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-   Chapter 3 - The Linear Model (http://book.caltech.edu/bookforum/forumdisplay.php?f=110)
-   -   Execise 3.12 (b), (c) (http://book.caltech.edu/bookforum/showthread.php?t=4880)

Fromdusktilldawn 03-27-2019 10:27 PM

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 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.

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?

htlin 04-04-2019 02:05 AM

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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