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-   -   Problem 1.10 (b) (http://book.caltech.edu/bookforum/showthread.php?t=4598)

NewtoML 04-01-2015 07:06 PM

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

yaser 04-02-2015 10:53 PM

Re: Problem 1.10 (b)
 
Quote:

Originally Posted by NewtoML (Post 11948)
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?

Exactly, and the output is limited to a binary value, too.

c.thejaswi 11-23-2015 09:13 AM

Re: Problem 1.10 (b)
 
Quote:

Originally Posted by yaser (Post 11949)
Exactly, and the output is limited to a binary value, too.

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?

c.thejaswi 11-23-2015 09:18 AM

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

memoweb 12-03-2015 05:59 AM

Re: Problem 1.10 (b)
 
Quote:

Originally Posted by c.thejaswi (Post 12207)
I am sorry for the confusion. I got it.
Thanks.

thanks

MaciekLeks 04-01-2016 06:32 AM

Re: Problem 1.10 (b)
 
IMHO, the answer to (b) is not an infinite number. As for part (a) the anwer is not \frac{1}{2}, 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 f. We also do not know the dimensionality of the datapoint in the input space \mathcal{X}=\{\mathbf{x}_{1,}\mathbf{x}_{2},...,\mathbf{x}_{N},\mathbf{x}_{N+1},...,\mathbf{x}_{N+M}\} but we know that this input space is fixed (all \mathbf{x}_{k} for 1\leq k\leq N+M are already set). In this case we have \mathcal{D} of size N generated in a deterministic way, and y_{n}=f(\mathbf{x}_{n})(1\leq n\leq N) is not affected by any noise. So, how many possible f:\mathcal{X}\mathcal{\rightarrow\mathcal{Y}} can 'generate' \mathcal{D}? The subtle point in this case is the assumption: “For a fixed \mathcal{D} of size N”, which means that \mathcal{D}=\{(\mathbf{x}_{1},y_{1}),(\mathbf{x}_{2},y_{2}),...,(\mathbf{x}_{N},y_{n})\} is already generated. We can calculate how many possible outputs y_{n} for 1\leq n\leq N we can get? Only 1. But there are remaining \{\mathbf{x}_{N+1},...,\mathbf{x}_{N+M}\} datapoints for which we can have 2^{M} possible values \{-1,1\}. So, the anwer is 2^{M}.

Am I wrong?


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