Execise 3.12 (b), (c)

[Fromdusktilldawn]

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 z-domain. Then these three points are separable by linear hypothesis in the z-domain, 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.


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?

[htlin]

Please note that the transformation in (3.12) allows you to use "specific" hyperbolas and ellipses, not every hyperbola. Hope this helps.