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We have established the idea of learning from exemplars.
Let us build our first prediction machine.
No interpolation. No extrapolation. Just memory.
The algorithm will be:
Given a new input x, find the past exemplar with the smallest distance from x. Return the output for that past exemplar.
If x is multidimensional  Then what is "nearby"?
Here is one possibility. Let x = (x_{1},..,x_{n}) and y = (y_{1},..,y_{n}). Define the Hamming distance between x and y as:
If x and y are vectors of binary digits, then the Hamming distance is just equal to the number of points at which x and y differ.
Now, our algorithm could be:
Given a new input x, find the past exemplar with the smallest Hamming distance from x. Return the output for that past exemplar.
The problem with this can be seen in the following.
Let the exemplars be:
Temperature 22 degrees, pressure 1000 millibars. No rain. Temperature 15 degrees, pressure 990 mb. Rain.Then the question is:
Temperature 17 degrees, pressure 1010 mb. Will there be rain?The Hamming distance from the first exemplar is 5 + 10, the distance from the second is 2 + 20. So we say it is nearest to the first exemplar.
However, we have just added together two different quantities  degrees and millibars! We could just as easily have said the exemplars were:
Temperature 22 degrees, pressure 1 bar. No rain. Temperature 15 degrees, pressure 0.990 bars. Rain.and the question:
Temperature 17 degrees, pressure 1.010 bars. Will there be rain?Then the distance is 5 + 0.010 from the first, 2 + 0.020 from the second. The second exemplar is closer.
Which is correct? Which one are we closer to?
The answer is really: If it rains, we must have been closer to the second exemplar. If it doesn't rain, we must be closer to the first. "Closer" is defined by observing the output, rather than predefined as in a distance metric.
The same applies to any other distance metric, such as the "real" distance metric  the distance between 2 points x = (x_{1},..,x_{n}) and y = (y_{1},..,y_{n}) in ndimensional Euclidean space:
(e.g. consider n=2 or n=3).
Again, we can change the scale of one axis without the other, and so change the definition of the "nearest" exemplar. It is just easier to show this with the Hamming distance.
Even if we do not have the above problem of heterogenous inputs, a distance metric is still a primitive form of predictor.
e.g. Consider classifying a
bitmap
of 100 x 100 pixel images.
The components of the input are all the same type of thing 
pixels that are, say, 0 (black) or 1 (white).
The distance metric is the number of pixels that are different.
This strategy of looking at distance from exemplars
is called template matching.
The basic problem with it is that it weighs every dimension (every pixel) as of equal importance.
In fact, matching the exemplar with the same haircut / background / ears is a reasonable first guess  after all, how does the machine know in advance that it should be looking at the mouth and eyes, and ignoring the other parts?
The problem is the system cannot learn this. The distance metric is fixed. It cannot learn that some parts of the input space are irrelevant.
Workarounds:
Does our algorithm have to compare
current input to all previous exemplars?
This will not scale well.
It will take a long time
to compare every pixel of the input with every pixel of a large number of
previous exemplars (and it must be compared to them all, every time).
And the list of exemplars to compare to
keeps getting larger.