Vlad Dumitrescu wrote:
Unfortunately, having more than one dimensions makes comparisons
impossible - if an ordering relation is defined over the domain, then
this domain is "one-dimensional" with regard to that relation.
In other words, one can't compare vectors, just scalars. So the
multi-dimensional "strength vector" has to be turned into a scalar (by
for example a weighted sum) and we're back where we started...
While that is true, as stated, it is also the case that this is exactly
the sort of thing at which artificial neural networks excel.
Given multiple "inputs" (a d-dimensional vector), the "squashing functions"
in the "layers" of the network in fact reduce the "output" to a single number.
Neural networks are excruciatingly well-documented, as is their use in
a wide variety of domains, even on the web. [By that I mean, you needn't
buy a book to discover how powerful "multi-layer perceptrons" can be.]
So, while it's correct that "one can't compare vectors, just scalars"
one can compare the _output_ of a vector-massaged-by-a-neural-net
against the _output_ of a vector-massaged-by-a-neural-net.
I think Vlad knows this, of course, as he said, 'the multi-dimensional
"strength vector" has to be turned into a scalar'. [I'm just here to
clarify one common method of doing that.]
The tough part is deciding what to measure, in your original vector.
I'm sure I don't know what variables one would use, but I agree with
Don: "the 2 numbers together would predict your chances of beating
another (2 dim) player more accurately that a 1 dimension system could."
I further agree: "And of course you could extend this." Which is to say,
there is no reason to stop at two. [Although there is something called
"the curse of dimensionality", which prevents a too-large dimensionality:
See <http://www.faqs.org/faqs/ai-faq/neural-nets/part2/section-13.html> .]
In practice, the "feature extraction" phase is crucial. Deciding what to
measure, finding the right number of items to measure, scaling them properly
(e.g., multiplying a vector element by a constant), not measuring irrelevant
items: All these things are exceedingly crucial (and often difficult) when
constructing a useful, effective, neural net.
So, I'm stumped.
In theory though, if one measures the right variables, and collects
enough data (sometimes a surprisingly small amount is sufficient!) a
neural network could be trained to recognize and predict the strength
of go-players; witness the (excruciatingly well-documented!) success
of neural networks in a wide variety of domains.
--
Rich
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