Numeric vectors are strictly more powerful as a representation than
predicates.
This is not really true...
A set of vectors is a relation, which is a predicate; I can do
any logical operation on them (given, e.g. a term constructor that is simply
a hash function along an axis). But they also have a metric, which can be
enormously useful in some situations.
Well, uncertain logical predicates also have a metric, the Jaccard metric
1 - P(A and B) / P(A or B)
http://en.wikipedia.org/wiki/Jaccard_index
And, there is a kind of averaging operator for logical predicates as
well -- revision,
which combines two different truth value estimates of the same
predicate, based on different
(possibly overlapping) evidence sets
Consider the "Copycat" operation (given A,B,C, find D such that A:B :: C:D for
some relations : and ::). (I call it "analogical quadrature".) This is
enormously useful. Copycat did it by using a funky search in a space defined
by a semantic net with variable link lengths, so as to simulate some of the
metric-ness of the semantic space in an ad hoc way. If the concepts are
represented as vectors IN A SPACE THAT CAPTURES THE SALIENT ATTRIBUTES,
analogical quadrature reduces to vector addition. This means I can afford to
search a lot of projections/transforms looking for good spaces. (I.e. because
testing them is cheap.)
Yes, copycat simulated a metric in an ad hoc way, because it lacked a
robust way of measuring
and utilizing uncertainty...
Anyway, the interesting thing in Hofstadter-ian analogy problems is the
identification of
WHAT THE SALIENT ATTRIBUTES ARE ... not the choice of representation itself.
If you represent concepts as probabilistic-logical combinations of the
salient attributes, you
will get the same advantages as in your numerical representation plus
more ;-)
There are two major problems arising in your numeric vector representation:
-- what are the relevant dimensions. For analogical quadrature to
reduce to vector addition,
you need to make sure the relevant dimensions are PROBABILISTICALLY
INDEPENDENT
in the context in question. But obviously, finding independent
dimensions may be very hard.
-- how are the numbers in the different dimensions normalized. Vector
addition treats all
components equally rather than weighting them; but if different
dimensions represent different
things they will not automatically be on the same scale. So you need to
do some clever
normalization or some clever weighting.
In a probabilistic framework, the search for contextually
probabilistically independent variables, and the weighting of different
aspects, is handled in a mathematically and conceptually coherent way.
How do you handle these two issues in your framework?
I fear you are taking a model of low-level vision and extrapolating it
into a model of cognition.
Whereas, I suspect that high-level vision actually has a lot to do with
probabilistic inference...
Similarly, you're a fan of Moravec's grids in robotics, but modern
robotics uses probabilistic occupancy
grids that work better ;-) ...
There are decent arguments that interconnections btw cortical columns
are representing conditional
probabilities in many cases ... if so then focusing on the numeric
vectors of multiple neural activations,
rather than the conditional probabilities they represent, may be
misdirected...
-- Ben
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