LFD Book Forum Execise 3.12 (b), (c)
 User Name Remember Me? Password
 FAQ Calendar Mark Forums Read

 Thread Tools Display Modes
#1
03-27-2019, 11:27 PM
 Fromdusktilldawn Junior Member Join Date: Sep 2017 Posts: 5
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.

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?
#2
04-04-2019, 03:05 AM
 htlin NTU Join Date: Aug 2009 Location: Taipei, Taiwan Posts: 601
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.
__________________
When one teaches, two learn.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off Forum Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home General     General Discussion of Machine Learning     Free Additional Material         Dynamic e-Chapters         Dynamic e-Appendices Course Discussions     Online LFD course         General comments on the course         Homework 1         Homework 2         Homework 3         Homework 4         Homework 5         Homework 6         Homework 7         Homework 8         The Final         Create New Homework Problems Book Feedback - Learning From Data     General comments on the book     Chapter 1 - The Learning Problem     Chapter 2 - Training versus Testing     Chapter 3 - The Linear Model     Chapter 4 - Overfitting     Chapter 5 - Three Learning Principles     e-Chapter 6 - Similarity Based Methods     e-Chapter 7 - Neural Networks     e-Chapter 8 - Support Vector Machines     e-Chapter 9 - Learning Aides     Appendix and Notation     e-Appendices

All times are GMT -7. The time now is 04:07 PM.

 Contact Us - LFD Book - Top