Execise 3.12 (b), (c)
In (b) we need to show that the growth function for the hypothesis set H_phi for 4 data points is less than 16.
I am not sure how to approach this question. In part (a), I can show that the growth function m_H_phi(3) = 8 by considering three points in x, arbitrarily placed, then I transform these points using phi into the zdomain. Then these three points are separable by linear hypothesis in the zdomain, hence they are separable in the original x domain per figure 3.6 I do not understand why the growth function of H_phi of 4 points is less than 16 in part (b). Note that H_phi is the set of hypotehsis h = sign(\tilde w Phi(x)). By exercise 3.11 this set contains hyperbolas, ellipses, straight (vertical lines), etc. Recall that the problem with linear hypothesis, H, is that it cannot separable the case in figure 2.1 c However, my new hypothesis set, H_phi, contains (per exercise 3.11) hyperbolas, ellipses, and straight lines. Therefore the case that was not separable in by linear hypothesis can simply be separated as shown in the diagram. http://i66.tinypic.com/29wk7x5.png Therefore the growth function of H_phi over 4 points has to equal 16. I cannot see a single configuration of 4 points on the plane where it cannot be separated by any of the function in H_phi. Where did I go wrong in my logic? 
Re: Execise 3.12 (b), (c)
Please note that the transformation in (3.12) allows you to use "specific" hyperbolas and ellipses, not every hyperbola. Hope this helps.

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