
#1




Problem 1.10 (b)
Hello,
I believe that the number of possible f that can generate D in a noiseless setting is infinite. For example, if I take a data set of 2 points, say (5,1) and (3,1), I can come up with any number of functions that will generate these two points. However, I'm confused as to how this reconciles with the example on p. 16 where the set of all possible target functions in the example is finite, namely 256. Is this because the input space X is limited to Boolean vectors in 3 dimensions? Thanks 
#2




Re: Problem 1.10 (b)
Exactly, and the output is limited to a binary value, too.
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#3




Re: Problem 1.10 (b)
Dear Professor,
However, it is no where mentioned that ${\cal X}$ is a space of binary strings, and "$f$" is a logical operation. Therefore, is it still true that number of $f'$s that can generate D is finite? 
#4




Re: Problem 1.10 (b)
I am sorry for the confusion. I got it.
Thanks. 
#5




Re: Problem 1.10 (b)

#6




Re: Problem 1.10 (b)
IMHO, the answer to (b) is not an infinite number. As for part (a) the anwer is not , so here the anwer is not as simple as it seems.
My way of thinking is as follows: We must recall that the assumption from (a) does not work here, and we still do not know function . We also do not know the dimensionality of the datapoint in the input space but we know that this input space is fixed (all for are already set). In this case we have of size N generated in a deterministic way, and () is not affected by any noise. So, how many possible can 'generate' ? The subtle point in this case is the assumption: “For a fixed of size ”, which means that is already generated. We can calculate how many possible outputs for we can get? Only 1. But there are remaining datapoints for which we can have possible values . So, the anwer is . Am I wrong? 
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