Now, I don't claim that the brain does this, as in Kohenen's maps
the connections tend to develop reciprocally.  However, if you
reran the session using order as well as distance, the connections
would probably not be reciprocal.  I see concept space as more of a 
directed graph, where connections can show different strengths based
on direction, as opposed to a Euclidian "undirected" graph.  This 
only makes sense, because B may imply A whereas A does not imply
B, and much of neural learning is based on predictive powers.

An interesting corrolary of this is that each person's conceptual
map may be unique: the overall configuration is set by the environment,
but details such as order of learning probably result in unique
"basis functions" for the network.  No one can tell, as long as the
output makes sense in terms of the input.  So if you want to utilize 
a person's conceptual space at more than a superficial level, you may 
have to do a unique mapping for each person.

--------------------------------------------------------------------------
| My life is Hardware,                    |                              | 
| my destiny is Software,                 |         Dave Stampe          |
| my CPU is Wetware...                    |                              | 
| Anybody got a SDB I can borrow?         | dstamp@watserv1.uwaterloo.ca |
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