#1




Q3, general question
Is this a trick question, or just a direct application of the concepts learned in class, and present in the book? If want to pick the smallest such that , this is the same thing as saying pick the smallest such as . The example of a polynomial transformation is discussed in detail in the book and the generic formula for the upper bound is provided. Is there anything in this example that begs for a tighter bound that that provided by the formula that appears in page 105 of the book?
I have an even more general question about the VC dimension of these nonlinear transformations. I understand that in the transformed space one needs to apply the dichotomy analysis to come up with the VC dimension and that (since we did it for the general linear case) that is . Yet there is the caveat that some of the dichotomies that allow for that VC dimension might not be valid points of the transformation. But isn't that the case that the likelihood that the vast majority of points that would allow for such VC dimension will NOT be valid points of the transformation because we are trying to generate independent points out of degrees of freedom. Thus, in most cases the VC dimension is likely to be closer to than to , unless one gets really lucky. 
#2




Re: Q3, general question
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#3




Re: Q3, general question
Thanks for the answer on the general question. The bottom line is that one has to be very careful with these transformations.
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#4




Re: Q3, general question
I got this one wrong...not because of the mathematical challenge, but because of the typographical challenge!
Anyone else with old eyes lose track of the little semicolons down among the little subscripts, and so miscount the number of features in the transformed space? Doh! 
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